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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)

  1. 1. Background on Semi-Topological Rings The exterior algebra of Ω 1 A / k over A is the differential graded (DG) k -ring � Ω i Ω A / k = A / k . i ≥ 0 The multiplication is super-commutative. The differential d extends the derivation (1.1). The DG ring Ω A / k is also called the de Rham complex of A . If A = k [ t ] , the polynomial ring in one variable, then Ω 1 A / k is a Example 1.2. 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 Ω 1 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 Ω 1 A / k is not even finitely generated! Amnon Yekutieli (BGU) TLFs and Residues 4 / 34

  2. 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 Ω 1 A / k . The closure of 0 is an A -submodule { 0 } . The associated separated (i.e. Hausdorff) module is Ω 1 , sep A / k := Ω 1 A / k / { 0 } . If we are lucky, the A -module Ω 1 , sep A / k has the expected properties. Continuing with Example 1.2, we take the t -adic topology on Example 1.3. A = k [[ t ]] . Then Ω 1 , sep A / k if free with basis d ( t ) . Amnon Yekutieli (BGU) TLFs and Residues 5 / 34

  3. 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 Ω 1 A / k . The closure of 0 is an A -submodule { 0 } . The associated separated (i.e. Hausdorff) module is Ω 1 , sep A / k := Ω 1 A / k / { 0 } . If we are lucky, the A -module Ω 1 , sep A / k has the expected properties. Continuing with Example 1.2, we take the t -adic topology on Example 1.3. A = k [[ t ]] . Then Ω 1 , sep A / k if free with basis d ( t ) . Amnon Yekutieli (BGU) TLFs and Residues 5 / 34

  4. 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 Ω 1 A / k . The closure of 0 is an A -submodule { 0 } . The associated separated (i.e. Hausdorff) module is Ω 1 , sep A / k := Ω 1 A / k / { 0 } . If we are lucky, the A -module Ω 1 , sep A / k has the expected properties. Continuing with Example 1.2, we take the t -adic topology on Example 1.3. A = k [[ t ]] . Then Ω 1 , sep A / k if free with basis d ( t ) . Amnon Yekutieli (BGU) TLFs and Residues 5 / 34

  5. 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 Ω 1 A / k . The closure of 0 is an A -submodule { 0 } . The associated separated (i.e. Hausdorff) module is Ω 1 , sep A / k := Ω 1 A / k / { 0 } . If we are lucky, the A -module Ω 1 , sep A / k has the expected properties. Continuing with Example 1.2, we take the t -adic topology on Example 1.3. A = k [[ t ]] . Then Ω 1 , sep A / k if free with basis d ( t ) . Amnon Yekutieli (BGU) TLFs and Residues 5 / 34

  6. 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 Ω 1 A / k . The closure of 0 is an A -submodule { 0 } . The associated separated (i.e. Hausdorff) module is Ω 1 , sep A / k := Ω 1 A / k / { 0 } . If we are lucky, the A -module Ω 1 , sep A / k has the expected properties. Continuing with Example 1.2, we take the t -adic topology on Example 1.3. A = k [[ t ]] . Then Ω 1 , sep A / k if free with basis d ( t ) . Amnon Yekutieli (BGU) TLFs and Residues 5 / 34

  7. 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 and β ( m , − ) : N → P 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

  8. 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 and β ( m , − ) : N → P 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

  9. 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 and β ( m , − ) : N → P 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

  10. 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”. Suppose A is a ST k -ring. Example 1.6. 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

  11. 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”. Suppose A is a ST k -ring. Example 1.6. 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

  12. 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”. Suppose A is a ST k -ring. Example 1.6. 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

  13. 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”. Suppose A is a ST k -ring. Example 1.6. 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

  14. 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”. Suppose A is a ST k -ring. Example 1.6. 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

  15. 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”. Suppose A is a ST k -ring. Example 1.6. 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

