Cohomologically Complete Complexes Amnon Yekutieli Department of - PowerPoint PPT Presentation

1. The Completion and Torsion Functors The a -torsion submodule of M is the submodule Γ a M := { m ∈ M | a i m = 0 for some i ≥ 1 } . This is an additive functor Γ a : Mod A → Mod A. The inclusion σ M : Γ a M → M is a natural transformation. The module M is said to be a -torsion if σ M is an isomorphism. The torsion functor Γ a is idempotent, in the following sense: for any module M , the module Γ a M is a -torsion. Amnon Yekutieli (BGU) Cohomologically Complete Complexes 5 / 36

1. The Completion and Torsion Functors The a -torsion submodule of M is the submodule Γ a M := { m ∈ M | a i m = 0 for some i ≥ 1 } . This is an additive functor Γ a : Mod A → Mod A. The inclusion σ M : Γ a M → M is a natural transformation. The module M is said to be a -torsion if σ M is an isomorphism. The torsion functor Γ a is idempotent, in the following sense: for any module M , the module Γ a M is a -torsion. Amnon Yekutieli (BGU) Cohomologically Complete Complexes 5 / 36

1. The Completion and Torsion Functors The a -torsion submodule of M is the submodule Γ a M := { m ∈ M | a i m = 0 for some i ≥ 1 } . This is an additive functor Γ a : Mod A → Mod A. The inclusion σ M : Γ a M → M is a natural transformation. The module M is said to be a -torsion if σ M is an isomorphism. The torsion functor Γ a is idempotent, in the following sense: for any module M , the module Γ a M is a -torsion. Amnon Yekutieli (BGU) Cohomologically Complete Complexes 5 / 36

1. The Completion and Torsion Functors This is not true, in general, for the completion functor Λ a . Indeed, there is an example of a module M whose a -adic completion Λ a M is not a -adically complete. See [Ye, Example 1.8]. The ring A is this example is A = K [ t 1 , t 2 , . . . ] , the polynomial ring in countably many variables over a field K , and a is the ideal generated by the variables. Fortunately we have: Theorem 1.1. If the ideal a is finitely generated, then for any A -module M , the a -adic completion Λ a M is a -adically complete. See [St], or [Ye, Corollary 3.6]. Amnon Yekutieli (BGU) Cohomologically Complete Complexes 6 / 36

1. The Completion and Torsion Functors This is not true, in general, for the completion functor Λ a . Indeed, there is an example of a module M whose a -adic completion Λ a M is not a -adically complete. See [Ye, Example 1.8]. The ring A is this example is A = K [ t 1 , t 2 , . . . ] , the polynomial ring in countably many variables over a field K , and a is the ideal generated by the variables. Fortunately we have: Theorem 1.1. If the ideal a is finitely generated, then for any A -module M , the a -adic completion Λ a M is a -adically complete. See [St], or [Ye, Corollary 3.6]. Amnon Yekutieli (BGU) Cohomologically Complete Complexes 6 / 36

1. The Completion and Torsion Functors This is not true, in general, for the completion functor Λ a . Indeed, there is an example of a module M whose a -adic completion Λ a M is not a -adically complete. See [Ye, Example 1.8]. The ring A is this example is A = K [ t 1 , t 2 , . . . ] , the polynomial ring in countably many variables over a field K , and a is the ideal generated by the variables. Fortunately we have: Theorem 1.1. If the ideal a is finitely generated, then for any A -module M , the a -adic completion Λ a M is a -adically complete. See [St], or [Ye, Corollary 3.6]. Amnon Yekutieli (BGU) Cohomologically Complete Complexes 6 / 36

1. The Completion and Torsion Functors This is not true, in general, for the completion functor Λ a . Indeed, there is an example of a module M whose a -adic completion Λ a M is not a -adically complete. See [Ye, Example 1.8]. The ring A is this example is A = K [ t 1 , t 2 , . . . ] , the polynomial ring in countably many variables over a field K , and a is the ideal generated by the variables. Fortunately we have: Theorem 1.1. If the ideal a is finitely generated, then for any A -module M , the a -adic completion Λ a M is a -adically complete. See [St], or [Ye, Corollary 3.6]. Amnon Yekutieli (BGU) Cohomologically Complete Complexes 6 / 36

1. The Completion and Torsion Functors This is not true, in general, for the completion functor Λ a . Indeed, there is an example of a module M whose a -adic completion Λ a M is not a -adically complete. See [Ye, Example 1.8]. The ring A is this example is A = K [ t 1 , t 2 , . . . ] , the polynomial ring in countably many variables over a field K , and a is the ideal generated by the variables. Fortunately we have: Theorem 1.1. If the ideal a is finitely generated, then for any A -module M , the a -adic completion Λ a M is a -adically complete. See [St], or [Ye, Corollary 3.6]. Amnon Yekutieli (BGU) Cohomologically Complete Complexes 6 / 36

