High Dimensional Topological Local Fields and Residues Amnon - - PowerPoint PPT Presentation

High Dimensional Topological Local Fields and Residues

Amnon Yekutieli

Department of Mathematics Ben Gurion University email: amyekut@math.bgu.ac.il Notes available at http://www.math.bgu.ac.il/~amyekut/lectures

(updated 12 January 2015) Amnon Yekutieli (BGU) TLFs and Residues 1 / 34

Outline

I will discuss several results, mostly mostly old ones from the paper [Ye1]. Here is the plan of my lecture.

• 1. Background on Semi-Topological Rings
• 2. High Dimensional Local Fields
• 3. Topological Local Fields
• 4. The Beilinson Completion
• 5. The Residue Functional

If time permits I will also talk about:

• 6. Some Applications of the Residue Functional

Amnon Yekutieli (BGU) TLFs and Residues 2 / 34

Outline

I will discuss several results, mostly mostly old ones from the paper [Ye1]. Here is the plan of my lecture.

• 1. Background on Semi-Topological Rings
• 2. High Dimensional Local Fields
• 3. Topological Local Fields
• 4. The Beilinson Completion
• 5. The Residue Functional

If time permits I will also talk about:

• 6. Some Applications of the Residue Functional

Amnon Yekutieli (BGU) TLFs and Residues 2 / 34

Outline

I will discuss several results, mostly mostly old ones from the paper [Ye1]. Here is the plan of my lecture.

• 1. Background on Semi-Topological Rings
• 2. High Dimensional Local Fields
• 3. Topological Local Fields
• 4. The Beilinson Completion
• 5. The Residue Functional

If time permits I will also talk about:

• 6. Some Applications of the Residue Functional

Amnon Yekutieli (BGU) TLFs and Residues 2 / 34

Outline

I will discuss several results, mostly mostly old ones from the paper [Ye1]. Here is the plan of my lecture.

• 1. Background on Semi-Topological Rings
• 2. High Dimensional Local Fields
• 3. Topological Local Fields
• 4. The Beilinson Completion
• 5. The Residue Functional

If time permits I will also talk about:

• 6. Some Applications of the Residue Functional

Amnon Yekutieli (BGU) TLFs and Residues 2 / 34

Outline

I will discuss several results, mostly mostly old ones from the paper [Ye1]. Here is the plan of my lecture.

• 1. Background on Semi-Topological Rings
• 2. High Dimensional Local Fields
• 3. Topological Local Fields
• 4. The Beilinson Completion
• 5. The Residue Functional

If time permits I will also talk about:

• 6. Some Applications of the Residue Functional

Amnon Yekutieli (BGU) TLFs and Residues 2 / 34

Outline

I will discuss several results, mostly mostly old ones from the paper [Ye1]. Here is the plan of my lecture.

• 1. Background on Semi-Topological Rings
• 2. High Dimensional Local Fields
• 3. Topological Local Fields
• 4. The Beilinson Completion
• 5. The Residue Functional

If time permits I will also talk about:

• 6. Some Applications of the Residue Functional

Amnon Yekutieli (BGU) TLFs and Residues 2 / 34

Outline

I will discuss several results, mostly mostly old ones from the paper [Ye1]. Here is the plan of my lecture.

• 1. Background on Semi-Topological Rings
• 2. High Dimensional Local Fields
• 3. Topological Local Fields
• 4. The Beilinson Completion
• 5. The Residue Functional

If time permits I will also talk about:

• 6. Some Applications of the Residue Functional

Amnon Yekutieli (BGU) TLFs and Residues 2 / 34

Outline

I will discuss several results, mostly mostly old ones from the paper [Ye1]. Here is the plan of my lecture.

• 1. Background on Semi-Topological Rings
• 2. High Dimensional Local Fields
• 3. Topological Local Fields
• 4. The Beilinson Completion
• 5. The Residue Functional

If time permits I will also talk about:

• 6. Some Applications of the Residue Functional

Amnon Yekutieli (BGU) TLFs and Residues 2 / 34

Outline

I will discuss several results, mostly mostly old ones from the paper [Ye1]. Here is the plan of my lecture.

• 1. Background on Semi-Topological Rings
• 2. High Dimensional Local Fields
• 3. Topological Local Fields
• 4. The Beilinson Completion
• 5. The Residue Functional

If time permits I will also talk about:

• 6. Some Applications of the Residue Functional

Amnon Yekutieli (BGU) TLFs and Residues 2 / 34

Outline

I will discuss several results, mostly mostly old ones from the paper [Ye1]. Here is the plan of my lecture.

• 1. Background on Semi-Topological Rings
• 2. High Dimensional Local Fields
• 3. Topological Local Fields
• 4. The Beilinson Completion
• 5. The Residue Functional

If time permits I will also talk about:

• 6. Some Applications of the Residue Functional

Amnon Yekutieli (BGU) TLFs and Residues 2 / 34

• 1. Background on Semi-Topological Rings
• 1. Background on Semi-Topological Rings

I will begin by recalling some definitions and constructions involving topologized rings. Let us fix a nonzero commutative ring k. By default all k-rings in the talk are

• commutative. They form a category Ringc k.

Suppose A is a k-ring. Recall that the module of differentials of A is the A-module Ω1

A/k, with its universal derivation

(1.1) d : A → Ω1

A/k.

Amnon Yekutieli (BGU) TLFs and Residues 3 / 34

• 1. Background on Semi-Topological Rings
• 1. Background on Semi-Topological Rings

I will begin by recalling some definitions and constructions involving topologized rings. Let us fix a nonzero commutative ring k. By default all k-rings in the talk are

• commutative. They form a category Ringc k.

Suppose A is a k-ring. Recall that the module of differentials of A is the A-module Ω1

A/k, with its universal derivation

(1.1) d : A → Ω1

A/k.

Amnon Yekutieli (BGU) TLFs and Residues 3 / 34

• 1. Background on Semi-Topological Rings
• 1. Background on Semi-Topological Rings

I will begin by recalling some definitions and constructions involving topologized rings. Let us fix a nonzero commutative ring k. By default all k-rings in the talk are

• commutative. They form a category Ringc k.

Suppose A is a k-ring. Recall that the module of differentials of A is the A-module Ω1

A/k, with its universal derivation

(1.1) d : A → Ω1

A/k.

Amnon Yekutieli (BGU) TLFs and Residues 3 / 34

• 1. Background on Semi-Topological Rings
• 1. Background on Semi-Topological Rings

I will begin by recalling some definitions and constructions involving topologized rings. Let us fix a nonzero commutative ring k. By default all k-rings in the talk are

• commutative. They form a category Ringc k.

Suppose A is a k-ring. Recall that the module of differentials of A is the A-module Ω1

A/k, with its universal derivation

(1.1) d : A → Ω1

A/k.

Amnon Yekutieli (BGU) TLFs and Residues 3 / 34

• 1. Background on Semi-Topological Rings

The exterior algebra of Ω1

A/k over A is the differential graded (DG) k-ring

ΩA/k =

• i≥0

Ωi

A/k.

The multiplication is super-commutative. The differential d extends the derivation (1.1). The DG ring ΩA/k is also called the de Rham complex of A. Example 1.2. If A = k[t], the polynomial ring in one variable, then Ω1

A/k is a

free A-module of rank 1. The differential form d(t) is a basis. Now take the ring of formal power series A = k[[t]]. The first guess would be that Ω1

A/k is a free A-module of rank 1 with basis d(t).

However, if k is a field of characteristic 0, this is false! The module Ω1

A/k is

not even finitely generated!

Amnon Yekutieli (BGU) TLFs and Residues 4 / 34

• 1. Background on Semi-Topological Rings

The exterior algebra of Ω1

A/k over A is the differential graded (DG) k-ring

ΩA/k =

• i≥0

Ωi

A/k.

The multiplication is super-commutative. The differential d extends the derivation (1.1). The DG ring ΩA/k is also called the de Rham complex of A. Example 1.2. If A = k[t], the polynomial ring in one variable, then Ω1

A/k is a

free A-module of rank 1. The differential form d(t) is a basis. Now take the ring of formal power series A = k[[t]]. The first guess would be that Ω1

A/k is a free A-module of rank 1 with basis d(t).

However, if k is a field of characteristic 0, this is false! The module Ω1

A/k is

not even finitely generated!

Amnon Yekutieli (BGU) TLFs and Residues 4 / 34

• 1. Background on Semi-Topological Rings

The exterior algebra of Ω1

A/k over A is the differential graded (DG) k-ring

ΩA/k =

• i≥0

Ωi

A/k.

The multiplication is super-commutative. The differential d extends the derivation (1.1). The DG ring ΩA/k is also called the de Rham complex of A. Example 1.2. If A = k[t], the polynomial ring in one variable, then Ω1

A/k is a

free A-module of rank 1. The differential form d(t) is a basis. Now take the ring of formal power series A = k[[t]]. The first guess would be that Ω1

A/k is a free A-module of rank 1 with basis d(t).

However, if k is a field of characteristic 0, this is false! The module Ω1

A/k is

not even finitely generated!

Amnon Yekutieli (BGU) TLFs and Residues 4 / 34

• 1. Background on Semi-Topological Rings

The exterior algebra of Ω1

A/k over A is the differential graded (DG) k-ring

ΩA/k =

• i≥0

Ωi

A/k.

The multiplication is super-commutative. The differential d extends the derivation (1.1). The DG ring ΩA/k is also called the de Rham complex of A. Example 1.2. If A = k[t], the polynomial ring in one variable, then Ω1

A/k is a

free A-module of rank 1. The differential form d(t) is a basis. Now take the ring of formal power series A = k[[t]]. The first guess would be that Ω1

A/k is a free A-module of rank 1 with basis d(t).

However, if k is a field of characteristic 0, this is false! The module Ω1

A/k is

not even finitely generated!

Amnon Yekutieli (BGU) TLFs and Residues 4 / 34

• 1. Background on Semi-Topological Rings

The exterior algebra of Ω1

A/k over A is the differential graded (DG) k-ring

ΩA/k =

• i≥0

Ωi

A/k.

The multiplication is super-commutative. The differential d extends the derivation (1.1). The DG ring ΩA/k is also called the de Rham complex of A. Example 1.2. If A = k[t], the polynomial ring in one variable, then Ω1

A/k is a

free A-module of rank 1. The differential form d(t) is a basis. Now take the ring of formal power series A = k[[t]]. The first guess would be that Ω1

A/k is a free A-module of rank 1 with basis d(t).

However, if k is a field of characteristic 0, this is false! The module Ω1

A/k is

not even finitely generated!

Amnon Yekutieli (BGU) TLFs and Residues 4 / 34

• 1. Background on Semi-Topological Rings

This “problem” is well-known, as well as its solution. Cf. [Se] or [EGA-IV, Section 20.3]. The general solution is this: put a suitable k-linear topology on the ring A. This will induce a k-linear topology on the module Ω1

A/k. The closure of 0 is

an A-submodule {0}. The associated separated (i.e. Hausdorff) module is Ω1,sep

A/k := Ω1 A/k / {0}.

If we are lucky, the A-module Ω1,sep

A/k has the expected properties.

Example 1.3. Continuing with Example 1.2, we take the t-adic topology on A = k[[t]]. Then Ω1,sep

A/k if free with basis d(t).

Amnon Yekutieli (BGU) TLFs and Residues 5 / 34

• 1. Background on Semi-Topological Rings

This “problem” is well-known, as well as its solution. Cf. [Se] or [EGA-IV, Section 20.3]. The general solution is this: put a suitable k-linear topology on the ring A. This will induce a k-linear topology on the module Ω1

A/k. The closure of 0 is

an A-submodule {0}. The associated separated (i.e. Hausdorff) module is Ω1,sep

A/k := Ω1 A/k / {0}.

If we are lucky, the A-module Ω1,sep

A/k has the expected properties.

Example 1.3. Continuing with Example 1.2, we take the t-adic topology on A = k[[t]]. Then Ω1,sep

A/k if free with basis d(t).

Amnon Yekutieli (BGU) TLFs and Residues 5 / 34

• 1. Background on Semi-Topological Rings

This “problem” is well-known, as well as its solution. Cf. [Se] or [EGA-IV, Section 20.3]. The general solution is this: put a suitable k-linear topology on the ring A. This will induce a k-linear topology on the module Ω1

A/k. The closure of 0 is

an A-submodule {0}. The associated separated (i.e. Hausdorff) module is Ω1,sep

A/k := Ω1 A/k / {0}.

If we are lucky, the A-module Ω1,sep

A/k has the expected properties.

Example 1.3. Continuing with Example 1.2, we take the t-adic topology on A = k[[t]]. Then Ω1,sep

A/k if free with basis d(t).

Amnon Yekutieli (BGU) TLFs and Residues 5 / 34

• 1. Background on Semi-Topological Rings

This “problem” is well-known, as well as its solution. Cf. [Se] or [EGA-IV, Section 20.3]. The general solution is this: put a suitable k-linear topology on the ring A. This will induce a k-linear topology on the module Ω1

A/k. The closure of 0 is

an A-submodule {0}. The associated separated (i.e. Hausdorff) module is Ω1,sep

A/k := Ω1 A/k / {0}.

If we are lucky, the A-module Ω1,sep

A/k has the expected properties.

Example 1.3. Continuing with Example 1.2, we take the t-adic topology on A = k[[t]]. Then Ω1,sep

A/k if free with basis d(t).

Amnon Yekutieli (BGU) TLFs and Residues 5 / 34

• 1. Background on Semi-Topological Rings

This “problem” is well-known, as well as its solution. Cf. [Se] or [EGA-IV, Section 20.3]. The general solution is this: put a suitable k-linear topology on the ring A. This will induce a k-linear topology on the module Ω1

A/k. The closure of 0 is

an A-submodule {0}. The associated separated (i.e. Hausdorff) module is Ω1,sep

A/k := Ω1 A/k / {0}.

If we are lucky, the A-module Ω1,sep

A/k has the expected properties.

Example 1.3. Continuing with Example 1.2, we take the t-adic topology on A = k[[t]]. Then Ω1,sep

A/k if free with basis d(t).

Amnon Yekutieli (BGU) TLFs and Residues 5 / 34

• 1. Background on Semi-Topological Rings

The rings that we will encounter (and I mean the topological local fields of dimensions ≥ 2) will have more complicated topologies. Suppose M, N, P are linearly topologized k-modules. A k-bilinear function β : M × N → P is called semi-continuous if for any m, n the functions β(m, −) : N → P and β(−, n) : M → P are continuous. Definition 1.4. A semi-topological k-ring is a k-ring A, endowed with a k-linear topology, such that multiplication A × A → A is a semi-continuous function.

Amnon Yekutieli (BGU) TLFs and Residues 6 / 34

• 1. Background on Semi-Topological Rings

The rings that we will encounter (and I mean the topological local fields of dimensions ≥ 2) will have more complicated topologies. Suppose M, N, P are linearly topologized k-modules. A k-bilinear function β : M × N → P is called semi-continuous if for any m, n the functions β(m, −) : N → P and β(−, n) : M → P are continuous. Definition 1.4. A semi-topological k-ring is a k-ring A, endowed with a k-linear topology, such that multiplication A × A → A is a semi-continuous function.

Amnon Yekutieli (BGU) TLFs and Residues 6 / 34

• 1. Background on Semi-Topological Rings

The rings that we will encounter (and I mean the topological local fields of dimensions ≥ 2) will have more complicated topologies. Suppose M, N, P are linearly topologized k-modules. A k-bilinear function β : M × N → P is called semi-continuous if for any m, n the functions β(m, −) : N → P and β(−, n) : M → P are continuous. Definition 1.4. A semi-topological k-ring is a k-ring A, endowed with a k-linear topology, such that multiplication A × A → A is a semi-continuous function.