  16. 1. Background on Semi-Topological Rings Let us denote by STRing c 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 Ω A / 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 � Ω sep Ω i , sep A / k = A / k . i ≥ 0 If f : A → B is a homomorphism in STRing c k , then there is an induced homomorphism of ST DG rings f : Ω sep A / k → Ω sep B / k . Amnon Yekutieli (BGU) TLFs and Residues 8 / 34

  17. 1. Background on Semi-Topological Rings Let us denote by STRing c 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 Ω A / 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 � Ω sep Ω i , sep A / k = A / k . i ≥ 0 If f : A → B is a homomorphism in STRing c k , then there is an induced homomorphism of ST DG rings f : Ω sep A / k → Ω sep B / k . Amnon Yekutieli (BGU) TLFs and Residues 8 / 34

  18. 1. Background on Semi-Topological Rings Let us denote by STRing c 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 Ω A / 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 � Ω sep Ω i , sep A / k = A / k . i ≥ 0 If f : A → B is a homomorphism in STRing c k , then there is an induced homomorphism of ST DG rings f : Ω sep A / k → Ω sep B / k . Amnon Yekutieli (BGU) TLFs and Residues 8 / 34

  19. 1. Background on Semi-Topological Rings Let us denote by STRing c 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 Ω A / 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 � Ω sep Ω i , sep A / k = A / k . i ≥ 0 If f : A → B is a homomorphism in STRing c k , then there is an induced homomorphism of ST DG rings f : Ω sep A / k → Ω sep B / k . Amnon Yekutieli (BGU) TLFs and Residues 8 / 34

  20. 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 � � O 1 ( K ) , . . . , O n ( K ) of complete DVRs, such that: ◮ The fraction field of O 1 ( K ) is K . ◮ The residue field k i ( K ) of O i ( K ) is the fraction field of O i + 1 ( K ) . ◮ All these rings and homomorphism are in the category of k -rings, and k → k n ( K ) is finite. Amnon Yekutieli (BGU) TLFs and Residues 9 / 34

  21. 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 � � O 1 ( K ) , . . . , O n ( K ) of complete DVRs, such that: ◮ The fraction field of O 1 ( K ) is K . ◮ The residue field k i ( K ) of O i ( K ) is the fraction field of O i + 1 ( K ) . ◮ All these rings and homomorphism are in the category of k -rings, and k → k n ( K ) is finite. Amnon Yekutieli (BGU) TLFs and Residues 9 / 34

  22. 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 � � O 1 ( K ) , . . . , O n ( K ) of complete DVRs, such that: ◮ The fraction field of O 1 ( K ) is K . ◮ The residue field k i ( K ) of O i ( K ) is the fraction field of O i + 1 ( K ) . ◮ All these rings and homomorphism are in the category of k -rings, and k → k n ( K ) is finite. Amnon Yekutieli (BGU) TLFs and Residues 9 / 34

  23. 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 � � O 1 ( K ) , . . . , O n ( K ) of complete DVRs, such that: ◮ The fraction field of O 1 ( K ) is K . ◮ The residue field k i ( K ) of O i ( K ) is the fraction field of O i + 1 ( K ) . ◮ All these rings and homomorphism are in the category of k -rings, and k → k n ( K ) is finite. Amnon Yekutieli (BGU) TLFs and Residues 9 / 34

  24. 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 � � O 1 ( K ) , . . . , O n ( K ) of complete DVRs, such that: ◮ The fraction field of O 1 ( K ) is K . ◮ The residue field k i ( K ) of O i ( K ) is the fraction field of O i + 1 ( K ) . ◮ All these rings and homomorphism are in the category of k -rings, and k → k n ( K ) is finite. Amnon Yekutieli (BGU) TLFs and Residues 9 / 34

  25. 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 � � O 1 ( K ) , . . . , O n ( K ) of complete DVRs, such that: ◮ The fraction field of O 1 ( K ) is K . ◮ The residue field k i ( K ) of O i ( K ) is the fraction field of O i + 1 ( K ) . ◮ All these rings and homomorphism are in the category of k -rings, and k → k n ( K ) is finite. Amnon Yekutieli (BGU) TLFs and Residues 9 / 34

  26. 2. High Dimensional Local Fields Here is the picture for n = 2. Amnon Yekutieli (BGU) TLFs and Residues 10 / 34

  27. 2. High Dimensional Local Fields Here is the picture for n = 2. K Amnon Yekutieli (BGU) TLFs and Residues 10 / 34

  28. 2. High Dimensional Local Fields Here is the picture for n = 2. � K O 1 ( K ) � Amnon Yekutieli (BGU) TLFs and Residues 10 / 34

  29. � �� 2. High Dimensional Local Fields Here is the picture for n = 2. O 1 ( K ) � K k 1 ( K ) Amnon Yekutieli (BGU) TLFs and Residues 10 / 34

  30. � �� 2. High Dimensional Local Fields Here is the picture for n = 2. O 1 ( K ) � K � k 1 ( K ) O 2 ( K ) � Amnon Yekutieli (BGU) TLFs and Residues 10 / 34

  31. � �� � �� 2. High Dimensional Local Fields Here is the picture for n = 2. O 1 ( K ) � K O 2 ( K ) � k 1 ( K ) k 2 ( K ) Amnon Yekutieli (BGU) TLFs and Residues 10 / 34

  32. � � �� � �� � � 2. High Dimensional Local Fields Here is the picture for n = 2. O 1 ( K ) � K O 2 ( K ) � k 1 ( K ) k k 2 ( K ) Amnon Yekutieli (BGU) TLFs and Residues 10 / 34

  33. � � �� � �� � � � 2. High Dimensional Local Fields Here is the picture for n = 2. O 1 ( K ) � K O 2 ( K ) � k 1 ( K ) k finite k 2 ( K ) Amnon Yekutieli (BGU) TLFs and Residues 10 / 34

  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 � Q p and F p (( 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 ( O 1 ( K )) ⊂ O 1 ( L ) . ◮ The induced k -ring homomorphism f : O 1 ( K ) → O 1 ( L ) is a local homomorphism. ◮ The induced k -ring homomorphism ¯ f : k 1 ( K ) → k 1 ( L ) is a morphism of ( n − 1 ) -dimensional local fields over k . Amnon Yekutieli (BGU) TLFs and Residues 11 / 34

  35. 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 � Q p and F p (( 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 ( O 1 ( K )) ⊂ O 1 ( L ) . ◮ The induced k -ring homomorphism f : O 1 ( K ) → O 1 ( L ) is a local homomorphism. ◮ The induced k -ring homomorphism ¯ f : k 1 ( K ) → k 1 ( L ) is a morphism of ( n − 1 ) -dimensional local fields over k . Amnon Yekutieli (BGU) TLFs and Residues 11 / 34

  36. 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 � Q p and F p (( 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 ( O 1 ( K )) ⊂ O 1 ( L ) . ◮ The induced k -ring homomorphism f : O 1 ( K ) → O 1 ( L ) is a local homomorphism. ◮ The induced k -ring homomorphism ¯ f : k 1 ( K ) → k 1 ( L ) is a morphism of ( n − 1 ) -dimensional local fields over k . Amnon Yekutieli (BGU) TLFs and Residues 11 / 34

  37. 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 � Q p and F p (( 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 ( O 1 ( K )) ⊂ O 1 ( L ) . ◮ The induced k -ring homomorphism f : O 1 ( K ) → O 1 ( L ) is a local homomorphism. ◮ The induced k -ring homomorphism ¯ f : k 1 ( K ) → k 1 ( L ) is a morphism of ( n − 1 ) -dimensional local fields over k . Amnon Yekutieli (BGU) TLFs and Residues 11 / 34

  38. 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 � Q p and F p (( 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 ( O 1 ( K )) ⊂ O 1 ( L ) . ◮ The induced k -ring homomorphism f : O 1 ( K ) → O 1 ( L ) is a local homomorphism. ◮ The induced k -ring homomorphism ¯ f : k 1 ( K ) → k 1 ( L ) is a morphism of ( n − 1 ) -dimensional local fields over k . Amnon Yekutieli (BGU) TLFs and Residues 11 / 34