Amnon Yekutieli (BGU) TLFs and Residues 6 / 34

• 1. Background on Semi-Topological Rings

Definition 1.5. Let A be a semi-topological k-ring. A semi-topological A-module is an A-module M, endowed with a k-linear topology, such that multiplication A × M → M is a semi-continuous function. We write “ST” as an abbreviation for “semi-topological”. Example 1.6. Suppose A is a ST k-ring. The ring A[[t]] of formal power series is isomorphic, as A-module, to

i∈N A,

and we give it the product topology. The ring A((t)) of formal Laurent series is isomorphic, as A-module, to A[[t]] ⊕

• i∈N A
• ,

and we give it the direct sum topology. Both A[[t]] and A((t)) are ST k-rings.

Amnon Yekutieli (BGU) TLFs and Residues 7 / 34

• 1. Background on Semi-Topological Rings

Definition 1.5. Let A be a semi-topological k-ring. A semi-topological A-module is an A-module M, endowed with a k-linear topology, such that multiplication A × M → M is a semi-continuous function. We write “ST” as an abbreviation for “semi-topological”. Example 1.6. Suppose A is a ST k-ring. The ring A[[t]] of formal power series is isomorphic, as A-module, to

i∈N A,

and we give it the product topology. The ring A((t)) of formal Laurent series is isomorphic, as A-module, to A[[t]] ⊕

• i∈N A
• ,

and we give it the direct sum topology. Both A[[t]] and A((t)) are ST k-rings.

Amnon Yekutieli (BGU) TLFs and Residues 7 / 34

• 1. Background on Semi-Topological Rings

Definition 1.5. Let A be a semi-topological k-ring. A semi-topological A-module is an A-module M, endowed with a k-linear topology, such that multiplication A × M → M is a semi-continuous function. We write “ST” as an abbreviation for “semi-topological”. Example 1.6. Suppose A is a ST k-ring. The ring A[[t]] of formal power series is isomorphic, as A-module, to

i∈N A,

and we give it the product topology. The ring A((t)) of formal Laurent series is isomorphic, as A-module, to A[[t]] ⊕

• i∈N A
• ,

and we give it the direct sum topology. Both A[[t]] and A((t)) are ST k-rings.

Amnon Yekutieli (BGU) TLFs and Residues 7 / 34

• 1. Background on Semi-Topological Rings

Definition 1.5. Let A be a semi-topological k-ring. A semi-topological A-module is an A-module M, endowed with a k-linear topology, such that multiplication A × M → M is a semi-continuous function. We write “ST” as an abbreviation for “semi-topological”. Example 1.6. Suppose A is a ST k-ring. The ring A[[t]] of formal power series is isomorphic, as A-module, to

i∈N A,

and we give it the product topology. The ring A((t)) of formal Laurent series is isomorphic, as A-module, to A[[t]] ⊕

• i∈N A
• ,

and we give it the direct sum topology. Both A[[t]] and A((t)) are ST k-rings.

Amnon Yekutieli (BGU) TLFs and Residues 7 / 34

• 1. Background on Semi-Topological Rings

Definition 1.5. Let A be a semi-topological k-ring. A semi-topological A-module is an A-module M, endowed with a k-linear topology, such that multiplication A × M → M is a semi-continuous function. We write “ST” as an abbreviation for “semi-topological”. Example 1.6. Suppose A is a ST k-ring. The ring A[[t]] of formal power series is isomorphic, as A-module, to

i∈N A,

and we give it the product topology. The ring A((t)) of formal Laurent series is isomorphic, as A-module, to A[[t]] ⊕

• i∈N A
• ,

and we give it the direct sum topology. Both A[[t]] and A((t)) are ST k-rings.

Amnon Yekutieli (BGU) TLFs and Residues 7 / 34

• 1. Background on Semi-Topological Rings

Definition 1.5. Let A be a semi-topological k-ring. A semi-topological A-module is an A-module M, endowed with a k-linear topology, such that multiplication A × M → M is a semi-continuous function. We write “ST” as an abbreviation for “semi-topological”. Example 1.6. Suppose A is a ST k-ring. The ring A[[t]] of formal power series is isomorphic, as A-module, to

i∈N A,

and we give it the product topology. The ring A((t)) of formal Laurent series is isomorphic, as A-module, to A[[t]] ⊕

• i∈N A
• ,

and we give it the direct sum topology. Both A[[t]] and A((t)) are ST k-rings.

Amnon Yekutieli (BGU) TLFs and Residues 7 / 34

• 1. Background on Semi-Topological Rings

Let us denote by STRingc k the category of commutative ST k-rings. The morphisms are the continuous k-ring homomorphisms. Let A be an ST k-ring. Then the DG ring of differentials ΩA/k has an induced topology, making it into an ST DG k-ring. Passing to the associated separated object we get the DG ring of separated differentials Ωsep

A/k =

• i≥0

Ωi,sep

A/k .

If f : A → B is a homomorphism in STRingc k, then there is an induced homomorphism of ST DG rings f : Ωsep

A/k → Ωsep B/k.

Amnon Yekutieli (BGU) TLFs and Residues 8 / 34

• 1. Background on Semi-Topological Rings

Let us denote by STRingc k the category of commutative ST k-rings. The morphisms are the continuous k-ring homomorphisms. Let A be an ST k-ring. Then the DG ring of differentials ΩA/k has an induced topology, making it into an ST DG k-ring. Passing to the associated separated object we get the DG ring of separated differentials Ωsep

A/k =

• i≥0

Ωi,sep

A/k .

If f : A → B is a homomorphism in STRingc k, then there is an induced homomorphism of ST DG rings f : Ωsep

A/k → Ωsep B/k.

Amnon Yekutieli (BGU) TLFs and Residues 8 / 34

• 1. Background on Semi-Topological Rings

Let us denote by STRingc k the category of commutative ST k-rings. The morphisms are the continuous k-ring homomorphisms. Let A be an ST k-ring. Then the DG ring of differentials ΩA/k has an induced topology, making it into an ST DG k-ring. Passing to the associated separated object we get the DG ring of separated differentials Ωsep

A/k =

• i≥0

Ωi,sep

A/k .

If f : A → B is a homomorphism in STRingc k, then there is an induced homomorphism of ST DG rings f : Ωsep

A/k → Ωsep B/k.

Amnon Yekutieli (BGU) TLFs and Residues 8 / 34

• 1. Background on Semi-Topological Rings

Let us denote by STRingc k the category of commutative ST k-rings. The morphisms are the continuous k-ring homomorphisms. Let A be an ST k-ring. Then the DG ring of differentials ΩA/k has an induced topology, making it into an ST DG k-ring. Passing to the associated separated object we get the DG ring of separated differentials Ωsep

A/k =

• i≥0

Ωi,sep

A/k .

If f : A → B is a homomorphism in STRingc k, then there is an induced homomorphism of ST DG rings f : Ωsep

A/k → Ωsep B/k.

Amnon Yekutieli (BGU) TLFs and Residues 8 / 34

• 2. High Dimensional Local Fields
• 2. High Dimensional Local Fields

This concept, introduced by Parshin [Pa1, Pa2] and Kato [Ka], was already explained in earlier talks. I will quickly recap, and introduce notation. Definition 2.1. An n-dimensional local field over k is a field K, together with a sequence

• O1(K), . . . , On(K)
• f complete DVRs, such that:

◮ The fraction field of O1(K) is K. ◮ The residue field ki(K) of Oi(K) is the fraction field of Oi+1(K). ◮ All these rings and homomorphism are in the category of k-rings, and

k → kn(K) is finite.

Amnon Yekutieli (BGU) TLFs and Residues 9 / 34

• 2. High Dimensional Local Fields
• 2. High Dimensional Local Fields

This concept, introduced by Parshin [Pa1, Pa2] and Kato [Ka], was already explained in earlier talks. I will quickly recap, and introduce notation. Definition 2.1. An n-dimensional local field over k is a field K, together with a sequence

• O1(K), . . . , On(K)
• f complete DVRs, such that:

◮ The fraction field of O1(K) is K. ◮ The residue field ki(K) of Oi(K) is the fraction field of Oi+1(K). ◮ All these rings and homomorphism are in the category of k-rings, and

k → kn(K) is finite.

Amnon Yekutieli (BGU) TLFs and Residues 9 / 34

• 2. High Dimensional Local Fields
• 2. High Dimensional Local Fields

This concept, introduced by Parshin [Pa1, Pa2] and Kato [Ka], was already explained in earlier talks. I will quickly recap, and introduce notation. Definition 2.1. An n-dimensional local field over k is a field K, together with a sequence

• O1(K), . . . , On(K)
• f complete DVRs, such that:

◮ The fraction field of O1(K) is K. ◮ The residue field ki(K) of Oi(K) is the fraction field of Oi+1(K). ◮ All these rings and homomorphism are in the category of k-rings, and

k → kn(K) is finite.

Amnon Yekutieli (BGU) TLFs and Residues 9 / 34

• 2. High Dimensional Local Fields
• 2. High Dimensional Local Fields

This concept, introduced by Parshin [Pa1, Pa2] and Kato [Ka], was already explained in earlier talks. I will quickly recap, and introduce notation. Definition 2.1. An n-dimensional local field over k is a field K, together with a sequence

• O1(K), . . . , On(K)
• f complete DVRs, such that:

◮ The fraction field of O1(K) is K. ◮ The residue field ki(K) of Oi(K) is the fraction field of Oi+1(K). ◮ All these rings and homomorphism are in the category of k-rings, and

k → kn(K) is finite.

Amnon Yekutieli (BGU) TLFs and Residues 9 / 34

• 2. High Dimensional Local Fields
• 2. High Dimensional Local Fields

This concept, introduced by Parshin [Pa1, Pa2] and Kato [Ka], was already explained in earlier talks. I will quickly recap, and introduce notation. Definition 2.1. An n-dimensional local field over k is a field K, together with a sequence

• O1(K), . . . , On(K)
• f complete DVRs, such that:

◮ The fraction field of O1(K) is K. ◮ The residue field ki(K) of Oi(K) is the fraction field of Oi+1(K). ◮ All these rings and homomorphism are in the category of k-rings, and

k → kn(K) is finite.

Amnon Yekutieli (BGU) TLFs and Residues 9 / 34

• 2. High Dimensional Local Fields
• 2. High Dimensional Local Fields

This concept, introduced by Parshin [Pa1, Pa2] and Kato [Ka], was already explained in earlier talks. I will quickly recap, and introduce notation. Definition 2.1. An n-dimensional local field over k is a field K, together with a sequence

• O1(K), . . . , On(K)
• f complete DVRs, such that:

◮ The fraction field of O1(K) is K. ◮ The residue field ki(K) of Oi(K) is the fraction field of Oi+1(K). ◮ All these rings and homomorphism are in the category of k-rings, and

k → kn(K) is finite.

Amnon Yekutieli (BGU) TLFs and Residues 9 / 34

• 2. High Dimensional Local Fields

Here is the picture for n = 2.

Amnon Yekutieli (BGU) TLFs and Residues 10 / 34

• 2. High Dimensional Local Fields

Here is the picture for n = 2. K

Amnon Yekutieli (BGU) TLFs and Residues 10 / 34

• 2. High Dimensional Local Fields

Here is the picture for n = 2. O1(K)

K

Amnon Yekutieli (BGU) TLFs and Residues 10 / 34

• 2. High Dimensional Local Fields

Here is the picture for n = 2. O1(K)

• K

k1(K)

Amnon Yekutieli (BGU) TLFs and Residues 10 / 34

• 2. High Dimensional Local Fields

Here is the picture for n = 2. O1(K)

• K

O2(K)

k1(K)

Amnon Yekutieli (BGU) TLFs and Residues 10 / 34

• 2. High Dimensional Local Fields

Here is the picture for n = 2. O1(K)

• K

O2(K)

• k1(K)

k2(K)

Amnon Yekutieli (BGU) TLFs and Residues 10 / 34

• 2. High Dimensional Local Fields

Here is the picture for n = 2. O1(K)

• K

k

• O2(K)
• k1(K)

k2(K)

Amnon Yekutieli (BGU) TLFs and Residues 10 / 34

• 2. High Dimensional Local Fields

Here is the picture for n = 2. O1(K)

• K

k

• finite
• O2(K)
• k1(K)

k2(K)

Amnon Yekutieli (BGU) TLFs and Residues 10 / 34

• 2. High Dimensional Local Fields

A 0-dimensional local field over k is just a field K finite over k. Example 2.2. Take k := Z. The fields Qp and Fp((t)) are 1-dimensional local fields over Z. Definition 2.3. Let K and L be local fields over k of dimension n ≥ 1. A morphism of local fields f : K → L is a k-ring homomorphism such that the following conditions hold:

◮ f (O1(K)) ⊂ O1(L). ◮ The induced k-ring homomorphism f : O1(K) → O1(L) is a local

homomorphism.

◮ The induced k-ring homomorphism ¯

f : k1(K) → k1(L) is a morphism of (n − 1)-dimensional local fields over k.

Amnon Yekutieli (BGU) TLFs and Residues 11 / 34

• 2. High Dimensional Local Fields

A 0-dimensional local field over k is just a field K finite over k. Example 2.2. Take k := Z. The fields Qp and Fp((t)) are 1-dimensional local fields over Z. Definition 2.3. Let K and L be local fields over k of dimension n ≥ 1. A morphism of local fields f : K → L is a k-ring homomorphism such that the following conditions hold:

◮ f (O1(K)) ⊂ O1(L). ◮ The induced k-ring homomorphism f : O1(K) → O1(L) is a local

homomorphism.

◮ The induced k-ring homomorphism ¯

f : k1(K) → k1(L) is a morphism of (n − 1)-dimensional local fields over k.

Amnon Yekutieli (BGU) TLFs and Residues 11 / 34

• 2. High Dimensional Local Fields

A 0-dimensional local field over k is just a field K finite over k. Example 2.2. Take k := Z. The fields Qp and Fp((t)) are 1-dimensional local fields over Z. Definition 2.3. Let K and L be local fields over k of dimension n ≥ 1. A morphism of local fields f : K → L is a k-ring homomorphism such that the following conditions hold:

◮ f (O1(K)) ⊂ O1(L). ◮ The induced k-ring homomorphism f : O1(K) → O1(L) is a local

homomorphism.

◮ The induced k-ring homomorphism ¯

f : k1(K) → k1(L) is a morphism of (n − 1)-dimensional local fields over k.

Amnon Yekutieli (BGU) TLFs and Residues 11 / 34

• 2. High Dimensional Local Fields

A 0-dimensional local field over k is just a field K finite over k. Example 2.2. Take k := Z. The fields Qp and Fp((t)) are 1-dimensional local fields over Z. Definition 2.3. Let K and L be local fields over k of dimension n ≥ 1. A morphism of local fields f : K → L is a k-ring homomorphism such that the following conditions hold:

◮ f (O1(K)) ⊂ O1(L). ◮ The induced k-ring homomorphism f : O1(K) → O1(L) is a local

homomorphism.

◮ The induced k-ring homomorphism ¯

f : k1(K) → k1(L) is a morphism of (n − 1)-dimensional local fields over k.

Amnon Yekutieli (BGU) TLFs and Residues 11 / 34

• 2. High Dimensional Local Fields

A 0-dimensional local field over k is just a field K finite over k. Example 2.2. Take k := Z. The fields Qp and Fp((t)) are 1-dimensional local fields over Z. Definition 2.3. Let K and L be local fields over k of dimension n ≥ 1. A morphism of local fields f : K → L is a k-ring homomorphism such that the following conditions hold:

◮ f (O1(K)) ⊂ O1(L). ◮ The induced k-ring homomorphism f : O1(K) → O1(L) is a local

homomorphism.