  39. 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 � Q p and F p (( 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 ( O 1 ( K )) ⊂ O 1 ( L ) . ◮ The induced k -ring homomorphism f : O 1 ( K ) → O 1 ( L ) is a local homomorphism. ◮ The induced k -ring homomorphism ¯ f : k 1 ( K ) → k 1 ( L ) is a morphism of ( n − 1 ) -dimensional local fields over k . Amnon Yekutieli (BGU) TLFs and Residues 11 / 34

  40. 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 � Q p and F p (( 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 ( O 1 ( K )) ⊂ O 1 ( L ) . ◮ The induced k -ring homomorphism f : O 1 ( K ) → O 1 ( L ) is a local homomorphism. ◮ The induced k -ring homomorphism ¯ f : k 1 ( K ) → k 1 ( L ) is a morphism of ( n − 1 ) -dimensional local fields over k . Amnon Yekutieli (BGU) TLFs and Residues 11 / 34

  41. 2. High Dimensional Local Fields Let us denote by LF n k the category of n -dimensional local fields over k . It is not hard to show that any homomorphism K → L in LF n 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 LF n k is a full subcategory. See [Ye1] for details. If k is a field, then the field of Laurent series K := k (( t 2 )) is a Example 2.4. 1-dimensional local field. The field of iterated Laurent series L := K (( t 1 )) = k (( t 2 ))(( t 1 )) is a 2-dimensional local field. The inclusions k → K → L are morphisms in LF k . Amnon Yekutieli (BGU) TLFs and Residues 12 / 34

  42. 2. High Dimensional Local Fields Let us denote by LF n k the category of n -dimensional local fields over k . It is not hard to show that any homomorphism K → L in LF n 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 LF n k is a full subcategory. See [Ye1] for details. If k is a field, then the field of Laurent series K := k (( t 2 )) is a Example 2.4. 1-dimensional local field. The field of iterated Laurent series L := K (( t 1 )) = k (( t 2 ))(( t 1 )) is a 2-dimensional local field. The inclusions k → K → L are morphisms in LF k . Amnon Yekutieli (BGU) TLFs and Residues 12 / 34

  43. 2. High Dimensional Local Fields Let us denote by LF n k the category of n -dimensional local fields over k . It is not hard to show that any homomorphism K → L in LF n 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 LF n k is a full subcategory. See [Ye1] for details. If k is a field, then the field of Laurent series K := k (( t 2 )) is a Example 2.4. 1-dimensional local field. The field of iterated Laurent series L := K (( t 1 )) = k (( t 2 ))(( t 1 )) is a 2-dimensional local field. The inclusions k → K → L are morphisms in LF k . Amnon Yekutieli (BGU) TLFs and Residues 12 / 34

  44. 2. High Dimensional Local Fields Let us denote by LF n k the category of n -dimensional local fields over k . It is not hard to show that any homomorphism K → L in LF n 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 LF n k is a full subcategory. See [Ye1] for details. If k is a field, then the field of Laurent series K := k (( t 2 )) is a Example 2.4. 1-dimensional local field. The field of iterated Laurent series L := K (( t 1 )) = k (( t 2 ))(( t 1 )) is a 2-dimensional local field. The inclusions k → K → L are morphisms in LF k . Amnon Yekutieli (BGU) TLFs and Residues 12 / 34

  45. 2. High Dimensional Local Fields Let us denote by LF n k the category of n -dimensional local fields over k . It is not hard to show that any homomorphism K → L in LF n 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 LF n k is a full subcategory. See [Ye1] for details. If k is a field, then the field of Laurent series K := k (( t 2 )) is a Example 2.4. 1-dimensional local field. The field of iterated Laurent series L := K (( t 1 )) = k (( t 2 ))(( t 1 )) is a 2-dimensional local field. The inclusions k → K → L are morphisms in LF k . Amnon Yekutieli (BGU) TLFs and Residues 12 / 34

  46. 2. High Dimensional Local Fields Let us denote by LF n k the category of n -dimensional local fields over k . It is not hard to show that any homomorphism K → L in LF n 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 LF n k is a full subcategory. See [Ye1] for details. If k is a field, then the field of Laurent series K := k (( t 2 )) is a Example 2.4. 1-dimensional local field. The field of iterated Laurent series L := K (( t 1 )) = k (( t 2 ))(( t 1 )) is a 2-dimensional local field. The inclusions k → K → L are morphisms in LF k . Amnon Yekutieli (BGU) TLFs and Residues 12 / 34

  47. 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 ′ := k n ( 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

  48. 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 ′ := k n ( 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

  49. 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 ′ := k n ( 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

  50. 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 ′ := k n ( 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

  51. 3. Topological Local Fields Let k ′ be a finite field extension of k , and let t = ( t 1 , . . . , t n ) be a sequence of variables. We denote by k ′ (( t )) = k ′ (( t 1 , . . . , t n )) := k ′ (( t n )) · · · (( t 1 )) the iterated field of Laurent series. The field k ′ (( t )) has a canonical structure of n -dimensional local field over k . The DVRs are � � k ′ (( t )) := k ′ (( t i + 1 , . . . , t n ))[[ t i ]] , O i and the residue fields are � � k ′ (( t )) := k ′ (( t i + 1 , . . . , t n )) . k i Amnon Yekutieli (BGU) TLFs and Residues 14 / 34

  52. 3. Topological Local Fields Let k ′ be a finite field extension of k , and let t = ( t 1 , . . . , t n ) be a sequence of variables. We denote by k ′ (( t )) = k ′ (( t 1 , . . . , t n )) := k ′ (( t n )) · · · (( t 1 )) the iterated field of Laurent series. The field k ′ (( t )) has a canonical structure of n -dimensional local field over k . The DVRs are � � k ′ (( t )) := k ′ (( t i + 1 , . . . , t n ))[[ t i ]] , O i and the residue fields are � � k ′ (( t )) := k ′ (( t i + 1 , . . . , t n )) . k i Amnon Yekutieli (BGU) TLFs and Residues 14 / 34

  53. 3. Topological Local Fields Let k ′ be a finite field extension of k , and let t = ( t 1 , . . . , t n ) be a sequence of variables. We denote by k ′ (( t )) = k ′ (( t 1 , . . . , t n )) := k ′ (( t n )) · · · (( t 1 )) the iterated field of Laurent series. The field k ′ (( t )) has a canonical structure of n -dimensional local field over k . The DVRs are � � k ′ (( t )) := k ′ (( t i + 1 , . . . , t n ))[[ t i ]] , O i and the residue fields are � � k ′ (( t )) := k ′ (( t i + 1 , . . . , t n )) . k i Amnon Yekutieli (BGU) TLFs and Residues 14 / 34

  54. 3. Topological Local Fields Let k ′ be a finite field extension of k , and let t = ( t 1 , . . . , t n ) be a sequence of variables. We denote by k ′ (( t )) = k ′ (( t 1 , . . . , t n )) := k ′ (( t n )) · · · (( t 1 )) the iterated field of Laurent series. The field k ′ (( t )) has a canonical structure of n -dimensional local field over k . The DVRs are � � k ′ (( t )) := k ′ (( t i + 1 , . . . , t n ))[[ t i ]] , O i and the residue fields are � � k ′ (( t )) := k ′ (( t i + 1 , . . . , t n )) . k i Amnon Yekutieli (BGU) TLFs and Residues 14 / 34