◮ The induced k-ring homomorphism ¯

f : k1(K) → k1(L) is a morphism of (n − 1)-dimensional local fields over k.

Amnon Yekutieli (BGU) TLFs and Residues 11 / 34

• 2. High Dimensional Local Fields

A 0-dimensional local field over k is just a field K finite over k. Example 2.2. Take k := Z. The fields Qp and Fp((t)) are 1-dimensional local fields over Z. Definition 2.3. Let K and L be local fields over k of dimension n ≥ 1. A morphism of local fields f : K → L is a k-ring homomorphism such that the following conditions hold:

◮ f (O1(K)) ⊂ O1(L). ◮ The induced k-ring homomorphism f : O1(K) → O1(L) is a local

homomorphism.

◮ The induced k-ring homomorphism ¯

f : k1(K) → k1(L) is a morphism of (n − 1)-dimensional local fields over k.

Amnon Yekutieli (BGU) TLFs and Residues 11 / 34

• 2. High Dimensional Local Fields

A 0-dimensional local field over k is just a field K finite over k. Example 2.2. Take k := Z. The fields Qp and Fp((t)) are 1-dimensional local fields over Z. Definition 2.3. Let K and L be local fields over k of dimension n ≥ 1. A morphism of local fields f : K → L is a k-ring homomorphism such that the following conditions hold:

◮ f (O1(K)) ⊂ O1(L). ◮ The induced k-ring homomorphism f : O1(K) → O1(L) is a local

homomorphism.

◮ The induced k-ring homomorphism ¯

f : k1(K) → k1(L) is a morphism of (n − 1)-dimensional local fields over k.

Amnon Yekutieli (BGU) TLFs and Residues 11 / 34

• 2. High Dimensional Local Fields

Let us denote by LFn k the category of n-dimensional local fields over k. It is not hard to show that any homomorphism K → L in LFn k is finite. Actually one can talk about a morphism of local fields f : K → L when dim(K) < dim(L); but the definition is more complicated. We get a category LF k, of which LFn k is a full subcategory. See [Ye1] for details. Example 2.4. If k is a field, then the field of Laurent series K := k((t2)) is a 1-dimensional local field. The field of iterated Laurent series L := K((t1)) = k((t2))((t1)) is a 2-dimensional local field. The inclusions k → K → L are morphisms in LF k.

Amnon Yekutieli (BGU) TLFs and Residues 12 / 34

• 2. High Dimensional Local Fields

Let us denote by LFn k the category of n-dimensional local fields over k. It is not hard to show that any homomorphism K → L in LFn k is finite. Actually one can talk about a morphism of local fields f : K → L when dim(K) < dim(L); but the definition is more complicated. We get a category LF k, of which LFn k is a full subcategory. See [Ye1] for details. Example 2.4. If k is a field, then the field of Laurent series K := k((t2)) is a 1-dimensional local field. The field of iterated Laurent series L := K((t1)) = k((t2))((t1)) is a 2-dimensional local field. The inclusions k → K → L are morphisms in LF k.

Amnon Yekutieli (BGU) TLFs and Residues 12 / 34

• 2. High Dimensional Local Fields

Let us denote by LFn k the category of n-dimensional local fields over k. It is not hard to show that any homomorphism K → L in LFn k is finite. Actually one can talk about a morphism of local fields f : K → L when dim(K) < dim(L); but the definition is more complicated. We get a category LF k, of which LFn k is a full subcategory. See [Ye1] for details. Example 2.4. If k is a field, then the field of Laurent series K := k((t2)) is a 1-dimensional local field. The field of iterated Laurent series L := K((t1)) = k((t2))((t1)) is a 2-dimensional local field. The inclusions k → K → L are morphisms in LF k.

Amnon Yekutieli (BGU) TLFs and Residues 12 / 34

• 2. High Dimensional Local Fields

Let us denote by LFn k the category of n-dimensional local fields over k. It is not hard to show that any homomorphism K → L in LFn k is finite. Actually one can talk about a morphism of local fields f : K → L when dim(K) < dim(L); but the definition is more complicated. We get a category LF k, of which LFn k is a full subcategory. See [Ye1] for details. Example 2.4. If k is a field, then the field of Laurent series K := k((t2)) is a 1-dimensional local field. The field of iterated Laurent series L := K((t1)) = k((t2))((t1)) is a 2-dimensional local field. The inclusions k → K → L are morphisms in LF k.

Amnon Yekutieli (BGU) TLFs and Residues 12 / 34

• 2. High Dimensional Local Fields

Let us denote by LFn k the category of n-dimensional local fields over k. It is not hard to show that any homomorphism K → L in LFn k is finite. Actually one can talk about a morphism of local fields f : K → L when dim(K) < dim(L); but the definition is more complicated. We get a category LF k, of which LFn k is a full subcategory. See [Ye1] for details. Example 2.4. If k is a field, then the field of Laurent series K := k((t2)) is a 1-dimensional local field. The field of iterated Laurent series L := K((t1)) = k((t2))((t1)) is a 2-dimensional local field. The inclusions k → K → L are morphisms in LF k.

Amnon Yekutieli (BGU) TLFs and Residues 12 / 34

• 2. High Dimensional Local Fields

Let us denote by LFn k the category of n-dimensional local fields over k. It is not hard to show that any homomorphism K → L in LFn k is finite. Actually one can talk about a morphism of local fields f : K → L when dim(K) < dim(L); but the definition is more complicated. We get a category LF k, of which LFn k is a full subcategory. See [Ye1] for details. Example 2.4. If k is a field, then the field of Laurent series K := k((t2)) is a 1-dimensional local field. The field of iterated Laurent series L := K((t1)) = k((t2))((t1)) is a 2-dimensional local field. The inclusions k → K → L are morphisms in LF k.

Amnon Yekutieli (BGU) TLFs and Residues 12 / 34

• 3. Topological Local Fields
• 3. Topological Local Fields

From here on k is a perfect field. Therefore all our local fields are now of equal characteristics. One of the reasons that we need this condition is as follows. Let K be an n-dimensional local field over k, with last residue field k′ := kn(K). Since k is perfect, the finite extension k → k′ is separable; or in other words, it is étale. An n-fold repeated application of formal lifting (also known as Hensel’s Lemma) shows that there is a canonical homomorphism k′ → K in the category of k-rings.

Amnon Yekutieli (BGU) TLFs and Residues 13 / 34

• 3. Topological Local Fields
• 3. Topological Local Fields

From here on k is a perfect field. Therefore all our local fields are now of equal characteristics. One of the reasons that we need this condition is as follows. Let K be an n-dimensional local field over k, with last residue field k′ := kn(K). Since k is perfect, the finite extension k → k′ is separable; or in other words, it is étale. An n-fold repeated application of formal lifting (also known as Hensel’s Lemma) shows that there is a canonical homomorphism k′ → K in the category of k-rings.

Amnon Yekutieli (BGU) TLFs and Residues 13 / 34

• 3. Topological Local Fields
• 3. Topological Local Fields

From here on k is a perfect field. Therefore all our local fields are now of equal characteristics. One of the reasons that we need this condition is as follows. Let K be an n-dimensional local field over k, with last residue field k′ := kn(K). Since k is perfect, the finite extension k → k′ is separable; or in other words, it is étale. An n-fold repeated application of formal lifting (also known as Hensel’s Lemma) shows that there is a canonical homomorphism k′ → K in the category of k-rings.

Amnon Yekutieli (BGU) TLFs and Residues 13 / 34

• 3. Topological Local Fields
• 3. Topological Local Fields

From here on k is a perfect field. Therefore all our local fields are now of equal characteristics. One of the reasons that we need this condition is as follows. Let K be an n-dimensional local field over k, with last residue field k′ := kn(K). Since k is perfect, the finite extension k → k′ is separable; or in other words, it is étale. An n-fold repeated application of formal lifting (also known as Hensel’s Lemma) shows that there is a canonical homomorphism k′ → K in the category of k-rings.

Amnon Yekutieli (BGU) TLFs and Residues 13 / 34

• 3. Topological Local Fields

Let k′ be a finite field extension of k, and let t = (t1, . . . , tn) be a sequence of variables. We denote by k′((t)) = k′((t1, . . . , tn)) := k′((tn)) · · · ((t1)) the iterated field of Laurent series. The field k′((t)) has a canonical structure of n-dimensional local field over k. The DVRs are Oi

• k′((t))
• := k′((ti+1, . . . , tn))[[ti]],

and the residue fields are ki

• k′((t))
• := k′((ti+1, . . . , tn)).

Amnon Yekutieli (BGU) TLFs and Residues 14 / 34

• 3. Topological Local Fields

Let k′ be a finite field extension of k, and let t = (t1, . . . , tn) be a sequence of variables. We denote by k′((t)) = k′((t1, . . . , tn)) := k′((tn)) · · · ((t1)) the iterated field of Laurent series. The field k′((t)) has a canonical structure of n-dimensional local field over k. The DVRs are Oi

• k′((t))
• := k′((ti+1, . . . , tn))[[ti]],

and the residue fields are ki

• k′((t))
• := k′((ti+1, . . . , tn)).

Amnon Yekutieli (BGU) TLFs and Residues 14 / 34

• 3. Topological Local Fields

Let k′ be a finite field extension of k, and let t = (t1, . . . , tn) be a sequence of variables. We denote by k′((t)) = k′((t1, . . . , tn)) := k′((tn)) · · · ((t1)) the iterated field of Laurent series. The field k′((t)) has a canonical structure of n-dimensional local field over k. The DVRs are Oi

• k′((t))
• := k′((ti+1, . . . , tn))[[ti]],

and the residue fields are ki

• k′((t))
• := k′((ti+1, . . . , tn)).

Amnon Yekutieli (BGU) TLFs and Residues 14 / 34

• 3. Topological Local Fields

Let k′ be a finite field extension of k, and let t = (t1, . . . , tn) be a sequence of variables. We denote by k′((t)) = k′((t1, . . . , tn)) := k′((tn)) · · · ((t1)) the iterated field of Laurent series. The field k′((t)) has a canonical structure of n-dimensional local field over k. The DVRs are Oi

• k′((t))
• := k′((ti+1, . . . , tn))[[ti]],

and the residue fields are ki

• k′((t))
• := k′((ti+1, . . . , tn)).

Amnon Yekutieli (BGU) TLFs and Residues 14 / 34

• 3. Topological Local Fields

Let k′ be a finite field extension of k, and let t = (t1, . . . , tn) be a sequence of variables. We denote by k′((t)) = k′((t1, . . . , tn)) := k′((tn)) · · · ((t1)) the iterated field of Laurent series. The field k′((t)) has a canonical structure of n-dimensional local field over k. The DVRs are Oi

• k′((t))
• := k′((ti+1, . . . , tn))[[ti]],

and the residue fields are ki

• k′((t))
• := k′((ti+1, . . . , tn)).

Amnon Yekutieli (BGU) TLFs and Residues 14 / 34

• 3. Topological Local Fields

The field k′((t)) has a topology on it, starting from the discrete topology on k′, and performing the operations of Example 1.6 recursively. This topology makes k′((t)) into a ST k-ring. We call k′((t)) the standard n-dimensional topological local field with last residue field k′.

Amnon Yekutieli (BGU) TLFs and Residues 15 / 34

• 3. Topological Local Fields

The field k′((t)) has a topology on it, starting from the discrete topology on k′, and performing the operations of Example 1.6 recursively. This topology makes k′((t)) into a ST k-ring. We call k′((t)) the standard n-dimensional topological local field with last residue field k′.

Amnon Yekutieli (BGU) TLFs and Residues 15 / 34

• 3. Topological Local Fields

The field k′((t)) has a topology on it, starting from the discrete topology on k′, and performing the operations of Example 1.6 recursively. This topology makes k′((t)) into a ST k-ring. We call k′((t)) the standard n-dimensional topological local field with last residue field k′.

Amnon Yekutieli (BGU) TLFs and Residues 15 / 34

• 3. Topological Local Fields

Definition 3.1. ([Ye1]) An n-dimensional topological local field over k is a field K, together with: (a) A structure {Oi(K)}n

i=1 of n-dimensional local field over k.

(b) A topology, making K a semi-topological k-ring. The condition is this: (P) There a bijection f : k′((t)) ≃ − → K from the standard n-dimensional topological local field with last residue field k′ := kn(K), such that:

(i) f is an isomorphism in LFn k (i.e. it respects the valuations). (ii) f is an isomorphism in STRingc k (i.e. it respects the topologies).

Such an isomorphism f is called a parametrization of K.

Amnon Yekutieli (BGU) TLFs and Residues 16 / 34

• 3. Topological Local Fields

Definition 3.1. ([Ye1]) An n-dimensional topological local field over k is a field K, together with: (a) A structure {Oi(K)}n

i=1 of n-dimensional local field over k.

(b) A topology, making K a semi-topological k-ring. The condition is this: (P) There a bijection f : k′((t)) ≃ − → K from the standard n-dimensional topological local field with last residue field k′ := kn(K), such that:

(i) f is an isomorphism in LFn k (i.e. it respects the valuations). (ii) f is an isomorphism in STRingc k (i.e. it respects the topologies).

Such an isomorphism f is called a parametrization of K.

Amnon Yekutieli (BGU) TLFs and Residues 16 / 34

• 3. Topological Local Fields

Definition 3.1. ([Ye1]) An n-dimensional topological local field over k is a field K, together with: (a) A structure {Oi(K)}n

i=1 of n-dimensional local field over k.

(b) A topology, making K a semi-topological k-ring. The condition is this: (P) There a bijection f : k′((t)) ≃ − → K from the standard n-dimensional topological local field with last residue field k′ := kn(K), such that:

(i) f is an isomorphism in LFn k (i.e. it respects the valuations). (ii) f is an isomorphism in STRingc k (i.e. it respects the topologies).

Such an isomorphism f is called a parametrization of K.

Amnon Yekutieli (BGU) TLFs and Residues 16 / 34

• 3. Topological Local Fields

Definition 3.1. ([Ye1]) An n-dimensional topological local field over k is a field K, together with: (a) A structure {Oi(K)}n

i=1 of n-dimensional local field over k.

(b) A topology, making K a semi-topological k-ring. The condition is this: (P) There a bijection f : k′((t)) ≃ − → K from the standard n-dimensional topological local field with last residue field k′ := kn(K), such that:

(i) f is an isomorphism in LFn k (i.e. it respects the valuations). (ii) f is an isomorphism in STRingc k (i.e. it respects the topologies).

Such an isomorphism f is called a parametrization of K.

Amnon Yekutieli (BGU) TLFs and Residues 16 / 34

• 3. Topological Local Fields

Definition 3.1. ([Ye1]) An n-dimensional topological local field over k is a field K, together with: (a) A structure {Oi(K)}n

i=1 of n-dimensional local field over k.

(b) A topology, making K a semi-topological k-ring. The condition is this: (P) There a bijection f : k′((t)) ≃ − → K from the standard n-dimensional topological local field with last residue field k′ := kn(K), such that:

(i) f is an isomorphism in LFn k (i.e. it respects the valuations). (ii) f is an isomorphism in STRingc k (i.e. it respects the topologies).

Such an isomorphism f is called a parametrization of K.

Amnon Yekutieli (BGU) TLFs and Residues 16 / 34

• 3. Topological Local Fields

Definition 3.1. ([Ye1]) An n-dimensional topological local field over k is a field K, together with: (a) A structure {Oi(K)}n

i=1 of n-dimensional local field over k.