  55. 3. Topological Local Fields Let k ′ be a finite field extension of k , and let t = ( t 1 , . . . , t n ) be a sequence of variables. We denote by k ′ (( t )) = k ′ (( t 1 , . . . , t n )) := k ′ (( t n )) · · · (( t 1 )) the iterated field of Laurent series. The field k ′ (( t )) has a canonical structure of n -dimensional local field over k . The DVRs are � � k ′ (( t )) := k ′ (( t i + 1 , . . . , t n ))[[ t i ]] , O i and the residue fields are � � k ′ (( t )) := k ′ (( t i + 1 , . . . , t n )) . k i Amnon Yekutieli (BGU) TLFs and Residues 14 / 34

  56. 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

  57. 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

  58. 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

  59. 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 {O i ( 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 ′ := k n ( K ) , such that: (i) f is an isomorphism in LF n k (i.e. it respects the valuations). (ii) f is an isomorphism in STRing c 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

  60. 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 {O i ( 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 ′ := k n ( K ) , such that: (i) f is an isomorphism in LF n k (i.e. it respects the valuations). (ii) f is an isomorphism in STRing c 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

  61. 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 {O i ( 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 ′ := k n ( K ) , such that: (i) f is an isomorphism in LF n k (i.e. it respects the valuations). (ii) f is an isomorphism in STRing c 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

  62. 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 {O i ( 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 ′ := k n ( K ) , such that: (i) f is an isomorphism in LF n k (i.e. it respects the valuations). (ii) f is an isomorphism in STRing c 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

  63. 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 {O i ( 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 ′ := k n ( K ) , such that: (i) f is an isomorphism in LF n k (i.e. it respects the valuations). (ii) f is an isomorphism in STRing c 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

  64. 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 {O i ( 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 ′ := k n ( K ) , such that: (i) f is an isomorphism in LF n k (i.e. it respects the valuations). (ii) f is an isomorphism in STRing c 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

  65. 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 {O i ( 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 ′ := k n ( K ) , such that: (i) f is an isomorphism in LF n k (i.e. it respects the valuations). (ii) f is an isomorphism in STRing c 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

  66. 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 {O i ( 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 ′ := k n ( K ) , such that: (i) f is an isomorphism in LF n k (i.e. it respects the valuations). (ii) f is an isomorphism in STRing c 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

  67. 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 {O i ( 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 ′ := k n ( K ) , such that: (i) f is an isomorphism in LF n k (i.e. it respects the valuations). (ii) f is an isomorphism in STRing c 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

  68. 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 K sep = 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

  69. 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 K sep = 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

  70. 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 K sep = 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

  71. 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 K sep = 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

  72. 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 K sep = 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

  73. 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 K sep = 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

  74. 3. Topological Local Fields Let K be an n -dimensional TLF. A system of uniformizers in K is a sequence ( a 1 , . . . , a n ) of elements of O 1 ( K ) , such that a 1 generates the maximal ideal of O 1 ( K ) , and if n ≥ 2, the sequence (¯ a n ) , which is the image of a 2 , . . . , ¯ ( a 2 , . . . , a n ) under the canonical surjection O 1 ( K ) ։ k 1 ( K ) , is a system of uniformizers in k 1 ( 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 ( a 1 , . . . , a n ) be a system of uniformizers in K , let k ′ := k n ( K ) , and let σ : k ′ → K be the canonical lifting. Then σ extends uniquely to an isomorphism of TLFs f : k ′ (( t 1 , . . . , t n )) → K such that f ( t i ) = a i . Amnon Yekutieli (BGU) TLFs and Residues 18 / 34

  75. 3. Topological Local Fields Let K be an n -dimensional TLF. A system of uniformizers in K is a sequence ( a 1 , . . . , a n ) of elements of O 1 ( K ) , such that a 1 generates the maximal ideal of O 1 ( K ) , and if n ≥ 2, the sequence (¯ a n ) , which is the image of a 2 , . . . , ¯ ( a 2 , . . . , a n ) under the canonical surjection O 1 ( K ) ։ k 1 ( K ) , is a system of uniformizers in k 1 ( 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 ( a 1 , . . . , a n ) be a system of uniformizers in K , let k ′ := k n ( K ) , and let σ : k ′ → K be the canonical lifting. Then σ extends uniquely to an isomorphism of TLFs f : k ′ (( t 1 , . . . , t n )) → K such that f ( t i ) = a i . Amnon Yekutieli (BGU) TLFs and Residues 18 / 34