(b) A topology, making K a semi-topological k-ring. The condition is this: (P) There a bijection f : k′((t)) ≃ − → K from the standard n-dimensional topological local field with last residue field k′ := kn(K), such that:

(i) f is an isomorphism in LFn k (i.e. it respects the valuations). (ii) f is an isomorphism in STRingc k (i.e. it respects the topologies).

Such an isomorphism f is called a parametrization of K.

Amnon Yekutieli (BGU) TLFs and Residues 16 / 34

• 3. Topological Local Fields

Definition 3.1. ([Ye1]) An n-dimensional topological local field over k is a field K, together with: (a) A structure {Oi(K)}n

i=1 of n-dimensional local field over k.

(b) A topology, making K a semi-topological k-ring. The condition is this: (P) There a bijection f : k′((t)) ≃ − → K from the standard n-dimensional topological local field with last residue field k′ := kn(K), such that:

(i) f is an isomorphism in LFn k (i.e. it respects the valuations). (ii) f is an isomorphism in STRingc k (i.e. it respects the topologies).

Such an isomorphism f is called a parametrization of K.

Amnon Yekutieli (BGU) TLFs and Residues 16 / 34

• 3. Topological Local Fields

Definition 3.1. ([Ye1]) An n-dimensional topological local field over k is a field K, together with: (a) A structure {Oi(K)}n

i=1 of n-dimensional local field over k.

(b) A topology, making K a semi-topological k-ring. The condition is this: (P) There a bijection f : k′((t)) ≃ − → K from the standard n-dimensional topological local field with last residue field k′ := kn(K), such that:

(i) f is an isomorphism in LFn k (i.e. it respects the valuations). (ii) f is an isomorphism in STRingc k (i.e. it respects the topologies).

Such an isomorphism f is called a parametrization of K.

Amnon Yekutieli (BGU) TLFs and Residues 16 / 34

• 3. Topological Local Fields

Definition 3.1. ([Ye1]) An n-dimensional topological local field over k is a field K, together with: (a) A structure {Oi(K)}n

i=1 of n-dimensional local field over k.

(b) A topology, making K a semi-topological k-ring. The condition is this: (P) There a bijection f : k′((t)) ≃ − → K from the standard n-dimensional topological local field with last residue field k′ := kn(K), such that:

(i) f is an isomorphism in LFn k (i.e. it respects the valuations). (ii) f is an isomorphism in STRingc k (i.e. it respects the topologies).

Such an isomorphism f is called a parametrization of K.

Amnon Yekutieli (BGU) TLFs and Residues 16 / 34

• 3. Topological Local Fields

The parametrization f is not part of the structure of K; it is required to exist, but (as we shall soon see) there are many distinct parametrizations. We use the abbreviation “TLF” for “topological local field”. Here are some basic facts about TLFs. As a ST k-module, each TLF K is complete. This means that the canonical homomorphism K → lim

←U K/U,

where U runs over all open k-submodules of K, is bijective. In particular K is separated, so that Ksep = K. If dim(K) ≥ 2, then K is not a metrizable topological space, and it is not a topological ring (only ST).

Amnon Yekutieli (BGU) TLFs and Residues 17 / 34

• 3. Topological Local Fields

The parametrization f is not part of the structure of K; it is required to exist, but (as we shall soon see) there are many distinct parametrizations. We use the abbreviation “TLF” for “topological local field”. Here are some basic facts about TLFs. As a ST k-module, each TLF K is complete. This means that the canonical homomorphism K → lim

←U K/U,

where U runs over all open k-submodules of K, is bijective. In particular K is separated, so that Ksep = K. If dim(K) ≥ 2, then K is not a metrizable topological space, and it is not a topological ring (only ST).

Amnon Yekutieli (BGU) TLFs and Residues 17 / 34

• 3. Topological Local Fields

The parametrization f is not part of the structure of K; it is required to exist, but (as we shall soon see) there are many distinct parametrizations. We use the abbreviation “TLF” for “topological local field”. Here are some basic facts about TLFs. As a ST k-module, each TLF K is complete. This means that the canonical homomorphism K → lim

←U K/U,

where U runs over all open k-submodules of K, is bijective. In particular K is separated, so that Ksep = K. If dim(K) ≥ 2, then K is not a metrizable topological space, and it is not a topological ring (only ST).

Amnon Yekutieli (BGU) TLFs and Residues 17 / 34

• 3. Topological Local Fields

The parametrization f is not part of the structure of K; it is required to exist, but (as we shall soon see) there are many distinct parametrizations. We use the abbreviation “TLF” for “topological local field”. Here are some basic facts about TLFs. As a ST k-module, each TLF K is complete. This means that the canonical homomorphism K → lim

←U K/U,

where U runs over all open k-submodules of K, is bijective. In particular K is separated, so that Ksep = K. If dim(K) ≥ 2, then K is not a metrizable topological space, and it is not a topological ring (only ST).

Amnon Yekutieli (BGU) TLFs and Residues 17 / 34

• 3. Topological Local Fields

The parametrization f is not part of the structure of K; it is required to exist, but (as we shall soon see) there are many distinct parametrizations. We use the abbreviation “TLF” for “topological local field”. Here are some basic facts about TLFs. As a ST k-module, each TLF K is complete. This means that the canonical homomorphism K → lim

←U K/U,

where U runs over all open k-submodules of K, is bijective. In particular K is separated, so that Ksep = K. If dim(K) ≥ 2, then K is not a metrizable topological space, and it is not a topological ring (only ST).

Amnon Yekutieli (BGU) TLFs and Residues 17 / 34

• 3. Topological Local Fields

The parametrization f is not part of the structure of K; it is required to exist, but (as we shall soon see) there are many distinct parametrizations. We use the abbreviation “TLF” for “topological local field”. Here are some basic facts about TLFs. As a ST k-module, each TLF K is complete. This means that the canonical homomorphism K → lim

←U K/U,

where U runs over all open k-submodules of K, is bijective. In particular K is separated, so that Ksep = K. If dim(K) ≥ 2, then K is not a metrizable topological space, and it is not a topological ring (only ST).

Amnon Yekutieli (BGU) TLFs and Residues 17 / 34

• 3. Topological Local Fields

Let K be an n-dimensional TLF. A system of uniformizers in K is a sequence (a1, . . . , an) of elements of O1(K), such that a1 generates the maximal ideal

• f O1(K), and if n ≥ 2, the sequence (¯

a2, . . . , ¯ an), which is the image of (a2, . . . , an) under the canonical surjection O1(K) ։ k1(K), is a system of uniformizers in k1(K). The next theorem tells us what are all the possible parametrizations of a TLF. Theorem 3.2. ([Ye1]) Let K be an n-dimensional TLF over k, let (a1, . . . , an) be a system of uniformizers in K, let k′ := kn(K), and let σ : k′ → K be the canonical lifting. Then σ extends uniquely to an isomorphism of TLFs f : k′((t1, . . . , tn)) → K such that f (ti) = ai.

Amnon Yekutieli (BGU) TLFs and Residues 18 / 34

• 3. Topological Local Fields

Let K be an n-dimensional TLF. A system of uniformizers in K is a sequence (a1, . . . , an) of elements of O1(K), such that a1 generates the maximal ideal

• f O1(K), and if n ≥ 2, the sequence (¯

a2, . . . , ¯ an), which is the image of (a2, . . . , an) under the canonical surjection O1(K) ։ k1(K), is a system of uniformizers in k1(K). The next theorem tells us what are all the possible parametrizations of a TLF. Theorem 3.2. ([Ye1]) Let K be an n-dimensional TLF over k, let (a1, . . . , an) be a system of uniformizers in K, let k′ := kn(K), and let σ : k′ → K be the canonical lifting. Then σ extends uniquely to an isomorphism of TLFs f : k′((t1, . . . , tn)) → K such that f (ti) = ai.

Amnon Yekutieli (BGU) TLFs and Residues 18 / 34

• 3. Topological Local Fields

Let K be an n-dimensional TLF. A system of uniformizers in K is a sequence (a1, . . . , an) of elements of O1(K), such that a1 generates the maximal ideal

• f O1(K), and if n ≥ 2, the sequence (¯

a2, . . . , ¯ an), which is the image of (a2, . . . , an) under the canonical surjection O1(K) ։ k1(K), is a system of uniformizers in k1(K). The next theorem tells us what are all the possible parametrizations of a TLF. Theorem 3.2. ([Ye1]) Let K be an n-dimensional TLF over k, let (a1, . . . , an) be a system of uniformizers in K, let k′ := kn(K), and let σ : k′ → K be the canonical lifting. Then σ extends uniquely to an isomorphism of TLFs f : k′((t1, . . . , tn)) → K such that f (ti) = ai.

Amnon Yekutieli (BGU) TLFs and Residues 18 / 34

• 3. Topological Local Fields

Let K be an n-dimensional TLF. A system of uniformizers in K is a sequence (a1, . . . , an) of elements of O1(K), such that a1 generates the maximal ideal

• f O1(K), and if n ≥ 2, the sequence (¯

a2, . . . , ¯ an), which is the image of (a2, . . . , an) under the canonical surjection O1(K) ։ k1(K), is a system of uniformizers in k1(K). The next theorem tells us what are all the possible parametrizations of a TLF. Theorem 3.2. ([Ye1]) Let K be an n-dimensional TLF over k, let (a1, . . . , an) be a system of uniformizers in K, let k′ := kn(K), and let σ : k′ → K be the canonical lifting. Then σ extends uniquely to an isomorphism of TLFs f : k′((t1, . . . , tn)) → K such that f (ti) = ai.

Amnon Yekutieli (BGU) TLFs and Residues 18 / 34

• 3. Topological Local Fields

Let K be an n-dimensional TLF. A system of uniformizers in K is a sequence (a1, . . . , an) of elements of O1(K), such that a1 generates the maximal ideal

• f O1(K), and if n ≥ 2, the sequence (¯

a2, . . . , ¯ an), which is the image of (a2, . . . , an) under the canonical surjection O1(K) ։ k1(K), is a system of uniformizers in k1(K). The next theorem tells us what are all the possible parametrizations of a TLF. Theorem 3.2. ([Ye1]) Let K be an n-dimensional TLF over k, let (a1, . . . , an) be a system of uniformizers in K, let k′ := kn(K), and let σ : k′ → K be the canonical lifting. Then σ extends uniquely to an isomorphism of TLFs f : k′((t1, . . . , tn)) → K such that f (ti) = ai.

Amnon Yekutieli (BGU) TLFs and Residues 18 / 34

• 3. Topological Local Fields

Let K be an n-dimensional TLF. A system of uniformizers in K is a sequence (a1, . . . , an) of elements of O1(K), such that a1 generates the maximal ideal

• f O1(K), and if n ≥ 2, the sequence (¯

a2, . . . , ¯ an), which is the image of (a2, . . . , an) under the canonical surjection O1(K) ։ k1(K), is a system of uniformizers in k1(K). The next theorem tells us what are all the possible parametrizations of a TLF. Theorem 3.2. ([Ye1]) Let K be an n-dimensional TLF over k, let (a1, . . . , an) be a system of uniformizers in K, let k′ := kn(K), and let σ : k′ → K be the canonical lifting. Then σ extends uniquely to an isomorphism of TLFs f : k′((t1, . . . , tn)) → K such that f (ti) = ai.

Amnon Yekutieli (BGU) TLFs and Residues 18 / 34

• 3. Topological Local Fields

Definition 3.3. Let K and L be n-dimensional TLFs. A morphism of TLFs f : K → L is morphism of local fields which is also continuous. We denote by TLFn k the category of n-dimensional TLFs over k. There is a bigger category TLF k, that allows morphisms f : K → L with dim(K) < dim(L). See Example 2.4. TLFn k is a full subcategory of TLF k. Consider the functor TLFn k → LFn k that forgets the topology. When n ≥ 2 and char(k) = 0 this forgetful functor is far from being an equivalence. In other words, any such local field K admits many distinct topologies, all satisfying condition (P). There is an example of this phenomenon in [Ye1].

Amnon Yekutieli (BGU) TLFs and Residues 19 / 34

• 3. Topological Local Fields

Definition 3.3. Let K and L be n-dimensional TLFs. A morphism of TLFs f : K → L is morphism of local fields which is also continuous. We denote by TLFn k the category of n-dimensional TLFs over k. There is a bigger category TLF k, that allows morphisms f : K → L with dim(K) < dim(L). See Example 2.4. TLFn k is a full subcategory of TLF k. Consider the functor TLFn k → LFn k that forgets the topology. When n ≥ 2 and char(k) = 0 this forgetful functor is far from being an equivalence. In other words, any such local field K admits many distinct topologies, all satisfying condition (P). There is an example of this phenomenon in [Ye1].

Amnon Yekutieli (BGU) TLFs and Residues 19 / 34

• 3. Topological Local Fields

Definition 3.3. Let K and L be n-dimensional TLFs. A morphism of TLFs f : K → L is morphism of local fields which is also continuous. We denote by TLFn k the category of n-dimensional TLFs over k. There is a bigger category TLF k, that allows morphisms f : K → L with dim(K) < dim(L). See Example 2.4. TLFn k is a full subcategory of TLF k. Consider the functor TLFn k → LFn k that forgets the topology. When n ≥ 2 and char(k) = 0 this forgetful functor is far from being an equivalence. In other words, any such local field K admits many distinct topologies, all satisfying condition (P). There is an example of this phenomenon in [Ye1].

Amnon Yekutieli (BGU) TLFs and Residues 19 / 34

• 3. Topological Local Fields

Definition 3.3. Let K and L be n-dimensional TLFs. A morphism of TLFs f : K → L is morphism of local fields which is also continuous. We denote by TLFn k the category of n-dimensional TLFs over k. There is a bigger category TLF k, that allows morphisms f : K → L with dim(K) < dim(L). See Example 2.4. TLFn k is a full subcategory of TLF k. Consider the functor TLFn k → LFn k that forgets the topology. When n ≥ 2 and char(k) = 0 this forgetful functor is far from being an equivalence. In other words, any such local field K admits many distinct topologies, all satisfying condition (P). There is an example of this phenomenon in [Ye1].

Amnon Yekutieli (BGU) TLFs and Residues 19 / 34

• 3. Topological Local Fields

Definition 3.3. Let K and L be n-dimensional TLFs. A morphism of TLFs f : K → L is morphism of local fields which is also continuous. We denote by TLFn k the category of n-dimensional TLFs over k. There is a bigger category TLF k, that allows morphisms f : K → L with dim(K) < dim(L). See Example 2.4. TLFn k is a full subcategory of TLF k. Consider the functor TLFn k → LFn k that forgets the topology. When n ≥ 2 and char(k) = 0 this forgetful functor is far from being an equivalence. In other words, any such local field K admits many distinct topologies, all satisfying condition (P). There is an example of this phenomenon in [Ye1].

Amnon Yekutieli (BGU) TLFs and Residues 19 / 34

• 3. Topological Local Fields

Definition 3.3. Let K and L be n-dimensional TLFs. A morphism of TLFs f : K → L is morphism of local fields which is also continuous. We denote by TLFn k the category of n-dimensional TLFs over k. There is a bigger category TLF k, that allows morphisms f : K → L with dim(K) < dim(L). See Example 2.4. TLFn k is a full subcategory of TLF k. Consider the functor TLFn k → LFn k that forgets the topology. When n ≥ 2 and char(k) = 0 this forgetful functor is far from being an equivalence. In other words, any such local field K admits many distinct topologies, all satisfying condition (P). There is an example of this phenomenon in [Ye1].

Amnon Yekutieli (BGU) TLFs and Residues 19 / 34

• 3. Topological Local Fields

Definition 3.3. Let K and L be n-dimensional TLFs. A morphism of TLFs f : K → L is morphism of local fields which is also continuous. We denote by TLFn k the category of n-dimensional TLFs over k. There is a bigger category TLF k, that allows morphisms f : K → L with dim(K) < dim(L). See Example 2.4. TLFn k is a full subcategory of TLF k. Consider the functor TLFn k → LFn k that forgets the topology. When n ≥ 2 and char(k) = 0 this forgetful functor is far from being an equivalence. In other words, any such local field K admits many distinct topologies, all satisfying condition (P). There is an example of this phenomenon in [Ye1].

Amnon Yekutieli (BGU) TLFs and Residues 19 / 34

• 3. Topological Local Fields

However: Theorem 3.4. ([Ye1]) If char(k) = p > 0, the forgetful functor TLFn k → LFn k is an equivalence. The rough idea of the proof is this: changing parametrizations involves Taylor series expansions (just like a change of coordinates in complex analytic geometry). The coefficients in these expansions are continuous differential

• perators.

It turns out that for TLFs in characteristic p, all differential operators are continuous.

Amnon Yekutieli (BGU) TLFs and Residues 20 / 34

• 3. Topological Local Fields

However: Theorem 3.4. ([Ye1]) If char(k) = p > 0, the forgetful functor TLFn k → LFn k is an equivalence. The rough idea of the proof is this: changing parametrizations involves Taylor series expansions (just like a change of coordinates in complex analytic geometry). The coefficients in these expansions are continuous differential

• perators.

It turns out that for TLFs in characteristic p, all differential operators are continuous.

Amnon Yekutieli (BGU) TLFs and Residues 20 / 34

• 3. Topological Local Fields

However: Theorem 3.4. ([Ye1]) If char(k) = p > 0, the forgetful functor TLFn k → LFn k is an equivalence. The rough idea of the proof is this: changing parametrizations involves Taylor series expansions (just like a change of coordinates in complex analytic geometry). The coefficients in these expansions are continuous differential

• perators.

It turns out that for TLFs in characteristic p, all differential operators are continuous.

Amnon Yekutieli (BGU) TLFs and Residues 20 / 34

• 3. Topological Local Fields

However: Theorem 3.4. ([Ye1]) If char(k) = p > 0, the forgetful functor TLFn k → LFn k is an equivalence. The rough idea of the proof is this: changing parametrizations involves Taylor series expansions (just like a change of coordinates in complex analytic geometry). The coefficients in these expansions are continuous differential

• perators.

It turns out that for TLFs in characteristic p, all differential operators are continuous.

Amnon Yekutieli (BGU) TLFs and Residues 20 / 34

• 4. The Beilinson Completion
• 4. The Beilinson Completion

Suppose X is a finite type k-scheme. By a chain of points in X we mean a sequence ξ = (x0, . . . , xn) of points such that xi is a specialization of xi−1. The chain ξ is saturated if each xi is an immediate specialization of xi−1. In [Be], Beilinson defined a completion operation, which is a special case of his higher adeles. Given a quasi-coherent sheaf M on X, and a chain ξ, the Beilinson completion of M along ξ is a k-module Mξ, gotten by an n-fold zig-zag of inverse and direct limits.

Amnon Yekutieli (BGU) TLFs and Residues 21 / 34

• 4. The Beilinson Completion
• 4. The Beilinson Completion

Suppose X is a finite type k-scheme. By a chain of points in X we mean a sequence ξ = (x0, . . . , xn) of points such that xi is a specialization of xi−1. The chain ξ is saturated if each xi is an immediate specialization of xi−1. In [Be], Beilinson defined a completion operation, which is a special case of his higher adeles. Given a quasi-coherent sheaf M on X, and a chain ξ, the Beilinson completion of M along ξ is a k-module Mξ, gotten by an n-fold zig-zag of inverse and direct limits.

Amnon Yekutieli (BGU) TLFs and Residues 21 / 34

• 4. The Beilinson Completion
• 4. The Beilinson Completion

Suppose X is a finite type k-scheme. By a chain of points in X we mean a sequence ξ = (x0, . . . , xn) of points such that xi is a specialization of xi−1. The chain ξ is saturated if each xi is an immediate specialization of xi−1. In [Be], Beilinson defined a completion operation, which is a special case of his higher adeles. Given a quasi-coherent sheaf M on X, and a chain ξ, the Beilinson completion of M along ξ is a k-module Mξ, gotten by an n-fold zig-zag of inverse and direct limits.

Amnon Yekutieli (BGU) TLFs and Residues 21 / 34

• 4. The Beilinson Completion
• 4. The Beilinson Completion

Suppose X is a finite type k-scheme. By a chain of points in X we mean a sequence ξ = (x0, . . . , xn) of points such that xi is a specialization of xi−1. The chain ξ is saturated if each xi is an immediate specialization of xi−1. In [Be], Beilinson defined a completion operation, which is a special case of his higher adeles. Given a quasi-coherent sheaf M on X, and a chain ξ, the Beilinson completion of M along ξ is a k-module Mξ, gotten by an n-fold zig-zag of inverse and direct limits.

Amnon Yekutieli (BGU) TLFs and Residues 21 / 34

• 4. The Beilinson Completion
• 4. The Beilinson Completion

Suppose X is a finite type k-scheme. By a chain of points in X we mean a sequence ξ = (x0, . . . , xn) of points such that xi is a specialization of xi−1. The chain ξ is saturated if each xi is an immediate specialization of xi−1. In [Be], Beilinson defined a completion operation, which is a special case of his higher adeles. Given a quasi-coherent sheaf M on X, and a chain ξ, the Beilinson completion of M along ξ is a k-module Mξ, gotten by an n-fold zig-zag of inverse and direct limits.

Amnon Yekutieli (BGU) TLFs and Residues 21 / 34

• 4. The Beilinson Completion
• 4. The Beilinson Completion

Suppose X is a finite type k-scheme. By a chain of points in X we mean a sequence ξ = (x0, . . . , xn) of points such that xi is a specialization of xi−1. The chain ξ is saturated if each xi is an immediate specialization of xi−1. In [Be], Beilinson defined a completion operation, which is a special case of his higher adeles. Given a quasi-coherent sheaf M on X, and a chain ξ, the Beilinson completion of M along ξ is a k-module Mξ, gotten by an n-fold zig-zag of inverse and direct limits.

Amnon Yekutieli (BGU) TLFs and Residues 21 / 34

• 4. The Beilinson Completion

The completion Mξ comes equipped with a topology, making it a ST k-module. Furthermore, the completion OX,ξ of the structure sheaf OX is a ST k-ring; and for any M, the completion Mξ is a ST OX,ξ-module. Example 4.1. If n = 0, so that ξ = (x0), we get OX,ξ = OX,x0, the mx0-adic completion of the local ring OX,x0, with the mx0-adic topology. Given a point x0 ∈ X, its residue field k(x0) can be viewed as a quasi-coherent sheaf, constant on the closed set {x0}. Theorem 4.2. ([Pa1], [Be], [Ye1]) Let X be a finite type k-scheme, and let ξ = (x0, . . . , xn) be a saturated chain in X, such that xn is a closed point. Then the Beilinson completion k(x0)ξ is a finite product of n-dimensional TLFs.

Amnon Yekutieli (BGU) TLFs and Residues 22 / 34

• 4. The Beilinson Completion

The completion Mξ comes equipped with a topology, making it a ST k-module. Furthermore, the completion OX,ξ of the structure sheaf OX is a ST k-ring; and for any M, the completion Mξ is a ST OX,ξ-module. Example 4.1. If n = 0, so that ξ = (x0), we get OX,ξ = OX,x0, the mx0-adic completion of the local ring OX,x0, with the mx0-adic topology. Given a point x0 ∈ X, its residue field k(x0) can be viewed as a quasi-coherent sheaf, constant on the closed set {x0}. Theorem 4.2. ([Pa1], [Be], [Ye1]) Let X be a finite type k-scheme, and let ξ = (x0, . . . , xn) be a saturated chain in X, such that xn is a closed point. Then the Beilinson completion k(x0)ξ is a finite product of n-dimensional TLFs.

Amnon Yekutieli (BGU) TLFs and Residues 22 / 34

• 4. The Beilinson Completion

The completion Mξ comes equipped with a topology, making it a ST k-module. Furthermore, the completion OX,ξ of the structure sheaf OX is a ST k-ring; and for any M, the completion Mξ is a ST OX,ξ-module. Example 4.1. If n = 0, so that ξ = (x0), we get OX,ξ = OX,x0, the mx0-adic completion of the local ring OX,x0, with the mx0-adic topology. Given a point x0 ∈ X, its residue field k(x0) can be viewed as a quasi-coherent sheaf, constant on the closed set {x0}. Theorem 4.2. ([Pa1], [Be], [Ye1]) Let X be a finite type k-scheme, and let ξ = (x0, . . . , xn) be a saturated chain in X, such that xn is a closed point. Then the Beilinson completion k(x0)ξ is a finite product of n-dimensional TLFs.

Amnon Yekutieli (BGU) TLFs and Residues 22 / 34

• 4. The Beilinson Completion

The completion Mξ comes equipped with a topology, making it a ST k-module. Furthermore, the completion OX,ξ of the structure sheaf OX is a ST k-ring; and for any M, the completion Mξ is a ST OX,ξ-module. Example 4.1. If n = 0, so that ξ = (x0), we get OX,ξ = OX,x0, the mx0-adic completion of the local ring OX,x0, with the mx0-adic topology. Given a point x0 ∈ X, its residue field k(x0) can be viewed as a quasi-coherent sheaf, constant on the closed set {x0}. Theorem 4.2. ([Pa1], [Be], [Ye1]) Let X be a finite type k-scheme, and let ξ = (x0, . . . , xn) be a saturated chain in X, such that xn is a closed point. Then the Beilinson completion k(x0)ξ is a finite product of n-dimensional TLFs.

Amnon Yekutieli (BGU) TLFs and Residues 22 / 34

• 4. The Beilinson Completion

The completion Mξ comes equipped with a topology, making it a ST k-module. Furthermore, the completion OX,ξ of the structure sheaf OX is a ST k-ring; and for any M, the completion Mξ is a ST OX,ξ-module. Example 4.1. If n = 0, so that ξ = (x0), we get OX,ξ = OX,x0, the mx0-adic completion of the local ring OX,x0, with the mx0-adic topology. Given a point x0 ∈ X, its residue field k(x0) can be viewed as a quasi-coherent sheaf, constant on the closed set {x0}. Theorem 4.2. ([Pa1], [Be], [Ye1]) Let X be a finite type k-scheme, and let ξ = (x0, . . . , xn) be a saturated chain in X, such that xn is a closed point. Then the Beilinson completion k(x0)ξ is a finite product of n-dimensional TLFs.

Amnon Yekutieli (BGU) TLFs and Residues 22 / 34

• 4. The Beilinson Completion

The completion Mξ comes equipped with a topology, making it a ST k-module. Furthermore, the completion OX,ξ of the structure sheaf OX is a ST k-ring; and for any M, the completion Mξ is a ST OX,ξ-module. Example 4.1. If n = 0, so that ξ = (x0), we get OX,ξ = OX,x0, the mx0-adic completion of the local ring OX,x0, with the mx0-adic topology. Given a point x0 ∈ X, its residue field k(x0) can be viewed as a quasi-coherent sheaf, constant on the closed set {x0}. Theorem 4.2. ([Pa1], [Be], [Ye1]) Let X be a finite type k-scheme, and let ξ = (x0, . . . , xn) be a saturated chain in X, such that xn is a closed point. Then the Beilinson completion k(x0)ξ is a finite product of n-dimensional TLFs.

Amnon Yekutieli (BGU) TLFs and Residues 22 / 34

• 4. The Beilinson Completion

The completion Mξ comes equipped with a topology, making it a ST k-module. Furthermore, the completion OX,ξ of the structure sheaf OX is a ST k-ring; and for any M, the completion Mξ is a ST OX,ξ-module. Example 4.1. If n = 0, so that ξ = (x0), we get OX,ξ = OX,x0, the mx0-adic completion of the local ring OX,x0, with the mx0-adic topology. Given a point x0 ∈ X, its residue field k(x0) can be viewed as a quasi-coherent sheaf, constant on the closed set {x0}. Theorem 4.2. ([Pa1], [Be], [Ye1]) Let X be a finite type k-scheme, and let ξ = (x0, . . . , xn) be a saturated chain in X, such that xn is a closed point. Then the Beilinson completion k(x0)ξ is a finite product of n-dimensional TLFs.

Amnon Yekutieli (BGU) TLFs and Residues 22 / 34

• 5. The Residue Functional
• 5. The Residue Functional

Let K be an n-dimensional TLF over k. Suppose a = (a1, . . . , an) is a system of uniformizers of K. The module of separated differential 1-forms Ω1,sep

K/k is a free K-module of rank

n, with basis

• d(a1), . . . , d(an)
• .

In degree n the module Ωn,sep

K/k is free of rank 1.

Any nonzero form α ∈ Ωn,sep

K/k determines an isomorphism of ST K-modules

K → Ωn,sep

K/k ,

b → b · α. The system of uniformizers a gives a very special nonzero n-form (5.1) dlog(a) := a−1

1

· d(a1) · · · a−1

n

· d(an).

Amnon Yekutieli (BGU) TLFs and Residues 23 / 34

• 5. The Residue Functional
• 5. The Residue Functional

Let K be an n-dimensional TLF over k. Suppose a = (a1, . . . , an) is a system of uniformizers of K. The module of separated differential 1-forms Ω1,sep

K/k is a free K-module of rank

n, with basis

• d(a1), . . . , d(an)
• .

In degree n the module Ωn,sep

K/k is free of rank 1.

Any nonzero form α ∈ Ωn,sep

K/k determines an isomorphism of ST K-modules

K → Ωn,sep

K/k ,

b → b · α. The system of uniformizers a gives a very special nonzero n-form (5.1) dlog(a) := a−1

1

· d(a1) · · · a−1

n

· d(an).

Amnon Yekutieli (BGU) TLFs and Residues 23 / 34

• 5. The Residue Functional
• 5. The Residue Functional

Let K be an n-dimensional TLF over k. Suppose a = (a1, . . . , an) is a system of uniformizers of K. The module of separated differential 1-forms Ω1,sep

K/k is a free K-module of rank

n, with basis

• d(a1), . . . , d(an)
• .

In degree n the module Ωn,sep

K/k is free of rank 1.

Any nonzero form α ∈ Ωn,sep

K/k determines an isomorphism of ST K-modules

K → Ωn,sep

K/k ,

b → b · α. The system of uniformizers a gives a very special nonzero n-form (5.1) dlog(a) := a−1

1

· d(a1) · · · a−1

n

· d(an).

Amnon Yekutieli (BGU) TLFs and Residues 23 / 34

• 5. The Residue Functional
• 5. The Residue Functional

Let K be an n-dimensional TLF over k. Suppose a = (a1, . . . , an) is a system of uniformizers of K. The module of separated differential 1-forms Ω1,sep

K/k is a free K-module of rank

n, with basis

• d(a1), . . . , d(an)
• .

In degree n the module Ωn,sep

K/k is free of rank 1.

Any nonzero form α ∈ Ωn,sep

K/k determines an isomorphism of ST K-modules

K → Ωn,sep

K/k ,

b → b · α. The system of uniformizers a gives a very special nonzero n-form (5.1) dlog(a) := a−1

1

· d(a1) · · · a−1

n

· d(an).