  76. 3. Topological Local Fields Let K be an n -dimensional TLF. A system of uniformizers in K is a sequence ( a 1 , . . . , a n ) of elements of O 1 ( K ) , such that a 1 generates the maximal ideal of O 1 ( K ) , and if n ≥ 2, the sequence (¯ a n ) , which is the image of a 2 , . . . , ¯ ( a 2 , . . . , a n ) under the canonical surjection O 1 ( K ) ։ k 1 ( K ) , is a system of uniformizers in k 1 ( 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 ( a 1 , . . . , a n ) be a system of uniformizers in K , let k ′ := k n ( K ) , and let σ : k ′ → K be the canonical lifting. Then σ extends uniquely to an isomorphism of TLFs f : k ′ (( t 1 , . . . , t n )) → K such that f ( t i ) = a i . Amnon Yekutieli (BGU) TLFs and Residues 18 / 34

  77. 3. Topological Local Fields Let K be an n -dimensional TLF. A system of uniformizers in K is a sequence ( a 1 , . . . , a n ) of elements of O 1 ( K ) , such that a 1 generates the maximal ideal of O 1 ( K ) , and if n ≥ 2, the sequence (¯ a n ) , which is the image of a 2 , . . . , ¯ ( a 2 , . . . , a n ) under the canonical surjection O 1 ( K ) ։ k 1 ( K ) , is a system of uniformizers in k 1 ( 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 ( a 1 , . . . , a n ) be a system of uniformizers in K , let k ′ := k n ( K ) , and let σ : k ′ → K be the canonical lifting. Then σ extends uniquely to an isomorphism of TLFs f : k ′ (( t 1 , . . . , t n )) → K such that f ( t i ) = a i . Amnon Yekutieli (BGU) TLFs and Residues 18 / 34

  78. 3. Topological Local Fields Let K be an n -dimensional TLF. A system of uniformizers in K is a sequence ( a 1 , . . . , a n ) of elements of O 1 ( K ) , such that a 1 generates the maximal ideal of O 1 ( K ) , and if n ≥ 2, the sequence (¯ a n ) , which is the image of a 2 , . . . , ¯ ( a 2 , . . . , a n ) under the canonical surjection O 1 ( K ) ։ k 1 ( K ) , is a system of uniformizers in k 1 ( 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 ( a 1 , . . . , a n ) be a system of uniformizers in K , let k ′ := k n ( K ) , and let σ : k ′ → K be the canonical lifting. Then σ extends uniquely to an isomorphism of TLFs f : k ′ (( t 1 , . . . , t n )) → K such that f ( t i ) = a i . Amnon Yekutieli (BGU) TLFs and Residues 18 / 34

  79. 3. Topological Local Fields Let K be an n -dimensional TLF. A system of uniformizers in K is a sequence ( a 1 , . . . , a n ) of elements of O 1 ( K ) , such that a 1 generates the maximal ideal of O 1 ( K ) , and if n ≥ 2, the sequence (¯ a n ) , which is the image of a 2 , . . . , ¯ ( a 2 , . . . , a n ) under the canonical surjection O 1 ( K ) ։ k 1 ( K ) , is a system of uniformizers in k 1 ( 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 ( a 1 , . . . , a n ) be a system of uniformizers in K , let k ′ := k n ( K ) , and let σ : k ′ → K be the canonical lifting. Then σ extends uniquely to an isomorphism of TLFs f : k ′ (( t 1 , . . . , t n )) → K such that f ( t i ) = a i . Amnon Yekutieli (BGU) TLFs and Residues 18 / 34

  80. 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 TLF n 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. TLF n k is a full subcategory of TLF k . Consider the functor TLF n k → LF n 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

  81. 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 TLF n 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. TLF n k is a full subcategory of TLF k . Consider the functor TLF n k → LF n 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

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