Amnon Yekutieli (BGU) TLFs and Residues 23 / 34

• 5. The Residue Functional
• 5. The Residue Functional

Let K be an n-dimensional TLF over k. Suppose a = (a1, . . . , an) is a system of uniformizers of K. The module of separated differential 1-forms Ω1,sep

K/k is a free K-module of rank

n, with basis

• d(a1), . . . , d(an)
• .

In degree n the module Ωn,sep

K/k is free of rank 1.

Any nonzero form α ∈ Ωn,sep

K/k determines an isomorphism of ST K-modules

K → Ωn,sep

K/k ,

b → b · α. The system of uniformizers a gives a very special nonzero n-form (5.1) dlog(a) := a−1

1

· d(a1) · · · a−1

n

· d(an).

Amnon Yekutieli (BGU) TLFs and Residues 23 / 34

• 5. The Residue Functional
• 5. The Residue Functional

Let K be an n-dimensional TLF over k. Suppose a = (a1, . . . , an) is a system of uniformizers of K. The module of separated differential 1-forms Ω1,sep

K/k is a free K-module of rank

n, with basis

• d(a1), . . . , d(an)
• .

In degree n the module Ωn,sep

K/k is free of rank 1.

Any nonzero form α ∈ Ωn,sep

K/k determines an isomorphism of ST K-modules

K → Ωn,sep

K/k ,

b → b · α. The system of uniformizers a gives a very special nonzero n-form (5.1) dlog(a) := a−1

1

· d(a1) · · · a−1

n

· d(an).

Amnon Yekutieli (BGU) TLFs and Residues 23 / 34

• 5. The Residue Functional
• 5. The Residue Functional

Let K be an n-dimensional TLF over k. Suppose a = (a1, . . . , an) is a system of uniformizers of K. The module of separated differential 1-forms Ω1,sep

K/k is a free K-module of rank

n, with basis

• d(a1), . . . , d(an)
• .

In degree n the module Ωn,sep

K/k is free of rank 1.

Any nonzero form α ∈ Ωn,sep

K/k determines an isomorphism of ST K-modules

K → Ωn,sep

K/k ,

b → b · α. The system of uniformizers a gives a very special nonzero n-form (5.1) dlog(a) := a−1

1

· d(a1) · · · a−1

n

· d(an).

Amnon Yekutieli (BGU) TLFs and Residues 23 / 34

• 5. The Residue Functional

Let K → L be a homomorphism in TLFn k. There is a trace homomorphism TrL/K : Ωn,sep

L/k → Ωn,sep K/k .

It is a nondegenerate K-linear homomorphism. By this I mean that the induced homomorphism Ωn,sep

L/k → HomK(L, Ωn,sep K/k )

is bijective. This trace is functorial: if L → M is another homomorphism in TLFn k, then TrM/K = TrL/K ◦ TrM/L . If k′ is a 0-dimensional TLF, then Ω0,sep

k′/k = k′, and Trk′/k = trk′/k, the usual

trace.

Amnon Yekutieli (BGU) TLFs and Residues 24 / 34

• 5. The Residue Functional

Let K → L be a homomorphism in TLFn k. There is a trace homomorphism TrL/K : Ωn,sep

L/k → Ωn,sep K/k .

It is a nondegenerate K-linear homomorphism. By this I mean that the induced homomorphism Ωn,sep

L/k → HomK(L, Ωn,sep K/k )

is bijective. This trace is functorial: if L → M is another homomorphism in TLFn k, then TrM/K = TrL/K ◦ TrM/L . If k′ is a 0-dimensional TLF, then Ω0,sep

k′/k = k′, and Trk′/k = trk′/k, the usual

trace.

Amnon Yekutieli (BGU) TLFs and Residues 24 / 34

• 5. The Residue Functional

Let K → L be a homomorphism in TLFn k. There is a trace homomorphism TrL/K : Ωn,sep

L/k → Ωn,sep K/k .

It is a nondegenerate K-linear homomorphism. By this I mean that the induced homomorphism Ωn,sep

L/k → HomK(L, Ωn,sep K/k )

is bijective. This trace is functorial: if L → M is another homomorphism in TLFn k, then TrM/K = TrL/K ◦ TrM/L . If k′ is a 0-dimensional TLF, then Ω0,sep

k′/k = k′, and Trk′/k = trk′/k, the usual

trace.

Amnon Yekutieli (BGU) TLFs and Residues 24 / 34

• 5. The Residue Functional

Let K → L be a homomorphism in TLFn k. There is a trace homomorphism TrL/K : Ωn,sep

L/k → Ωn,sep K/k .

It is a nondegenerate K-linear homomorphism. By this I mean that the induced homomorphism Ωn,sep

L/k → HomK(L, Ωn,sep K/k )

is bijective. This trace is functorial: if L → M is another homomorphism in TLFn k, then TrM/K = TrL/K ◦ TrM/L . If k′ is a 0-dimensional TLF, then Ω0,sep

k′/k = k′, and Trk′/k = trk′/k, the usual

trace.

Amnon Yekutieli (BGU) TLFs and Residues 24 / 34

• 5. The Residue Functional

Let K → L be a homomorphism in TLFn k. There is a trace homomorphism TrL/K : Ωn,sep

L/k → Ωn,sep K/k .

It is a nondegenerate K-linear homomorphism. By this I mean that the induced homomorphism Ωn,sep

L/k → HomK(L, Ωn,sep K/k )

is bijective. This trace is functorial: if L → M is another homomorphism in TLFn k, then TrM/K = TrL/K ◦ TrM/L . If k′ is a 0-dimensional TLF, then Ω0,sep

k′/k = k′, and Trk′/k = trk′/k, the usual

trace.

Amnon Yekutieli (BGU) TLFs and Residues 24 / 34

• 5. The Residue Functional

Theorem 5.2. ([Ye1]) Let K be an n-dimensional TLF over k. There is a k-linear homomorphism ResTLF

K/k : Ωn,sep K/k → k,

called the residue functional, with these properties.

• 1. Continuity: the homomorphism ResTLF

K/k is continuous.

• 2. Uniformization: let a = (a1, . . . , an) be a system of uniformizers for K,

and let k′ → K be the canonical lifting of the last residue field k′ := kn(K) into K. Then for any b ∈ k′ and any i1, . . . , in ∈ Z we have ResTLF

K/k

• b · ai1

1 · · · ain n · dlog(a)

• =
• trk′/k(b)

if i1 = · · · = in = 0

• therwise .

Amnon Yekutieli (BGU) TLFs and Residues 25 / 34

• 5. The Residue Functional

Theorem 5.2. ([Ye1]) Let K be an n-dimensional TLF over k. There is a k-linear homomorphism ResTLF

K/k : Ωn,sep K/k → k,

called the residue functional, with these properties.

• 1. Continuity: the homomorphism ResTLF

K/k is continuous.

• 2. Uniformization: let a = (a1, . . . , an) be a system of uniformizers for K,

and let k′ → K be the canonical lifting of the last residue field k′ := kn(K) into K. Then for any b ∈ k′ and any i1, . . . , in ∈ Z we have ResTLF

K/k

• b · ai1

1 · · · ain n · dlog(a)

• =
• trk′/k(b)

if i1 = · · · = in = 0

• therwise .

Amnon Yekutieli (BGU) TLFs and Residues 25 / 34

• 5. The Residue Functional

Theorem 5.2. ([Ye1]) Let K be an n-dimensional TLF over k. There is a k-linear homomorphism ResTLF

K/k : Ωn,sep K/k → k,

called the residue functional, with these properties.

• 1. Continuity: the homomorphism ResTLF

K/k is continuous.

• 2. Uniformization: let a = (a1, . . . , an) be a system of uniformizers for K,

and let k′ → K be the canonical lifting of the last residue field k′ := kn(K) into K. Then for any b ∈ k′ and any i1, . . . , in ∈ Z we have ResTLF

K/k

• b · ai1

1 · · · ain n · dlog(a)

• =
• trk′/k(b)

if i1 = · · · = in = 0

• therwise .

Amnon Yekutieli (BGU) TLFs and Residues 25 / 34

• 5. The Residue Functional

Theorem 5.2. ([Ye1]) Let K be an n-dimensional TLF over k. There is a k-linear homomorphism ResTLF

K/k : Ωn,sep K/k → k,

called the residue functional, with these properties.

• 1. Continuity: the homomorphism ResTLF

K/k is continuous.

• 2. Uniformization: let a = (a1, . . . , an) be a system of uniformizers for K,

and let k′ → K be the canonical lifting of the last residue field k′ := kn(K) into K. Then for any b ∈ k′ and any i1, . . . , in ∈ Z we have ResTLF

K/k

• b · ai1

1 · · · ain n · dlog(a)

• =
• trk′/k(b)

if i1 = · · · = in = 0

• therwise .

Amnon Yekutieli (BGU) TLFs and Residues 25 / 34

• 5. The Residue Functional

Theorem 5.2. ([Ye1]) Let K be an n-dimensional TLF over k. There is a k-linear homomorphism ResTLF

K/k : Ωn,sep K/k → k,

called the residue functional, with these properties.

• 1. Continuity: the homomorphism ResTLF

K/k is continuous.

• 2. Uniformization: let a = (a1, . . . , an) be a system of uniformizers for K,

and let k′ → K be the canonical lifting of the last residue field k′ := kn(K) into K. Then for any b ∈ k′ and any i1, . . . , in ∈ Z we have ResTLF

K/k

• b · ai1

1 · · · ain n · dlog(a)

• =
• trk′/k(b)

if i1 = · · · = in = 0

• therwise .

Amnon Yekutieli (BGU) TLFs and Residues 25 / 34

• 5. The Residue Functional

Theorem 5.2. ([Ye1]) Let K be an n-dimensional TLF over k. There is a k-linear homomorphism ResTLF

K/k : Ωn,sep K/k → k,

called the residue functional, with these properties.

• 1. Continuity: the homomorphism ResTLF

K/k is continuous.

• 2. Uniformization: let a = (a1, . . . , an) be a system of uniformizers for K,

and let k′ → K be the canonical lifting of the last residue field k′ := kn(K) into K. Then for any b ∈ k′ and any i1, . . . , in ∈ Z we have ResTLF

K/k

• b · ai1

1 · · · ain n · dlog(a)

• =
• trk′/k(b)

if i1 = · · · = in = 0

• therwise .

Amnon Yekutieli (BGU) TLFs and Residues 25 / 34

• 5. The Residue Functional

(cont.)

• 3. Functoriality: let K → L be a morphism in the category TLFn k. Then

ResTLF

L/k = ResTLF K/k ◦ TrTLF L/K .

• 4. Nondegeneracy: the residue pairing

−, −res : K × Ωn,sep

K/k → k ,

a, αres := ResTLF

K/k(a · α)

is a topological perfect pairing. Furthermore, the functional ResTLF

K/k is the uniquely determined by properties

(1) and (2).

Amnon Yekutieli (BGU) TLFs and Residues 26 / 34

• 5. The Residue Functional

(cont.)

• 3. Functoriality: let K → L be a morphism in the category TLFn k. Then

ResTLF

L/k = ResTLF K/k ◦ TrTLF L/K .

• 4. Nondegeneracy: the residue pairing

−, −res : K × Ωn,sep

K/k → k ,

a, αres := ResTLF

K/k(a · α)

is a topological perfect pairing. Furthermore, the functional ResTLF

K/k is the uniquely determined by properties

(1) and (2).

Amnon Yekutieli (BGU) TLFs and Residues 26 / 34

• 5. The Residue Functional

(cont.)

• 3. Functoriality: let K → L be a morphism in the category TLFn k. Then

ResTLF

L/k = ResTLF K/k ◦ TrTLF L/K .

• 4. Nondegeneracy: the residue pairing

−, −res : K × Ωn,sep

K/k → k ,

a, αres := ResTLF

K/k(a · α)

is a topological perfect pairing. Furthermore, the functional ResTLF

K/k is the uniquely determined by properties

(1) and (2).

Amnon Yekutieli (BGU) TLFs and Residues 26 / 34

• 5. The Residue Functional

By “topological perfect pairing” we mean that −, −res induces a bijection Homcont

k

(K, k) ∼ = Ωn,sep

K/k .

The residue functional is just a part of the bigger residue functor. For any homomorphism K → L in TLF k, with dim(K) = m and dim(L) = n, there is a homomorphism ResTLF

L/K : Ωsep L/k → Ωsep K/k.

It is a homomorphism of DG Ωsep

K/k-modules of degree m − n.

When m = n it is the trace homomorphism that was already mentioned.

Amnon Yekutieli (BGU) TLFs and Residues 27 / 34

• 5. The Residue Functional

By “topological perfect pairing” we mean that −, −res induces a bijection Homcont

k

(K, k) ∼ = Ωn,sep

K/k .

The residue functional is just a part of the bigger residue functor. For any homomorphism K → L in TLF k, with dim(K) = m and dim(L) = n, there is a homomorphism ResTLF

L/K : Ωsep L/k → Ωsep K/k.

It is a homomorphism of DG Ωsep

K/k-modules of degree m − n.

When m = n it is the trace homomorphism that was already mentioned.

Amnon Yekutieli (BGU) TLFs and Residues 27 / 34

• 5. The Residue Functional

By “topological perfect pairing” we mean that −, −res induces a bijection Homcont

k

(K, k) ∼ = Ωn,sep

K/k .

The residue functional is just a part of the bigger residue functor. For any homomorphism K → L in TLF k, with dim(K) = m and dim(L) = n, there is a homomorphism ResTLF

L/K : Ωsep L/k → Ωsep K/k.

It is a homomorphism of DG Ωsep

K/k-modules of degree m − n.

When m = n it is the trace homomorphism that was already mentioned.

Amnon Yekutieli (BGU) TLFs and Residues 27 / 34

• 5. The Residue Functional

By “topological perfect pairing” we mean that −, −res induces a bijection Homcont

k

(K, k) ∼ = Ωn,sep

K/k .

The residue functional is just a part of the bigger residue functor. For any homomorphism K → L in TLF k, with dim(K) = m and dim(L) = n, there is a homomorphism ResTLF

L/K : Ωsep L/k → Ωsep K/k.

It is a homomorphism of DG Ωsep

K/k-modules of degree m − n.

When m = n it is the trace homomorphism that was already mentioned.

Amnon Yekutieli (BGU) TLFs and Residues 27 / 34

• 5. The Residue Functional

By “topological perfect pairing” we mean that −, −res induces a bijection Homcont

k

(K, k) ∼ = Ωn,sep

K/k .

The residue functional is just a part of the bigger residue functor. For any homomorphism K → L in TLF k, with dim(K) = m and dim(L) = n, there is a homomorphism ResTLF

L/K : Ωsep L/k → Ωsep K/k.

It is a homomorphism of DG Ωsep

K/k-modules of degree m − n.

When m = n it is the trace homomorphism that was already mentioned.

Amnon Yekutieli (BGU) TLFs and Residues 27 / 34

• 5. The Residue Functional

The first attempt at a residue theory for higher local fields was by Parshin and his school. See the papers [Pa1], [Pa2], [Be], [Lo] and [Pa3]. However the concept of TLF was absent from their work, which resulted in ill-defined concepts. They had erroneously asserted that there is a residue functional defined on the category LFn k. This is false when char(k) = 0 and n ≥ 2. A counterexample to that appeared in [Ye1] (see [Ye7] for an elaborated version). Of course when char(k) = p > 0 the residue functional is well-defined on LFn k, by virtue of Theorem 3.4.

Amnon Yekutieli (BGU) TLFs and Residues 28 / 34

• 5. The Residue Functional

The first attempt at a residue theory for higher local fields was by Parshin and his school. See the papers [Pa1], [Pa2], [Be], [Lo] and [Pa3]. However the concept of TLF was absent from their work, which resulted in ill-defined concepts. They had erroneously asserted that there is a residue functional defined on the category LFn k. This is false when char(k) = 0 and n ≥ 2. A counterexample to that appeared in [Ye1] (see [Ye7] for an elaborated version). Of course when char(k) = p > 0 the residue functional is well-defined on LFn k, by virtue of Theorem 3.4.

Amnon Yekutieli (BGU) TLFs and Residues 28 / 34

• 5. The Residue Functional

The first attempt at a residue theory for higher local fields was by Parshin and his school. See the papers [Pa1], [Pa2], [Be], [Lo] and [Pa3]. However the concept of TLF was absent from their work, which resulted in ill-defined concepts. They had erroneously asserted that there is a residue functional defined on the category LFn k. This is false when char(k) = 0 and n ≥ 2. A counterexample to that appeared in [Ye1] (see [Ye7] for an elaborated version). Of course when char(k) = p > 0 the residue functional is well-defined on LFn k, by virtue of Theorem 3.4.

Amnon Yekutieli (BGU) TLFs and Residues 28 / 34

• 5. The Residue Functional

The first attempt at a residue theory for higher local fields was by Parshin and his school. See the papers [Pa1], [Pa2], [Be], [Lo] and [Pa3]. However the concept of TLF was absent from their work, which resulted in ill-defined concepts. They had erroneously asserted that there is a residue functional defined on the category LFn k. This is false when char(k) = 0 and n ≥ 2. A counterexample to that appeared in [Ye1] (see [Ye7] for an elaborated version). Of course when char(k) = p > 0 the residue functional is well-defined on LFn k, by virtue of Theorem 3.4.

Amnon Yekutieli (BGU) TLFs and Residues 28 / 34

• 5. The Residue Functional

The first attempt at a residue theory for higher local fields was by Parshin and his school. See the papers [Pa1], [Pa2], [Be], [Lo] and [Pa3]. However the concept of TLF was absent from their work, which resulted in ill-defined concepts. They had erroneously asserted that there is a residue functional defined on the category LFn k. This is false when char(k) = 0 and n ≥ 2. A counterexample to that appeared in [Ye1] (see [Ye7] for an elaborated version). Of course when char(k) = p > 0 the residue functional is well-defined on LFn k, by virtue of Theorem 3.4.

Amnon Yekutieli (BGU) TLFs and Residues 28 / 34

• 5. The Residue Functional

Let me say a few words on the relation with Milnor K groups. For a TLF L of dimension n, let KM

n (L) be its n-th Milnor group. There is a

group homomorphism dlog : KM

n (L) → Ωn,sep L/k .

See (5.1) Suppose K → L is a homomorphism of TLFs, with dim(K) = m. There should be a canonical group homomorphism (I did not check the details) ResM

L/K : KM n (L) tame symbol

− − − − − − − → KM

m

• kn−m(L)
• norm

− − − → KM

m (K).

In this situation the following diagram should be commutative: Kn(L)

dlog

• ResM

L/K

• Ωn,sep

L/k ResTLF

L/K

• Km(K)

dlog

Ωm,sep

K/k

Amnon Yekutieli (BGU) TLFs and Residues 29 / 34

• 5. The Residue Functional

Let me say a few words on the relation with Milnor K groups. For a TLF L of dimension n, let KM

n (L) be its n-th Milnor group. There is a

group homomorphism dlog : KM

n (L) → Ωn,sep L/k .

See (5.1) Suppose K → L is a homomorphism of TLFs, with dim(K) = m. There should be a canonical group homomorphism (I did not check the details) ResM

L/K : KM n (L) tame symbol

− − − − − − − → KM

m

• kn−m(L)
• norm

− − − → KM

m (K).

In this situation the following diagram should be commutative: Kn(L)

dlog

• ResM

L/K

• Ωn,sep

L/k ResTLF

L/K

• Km(K)

dlog

Ωm,sep

K/k

Amnon Yekutieli (BGU) TLFs and Residues 29 / 34

• 5. The Residue Functional

Let me say a few words on the relation with Milnor K groups. For a TLF L of dimension n, let KM

n (L) be its n-th Milnor group. There is a

group homomorphism dlog : KM

n (L) → Ωn,sep L/k .

See (5.1) Suppose K → L is a homomorphism of TLFs, with dim(K) = m. There should be a canonical group homomorphism (I did not check the details) ResM

L/K : KM n (L) tame symbol

− − − − − − − → KM

m

• kn−m(L)
• norm

− − − → KM

m (K).

In this situation the following diagram should be commutative: Kn(L)

dlog

• ResM

L/K

• Ωn,sep

L/k ResTLF

L/K

• Km(K)

dlog

Ωm,sep

K/k

Amnon Yekutieli (BGU) TLFs and Residues 29 / 34

• 5. The Residue Functional

Let me say a few words on the relation with Milnor K groups. For a TLF L of dimension n, let KM

n (L) be its n-th Milnor group. There is a

group homomorphism dlog : KM

n (L) → Ωn,sep L/k .

See (5.1) Suppose K → L is a homomorphism of TLFs, with dim(K) = m. There should be a canonical group homomorphism (I did not check the details) ResM

L/K : KM n (L) tame symbol

− − − − − − − → KM

m

• kn−m(L)
• norm

− − − → KM

m (K).

In this situation the following diagram should be commutative: Kn(L)

dlog

• ResM

L/K

• Ωn,sep

L/k ResTLF

L/K

• Km(K)

dlog

Ωm,sep

K/k

Amnon Yekutieli (BGU) TLFs and Residues 29 / 34

• 5. The Residue Functional

Let me say a few words on the relation with Milnor K groups. For a TLF L of dimension n, let KM

n (L) be its n-th Milnor group. There is a

group homomorphism dlog : KM

n (L) → Ωn,sep L/k .

See (5.1) Suppose K → L is a homomorphism of TLFs, with dim(K) = m. There should be a canonical group homomorphism (I did not check the details) ResM

L/K : KM n (L) tame symbol

− − − − − − − → KM

m

• kn−m(L)
• norm

− − − → KM

m (K).

In this situation the following diagram should be commutative: Kn(L)

dlog

• ResM

L/K

• Ωn,sep

L/k ResTLF

L/K

• Km(K)

dlog

Ωm,sep

K/k

Amnon Yekutieli (BGU) TLFs and Residues 29 / 34

• 6. Some Applications of the Residue Functional
• 6. Some Applications of the Residue Functional

Let X be a finite type k-scheme. On X there is the Grothendieck residue complex KX. It is a dualizing complex that has very special properties. If π : X → Spec X is the structural map, then – in terms of [RD] – KX is the Cousin complex representing the twisted inverse image π!(k). In the more modern terminology of [Ye4], [Ye5] and [Ye6], KX is called the rigid residue complex of X. It is not hard to construct the residue complex KX explicitly when X is a

• curve. This is classical.

The main result of [Ye1] is an explicit construction of KX for a high dimensional reduced scheme X, using TLF residues.

Amnon Yekutieli (BGU) TLFs and Residues 30 / 34

• 6. Some Applications of the Residue Functional
• 6. Some Applications of the Residue Functional

Let X be a finite type k-scheme. On X there is the Grothendieck residue complex KX. It is a dualizing complex that has very special properties. If π : X → Spec X is the structural map, then – in terms of [RD] – KX is the Cousin complex representing the twisted inverse image π!(k). In the more modern terminology of [Ye4], [Ye5] and [Ye6], KX is called the rigid residue complex of X. It is not hard to construct the residue complex KX explicitly when X is a

• curve. This is classical.

The main result of [Ye1] is an explicit construction of KX for a high dimensional reduced scheme X, using TLF residues.

Amnon Yekutieli (BGU) TLFs and Residues 30 / 34

• 6. Some Applications of the Residue Functional
• 6. Some Applications of the Residue Functional

Let X be a finite type k-scheme. On X there is the Grothendieck residue complex KX. It is a dualizing complex that has very special properties. If π : X → Spec X is the structural map, then – in terms of [RD] – KX is the Cousin complex representing the twisted inverse image π!(k). In the more modern terminology of [Ye4], [Ye5] and [Ye6], KX is called the rigid residue complex of X. It is not hard to construct the residue complex KX explicitly when X is a

• curve. This is classical.

The main result of [Ye1] is an explicit construction of KX for a high dimensional reduced scheme X, using TLF residues.

Amnon Yekutieli (BGU) TLFs and Residues 30 / 34

• 6. Some Applications of the Residue Functional
• 6. Some Applications of the Residue Functional

Let X be a finite type k-scheme. On X there is the Grothendieck residue complex KX. It is a dualizing complex that has very special properties. If π : X → Spec X is the structural map, then – in terms of [RD] – KX is the Cousin complex representing the twisted inverse image π!(k). In the more modern terminology of [Ye4], [Ye5] and [Ye6], KX is called the rigid residue complex of X. It is not hard to construct the residue complex KX explicitly when X is a

• curve. This is classical.

The main result of [Ye1] is an explicit construction of KX for a high dimensional reduced scheme X, using TLF residues.

Amnon Yekutieli (BGU) TLFs and Residues 30 / 34

• 6. Some Applications of the Residue Functional
• 6. Some Applications of the Residue Functional

Let X be a finite type k-scheme. On X there is the Grothendieck residue complex KX. It is a dualizing complex that has very special properties. If π : X → Spec X is the structural map, then – in terms of [RD] – KX is the Cousin complex representing the twisted inverse image π!(k). In the more modern terminology of [Ye4], [Ye5] and [Ye6], KX is called the rigid residue complex of X. It is not hard to construct the residue complex KX explicitly when X is a

• curve. This is classical.

The main result of [Ye1] is an explicit construction of KX for a high dimensional reduced scheme X, using TLF residues.

Amnon Yekutieli (BGU) TLFs and Residues 30 / 34

• 6. Some Applications of the Residue Functional
• 6. Some Applications of the Residue Functional

Let X be a finite type k-scheme. On X there is the Grothendieck residue complex KX. It is a dualizing complex that has very special properties. If π : X → Spec X is the structural map, then – in terms of [RD] – KX is the Cousin complex representing the twisted inverse image π!(k). In the more modern terminology of [Ye4], [Ye5] and [Ye6], KX is called the rigid residue complex of X. It is not hard to construct the residue complex KX explicitly when X is a

• curve. This is classical.

The main result of [Ye1] is an explicit construction of KX for a high dimensional reduced scheme X, using TLF residues.

Amnon Yekutieli (BGU) TLFs and Residues 30 / 34

• 6. Some Applications of the Residue Functional
• 6. Some Applications of the Residue Functional

Let X be a finite type k-scheme. On X there is the Grothendieck residue complex KX. It is a dualizing complex that has very special properties. If π : X → Spec X is the structural map, then – in terms of [RD] – KX is the Cousin complex representing the twisted inverse image π!(k). In the more modern terminology of [Ye4], [Ye5] and [Ye6], KX is called the rigid residue complex of X. It is not hard to construct the residue complex KX explicitly when X is a

• curve. This is classical.

The main result of [Ye1] is an explicit construction of KX for a high dimensional reduced scheme X, using TLF residues.

Amnon Yekutieli (BGU) TLFs and Residues 30 / 34

• 6. Some Applications of the Residue Functional

There is a very close relation between residues and differential operators. Let A be a commutative k-ring. We denote by DA/k the ring of differential

• perators of A. This is a noncommutative ring, and by definition A is a left

DA-module. If A is ST k-ring, we denote by Dcont

A/k the ring of continuous differential

• perators of A.

For a scheme X we have the sheaf of differential operators DX/k, and OX is a sheaf of left DX/k-modules. By the general theory of D-modules we know that when X is smooth of dimension n, the sheaf Ωn

X/k is a right DX/k-module.

Consider a saturated chain of points ξ = (x0, . . . , xm) in X, such that xm is a closed point. Let K := k(x0), the residue field, and let Kξ be its Beilinson completion. We know (by Theorem 4.2) that Kξ is a finite product of m-dimensional TLFs.

Amnon Yekutieli (BGU) TLFs and Residues 31 / 34

• 6. Some Applications of the Residue Functional

There is a very close relation between residues and differential operators. Let A be a commutative k-ring. We denote by DA/k the ring of differential

• perators of A. This is a noncommutative ring, and by definition A is a left

DA-module. If A is ST k-ring, we denote by Dcont

A/k the ring of continuous differential

• perators of A.

For a scheme X we have the sheaf of differential operators DX/k, and OX is a sheaf of left DX/k-modules. By the general theory of D-modules we know that when X is smooth of dimension n, the sheaf Ωn

X/k is a right DX/k-module.

Consider a saturated chain of points ξ = (x0, . . . , xm) in X, such that xm is a closed point. Let K := k(x0), the residue field, and let Kξ be its Beilinson completion. We know (by Theorem 4.2) that Kξ is a finite product of m-dimensional TLFs.

Amnon Yekutieli (BGU) TLFs and Residues 31 / 34

• 6. Some Applications of the Residue Functional

There is a very close relation between residues and differential operators. Let A be a commutative k-ring. We denote by DA/k the ring of differential

• perators of A. This is a noncommutative ring, and by definition A is a left

DA-module. If A is ST k-ring, we denote by Dcont

A/k the ring of continuous differential

• perators of A.

For a scheme X we have the sheaf of differential operators DX/k, and OX is a sheaf of left DX/k-modules. By the general theory of D-modules we know that when X is smooth of dimension n, the sheaf Ωn

X/k is a right DX/k-module.

Consider a saturated chain of points ξ = (x0, . . . , xm) in X, such that xm is a closed point. Let K := k(x0), the residue field, and let Kξ be its Beilinson completion. We know (by Theorem 4.2) that Kξ is a finite product of m-dimensional TLFs.

Amnon Yekutieli (BGU) TLFs and Residues 31 / 34

• 6. Some Applications of the Residue Functional

There is a very close relation between residues and differential operators. Let A be a commutative k-ring. We denote by DA/k the ring of differential

• perators of A. This is a noncommutative ring, and by definition A is a left

DA-module. If A is ST k-ring, we denote by Dcont

A/k the ring of continuous differential

• perators of A.

For a scheme X we have the sheaf of differential operators DX/k, and OX is a sheaf of left DX/k-modules. By the general theory of D-modules we know that when X is smooth of dimension n, the sheaf Ωn

X/k is a right DX/k-module.

Consider a saturated chain of points ξ = (x0, . . . , xm) in X, such that xm is a closed point. Let K := k(x0), the residue field, and let Kξ be its Beilinson completion. We know (by Theorem 4.2) that Kξ is a finite product of m-dimensional TLFs.

Amnon Yekutieli (BGU) TLFs and Residues 31 / 34

• 6. Some Applications of the Residue Functional

There is a very close relation between residues and differential operators. Let A be a commutative k-ring. We denote by DA/k the ring of differential

• perators of A. This is a noncommutative ring, and by definition A is a left

DA-module. If A is ST k-ring, we denote by Dcont

A/k the ring of continuous differential

• perators of A.

For a scheme X we have the sheaf of differential operators DX/k, and OX is a sheaf of left DX/k-modules. By the general theory of D-modules we know that when X is smooth of dimension n, the sheaf Ωn

X/k is a right DX/k-module.

Consider a saturated chain of points ξ = (x0, . . . , xm) in X, such that xm is a closed point. Let K := k(x0), the residue field, and let Kξ be its Beilinson completion. We know (by Theorem 4.2) that Kξ is a finite product of m-dimensional TLFs.

Amnon Yekutieli (BGU) TLFs and Residues 31 / 34

• 6. Some Applications of the Residue Functional

There is a very close relation between residues and differential operators. Let A be a commutative k-ring. We denote by DA/k the ring of differential

• perators of A. This is a noncommutative ring, and by definition A is a left

DA-module. If A is ST k-ring, we denote by Dcont

A/k the ring of continuous differential

• perators of A.

For a scheme X we have the sheaf of differential operators DX/k, and OX is a sheaf of left DX/k-modules. By the general theory of D-modules we know that when X is smooth of dimension n, the sheaf Ωn

X/k is a right DX/k-module.

Consider a saturated chain of points ξ = (x0, . . . , xm) in X, such that xm is a closed point. Let K := k(x0), the residue field, and let Kξ be its Beilinson completion. We know (by Theorem 4.2) that Kξ is a finite product of m-dimensional TLFs.

Amnon Yekutieli (BGU) TLFs and Residues 31 / 34

• 6. Some Applications of the Residue Functional

There is a very close relation between residues and differential operators. Let A be a commutative k-ring. We denote by DA/k the ring of differential

• perators of A. This is a noncommutative ring, and by definition A is a left

DA-module. If A is ST k-ring, we denote by Dcont

A/k the ring of continuous differential

• perators of A.

For a scheme X we have the sheaf of differential operators DX/k, and OX is a sheaf of left DX/k-modules. By the general theory of D-modules we know that when X is smooth of dimension n, the sheaf Ωn

X/k is a right DX/k-module.

Consider a saturated chain of points ξ = (x0, . . . , xm) in X, such that xm is a closed point. Let K := k(x0), the residue field, and let Kξ be its Beilinson completion. We know (by Theorem 4.2) that Kξ is a finite product of m-dimensional TLFs.

Amnon Yekutieli (BGU) TLFs and Residues 31 / 34

• 6. Some Applications of the Residue Functional

Since K is essentially smooth over k, of relative dimension m, the general formulas give Ωm

K/k a right DK/k-module structure.

For α ∈ Ωm

K/k and φ ∈ DK/k, this right action is denoted by

α ∗ φ ∈ Ωm

K/k.

The TLF Kξ is topologically smooth over k, of relative dimension m. So the general formulas give Ωm,sep

Kξ/k a right Dcont Kξ/k-module structure.

The canonical homomorphism K → Kξ is topologically étale in STRingc k. This implies that there is a canonical ring homomorphism DK/k → Dcont

Kξ/k,

and a canonical nondegenerate right DK/k-module homomorphism Ωm

K/k → Ωm,sep Kξ/k.

Amnon Yekutieli (BGU) TLFs and Residues 32 / 34

• 6. Some Applications of the Residue Functional

Since K is essentially smooth over k, of relative dimension m, the general formulas give Ωm

K/k a right DK/k-module structure.

For α ∈ Ωm

K/k and φ ∈ DK/k, this right action is denoted by

α ∗ φ ∈ Ωm

K/k.

The TLF Kξ is topologically smooth over k, of relative dimension m. So the general formulas give Ωm,sep

Kξ/k a right Dcont Kξ/k-module structure.

The canonical homomorphism K → Kξ is topologically étale in STRingc k. This implies that there is a canonical ring homomorphism DK/k → Dcont

Kξ/k,

and a canonical nondegenerate right DK/k-module homomorphism Ωm

K/k → Ωm,sep Kξ/k.

Amnon Yekutieli (BGU) TLFs and Residues 32 / 34

• 6. Some Applications of the Residue Functional

Since K is essentially smooth over k, of relative dimension m, the general formulas give Ωm

K/k a right DK/k-module structure.

For α ∈ Ωm

K/k and φ ∈ DK/k, this right action is denoted by

α ∗ φ ∈ Ωm

K/k.

The TLF Kξ is topologically smooth over k, of relative dimension m. So the general formulas give Ωm,sep

Kξ/k a right Dcont Kξ/k-module structure.

The canonical homomorphism K → Kξ is topologically étale in STRingc k. This implies that there is a canonical ring homomorphism DK/k → Dcont

Kξ/k,

and a canonical nondegenerate right DK/k-module homomorphism Ωm

K/k → Ωm,sep Kξ/k.

Amnon Yekutieli (BGU) TLFs and Residues 32 / 34

• 6. Some Applications of the Residue Functional

Since K is essentially smooth over k, of relative dimension m, the general formulas give Ωm

K/k a right DK/k-module structure.

For α ∈ Ωm

K/k and φ ∈ DK/k, this right action is denoted by

α ∗ φ ∈ Ωm

K/k.

The TLF Kξ is topologically smooth over k, of relative dimension m. So the general formulas give Ωm,sep

Kξ/k a right Dcont Kξ/k-module structure.

The canonical homomorphism K → Kξ is topologically étale in STRingc k. This implies that there is a canonical ring homomorphism DK/k → Dcont

Kξ/k,

and a canonical nondegenerate right DK/k-module homomorphism Ωm

K/k → Ωm,sep Kξ/k.

Amnon Yekutieli (BGU) TLFs and Residues 32 / 34

• 6. Some Applications of the Residue Functional

Since K is essentially smooth over k, of relative dimension m, the general formulas give Ωm

K/k a right DK/k-module structure.

For α ∈ Ωm

K/k and φ ∈ DK/k, this right action is denoted by

α ∗ φ ∈ Ωm

K/k.

The TLF Kξ is topologically smooth over k, of relative dimension m. So the general formulas give Ωm,sep

Kξ/k a right Dcont Kξ/k-module structure.

The canonical homomorphism K → Kξ is topologically étale in STRingc k. This implies that there is a canonical ring homomorphism DK/k → Dcont

Kξ/k,

and a canonical nondegenerate right DK/k-module homomorphism Ωm

K/k → Ωm,sep Kξ/k.

Amnon Yekutieli (BGU) TLFs and Residues 32 / 34

• 6. Some Applications of the Residue Functional

Now recall the residue pairing −, −res : Kξ × Ωn,sep

Kξ/k → k

from Theorem 5.2(4). Since this is a topological perfect pairing, any φ ∈ Dcont

Kξ/k, viewed as a

continuous k-linear homomorphism φ : Kξ → Kξ, has an adjoint operator (in the sense of functional analysis) φ∗ : Ωm,sep

Kξ/k → Ωm,sep Kξ/k.

Theorem 6.1. ([Ye2]) In this situation, for any α ∈ Ωm,sep

Kξ/k and φ ∈ Dcont Kξ/k we

have φ∗(α) = α ∗ φ ∈ Ωm,sep

Kξ/k.

In other words, the algebraic right D-module action on Ωm,sep

Kξ/k, which is

already defined on Ωm

K/k, coincides with the analytic adjoint action.

Amnon Yekutieli (BGU) TLFs and Residues 33 / 34

• 6. Some Applications of the Residue Functional

Now recall the residue pairing −, −res : Kξ × Ωn,sep

Kξ/k → k

from Theorem 5.2(4). Since this is a topological perfect pairing, any φ ∈ Dcont

Kξ/k, viewed as a

continuous k-linear homomorphism φ : Kξ → Kξ, has an adjoint operator (in the sense of functional analysis) φ∗ : Ωm,sep

Kξ/k → Ωm,sep Kξ/k.

Theorem 6.1. ([Ye2]) In this situation, for any α ∈ Ωm,sep

Kξ/k and φ ∈ Dcont Kξ/k we

have φ∗(α) = α ∗ φ ∈ Ωm,sep

Kξ/k.

In other words, the algebraic right D-module action on Ωm,sep

Kξ/k, which is

already defined on Ωm

K/k, coincides with the analytic adjoint action.

Amnon Yekutieli (BGU) TLFs and Residues 33 / 34

• 6. Some Applications of the Residue Functional

Now recall the residue pairing −, −res : Kξ × Ωn,sep

Kξ/k → k

from Theorem 5.2(4). Since this is a topological perfect pairing, any φ ∈ Dcont

Kξ/k, viewed as a

continuous k-linear homomorphism φ : Kξ → Kξ, has an adjoint operator (in the sense of functional analysis) φ∗ : Ωm,sep

Kξ/k → Ωm,sep Kξ/k.

Theorem 6.1. ([Ye2]) In this situation, for any α ∈ Ωm,sep

Kξ/k and φ ∈ Dcont Kξ/k we

have φ∗(α) = α ∗ φ ∈ Ωm,sep

Kξ/k.

In other words, the algebraic right D-module action on Ωm,sep

Kξ/k, which is

already defined on Ωm

K/k, coincides with the analytic adjoint action.

Amnon Yekutieli (BGU) TLFs and Residues 33 / 34

• 6. Some Applications of the Residue Functional

Now recall the residue pairing −, −res : Kξ × Ωn,sep

Kξ/k → k

from Theorem 5.2(4). Since this is a topological perfect pairing, any φ ∈ Dcont

Kξ/k, viewed as a

continuous k-linear homomorphism φ : Kξ → Kξ, has an adjoint operator (in the sense of functional analysis) φ∗ : Ωm,sep

Kξ/k → Ωm,sep Kξ/k.

Theorem 6.1. ([Ye2]) In this situation, for any α ∈ Ωm,sep

Kξ/k and φ ∈ Dcont Kξ/k we

have φ∗(α) = α ∗ φ ∈ Ωm,sep

Kξ/k.

In other words, the algebraic right D-module action on Ωm,sep

Kξ/k, which is

already defined on Ωm

K/k, coincides with the analytic adjoint action.

Amnon Yekutieli (BGU) TLFs and Residues 33 / 34

• 6. Some Applications of the Residue Functional

Theorem 6.1 implies: Theorem 6.2. ([Ye3]) The Grothendieck residue complex KX is a complex of right DX-modules. This tells us for instance that when X is an n-dimensional integral scheme, the dualizing sheaf ωX := H−n(KX) is a right DX-module. If X is smooth, so that ωX = Ωn

X/k, we recover the previous right DX-module

structure on this sheaf. ∼ END ∼

Amnon Yekutieli (BGU) TLFs and Residues 34 / 34

• 6. Some Applications of the Residue Functional

Theorem 6.1 implies: Theorem 6.2. ([Ye3]) The Grothendieck residue complex KX is a complex of right DX-modules. This tells us for instance that when X is an n-dimensional integral scheme, the dualizing sheaf ωX := H−n(KX) is a right DX-module. If X is smooth, so that ωX = Ωn

X/k, we recover the previous right DX-module

structure on this sheaf. ∼ END ∼

Amnon Yekutieli (BGU) TLFs and Residues 34 / 34

• 6. Some Applications of the Residue Functional

Theorem 6.1 implies: Theorem 6.2. ([Ye3]) The Grothendieck residue complex KX is a complex of right DX-modules. This tells us for instance that when X is an n-dimensional integral scheme, the dualizing sheaf ωX := H−n(KX) is a right DX-module. If X is smooth, so that ωX = Ωn

X/k, we recover the previous right DX-module

structure on this sheaf. ∼ END ∼

Amnon Yekutieli (BGU) TLFs and Residues 34 / 34

• 6. Some Applications of the Residue Functional

Theorem 6.1 implies: Theorem 6.2. ([Ye3]) The Grothendieck residue complex KX is a complex of right DX-modules. This tells us for instance that when X is an n-dimensional integral scheme, the dualizing sheaf ωX := H−n(KX) is a right DX-module. If X is smooth, so that ωX = Ωn

X/k, we recover the previous right DX-module

structure on this sheaf. ∼ END ∼

Amnon Yekutieli (BGU) TLFs and Residues 34 / 34

• 6. Some Applications of the Residue Functional

Theorem 6.1 implies: Theorem 6.2. ([Ye3]) The Grothendieck residue complex KX is a complex of right DX-modules. This tells us for instance that when X is an n-dimensional integral scheme, the dualizing sheaf ωX := H−n(KX) is a right DX-module. If X is smooth, so that ωX = Ωn

X/k, we recover the previous right DX-module

structure on this sheaf. ∼ END ∼

Amnon Yekutieli (BGU) TLFs and Residues 34 / 34

• 6. Some Applications of the Residue Functional

References [Be] A.A. Beilinson, Residues and adeles, Funkt. Anal. Pril. 14(1) (1980), 44-45; English trans. in Func. Anal. Appl. 14(1) (1980), 34-35. [Br]

• O. Braunling, On the local residue symbol in the style of Tate and

Beilinson, arXiv:1403.8142v2. [EGA-IV] A. Grothendieck and J. Dieudonné, “Éléments de Géometrie Algébrique.” Chapitre 0IV, Publ. Math. IHES 20 (1964); Chapitre IV,

• Publ. Math. IHES 32 (1967).

[Hu]

• A. Huber, On the Parshin-Beilinson Adeles for Schemes, Abh. Math.
• Sem. Univ. Hamburg 61 (1991), 249-273.

[Ka]

• K. Kato, A generalization of local class field theory by using K-groups

I, J. Fac. Sci. Univ. Tokyo Sec. IA 26 No. 2 (1979), 303-376. [Lo] V.G. Lomadze, On residues in algebraic geometry, Izv. Akad. Nauk SSSR Ser. Mat. 45 (1981), 1258-1287; English trans. in Math. USSR

• Izv. 19 (1982) no. 3, 495-520.

Amnon Yekutieli (BGU) TLFs and Residues 34 / 34

• 6. Some Applications of the Residue Functional

[Mo]

• M. Morrow, An introduction to higher dimensional local fields and

adeles, online at http://www.math.uni-bonn.de/people/morrow (see arXiv:1204.0586v2 for an older version). [Pa1]

• A. N. Parshin, On the Arithmetic of Two-Dimensional Schemes. I.

Distributions and Residues, Izv. Akad. Nauk SSSR Ser. Mat. Tom 40 (1976), No. 4. English translation Math. USSR Izvestija Vol. 10 (1976), No. 4. [Pa2]

• A. N. Parshin, Abelian coverings of arithmetical schemes (in russian),
• Dokl. Akad. Nauk. SSSR 243 (1978), 855-858.

[Pa3]

• A. N. Parshin, Chern classes, adeles and L-functions, J. Reine Angew.
• Math. 341 (1983), 174-192.

[RD]

• R. Hartshorne, “Residues and Duality,” Lecture Notes in Math. 20,

Springer-Verlag, Berlin, 1966. [Se] J.P. Serre, “Algebraic Groups and Class Fields”, Springer-Verlag, New York, 1988.

Amnon Yekutieli (BGU) TLFs and Residues 34 / 34

• 6. Some Applications of the Residue Functional

[Ta]

• J. Tate, Residues of differentials on curves, Ann. Sci. de l’E.N.S. serie

4, 1 (1968), 149-159. [Ye1]

• A. Yekutieli, “An Explicit Construction of the Grothendieck Residue

Complex”, Astérisque 208 (1992). [Ye2]

• A. Yekutieli, Traces and differential operators over Beilinson

completion algebras, Compositio Mathematica 99, no. 1 (1995), 59-97. [Ye3]

• A. Yekutieli, Residues and differential operators on schemes, Duke
• Math. J. 95 (1998), 305-341.

[Ye4]

• A. Yekutieli, Rigid Dualizing Complexes via Differential Graded

Algebras (Survey), in “Triangulated Categories”, LMS Lecture Note Series 375, 2010. [Ye5]

• A. Yekutieli, Rigidity, residues and duality for schemes, in

preparation.

Amnon Yekutieli (BGU) TLFs and Residues 34 / 34

• 6. Some Applications of the Residue Functional

[Ye6]

• A. Yekutieli, Rigidity, residues and duality for DM stacks, in

preparation. [Ye7]

• A. Yekutieli, Local Beilinson-Tate Operators, to appear in Algebra

and Number Theory, eprint arxiv arXiv:1406.6502.

Amnon Yekutieli (BGU) TLFs and Residues 34 / 34