2. Adically Projective Modules 2. Adically Projective Modules From now on we shall assume that A is noetherian, so in particular Theorem 1.1 holds: the a -adic completion functor Λ a is idempotent. Definition 2.1. An A module P is called a -adically free if P ∼ = Λ a Q for some free A -module Q . Note that an a -adically free module P is usually not free. Theorem 2.2. Let P be an a -adically free A -module. Then P is flat. See [St], or [Ye, Theorem 3.4]. Amnon Yekutieli (BGU) Cohomologically Complete Complexes 7 / 36

2. Adically Projective Modules 2. Adically Projective Modules From now on we shall assume that A is noetherian, so in particular Theorem 1.1 holds: the a -adic completion functor Λ a is idempotent. Definition 2.1. An A module P is called a -adically free if P ∼ = Λ a Q for some free A -module Q . Note that an a -adically free module P is usually not free. Theorem 2.2. Let P be an a -adically free A -module. Then P is flat. See [St], or [Ye, Theorem 3.4]. Amnon Yekutieli (BGU) Cohomologically Complete Complexes 7 / 36

2. Adically Projective Modules 2. Adically Projective Modules From now on we shall assume that A is noetherian, so in particular Theorem 1.1 holds: the a -adic completion functor Λ a is idempotent. Definition 2.1. An A module P is called a -adically free if P ∼ = Λ a Q for some free A -module Q . Note that an a -adically free module P is usually not free. Theorem 2.2. Let P be an a -adically free A -module. Then P is flat. See [St], or [Ye, Theorem 3.4]. Amnon Yekutieli (BGU) Cohomologically Complete Complexes 7 / 36

2. Adically Projective Modules 2. Adically Projective Modules From now on we shall assume that A is noetherian, so in particular Theorem 1.1 holds: the a -adic completion functor Λ a is idempotent. Definition 2.1. An A module P is called a -adically free if P ∼ = Λ a Q for some free A -module Q . Note that an a -adically free module P is usually not free. Theorem 2.2. Let P be an a -adically free A -module. Then P is flat. See [St], or [Ye, Theorem 3.4]. Amnon Yekutieli (BGU) Cohomologically Complete Complexes 7 / 36

2. Adically Projective Modules 2. Adically Projective Modules From now on we shall assume that A is noetherian, so in particular Theorem 1.1 holds: the a -adic completion functor Λ a is idempotent. Definition 2.1. An A module P is called a -adically free if P ∼ = Λ a Q for some free A -module Q . Note that an a -adically free module P is usually not free. Theorem 2.2. Let P be an a -adically free A -module. Then P is flat. See [St], or [Ye, Theorem 3.4]. Amnon Yekutieli (BGU) Cohomologically Complete Complexes 7 / 36

2. Adically Projective Modules 2. Adically Projective Modules From now on we shall assume that A is noetherian, so in particular Theorem 1.1 holds: the a -adic completion functor Λ a is idempotent. Definition 2.1. An A module P is called a -adically free if P ∼ = Λ a Q for some free A -module Q . Note that an a -adically free module P is usually not free. Theorem 2.2. Let P be an a -adically free A -module. Then P is flat. See [St], or [Ye, Theorem 3.4]. Amnon Yekutieli (BGU) Cohomologically Complete Complexes 7 / 36

2. Adically Projective Modules Let me say a few words on the structure of a -adically free A -modules. Take a set Z , and consider the set F fin ( Z, A ) of functions φ : Z → A with finite support. It is a free A -module with basis the set { δ z } z ∈ Z of delta functions. Of course any free A -module looks like this. The completion Λ a F fin ( Z, A ) is canonically isomorphic to the A -module F dec ( Z, � A ) of a -adically decaying functions φ : Z → � A , where � A := Λ a A . Theorem 2.2 is proved in [Ye] by showing that the functor M �→ F dec ( Z, M ) is exact on the category Mod f � A of finitely generated � A -modules, and that for any such M the canonical homomorphism A F dec ( Z, � M ⊗ � A ) → F dec ( Z, M ) is bijective. Amnon Yekutieli (BGU) Cohomologically Complete Complexes 8 / 36

2. Adically Projective Modules Let me say a few words on the structure of a -adically free A -modules. Take a set Z , and consider the set F fin ( Z, A ) of functions φ : Z → A with finite support. It is a free A -module with basis the set { δ z } z ∈ Z of delta functions. Of course any free A -module looks like this. The completion Λ a F fin ( Z, A ) is canonically isomorphic to the A -module F dec ( Z, � A ) of a -adically decaying functions φ : Z → � A , where � A := Λ a A . Theorem 2.2 is proved in [Ye] by showing that the functor M �→ F dec ( Z, M ) is exact on the category Mod f � A of finitely generated � A -modules, and that for any such M the canonical homomorphism A F dec ( Z, � M ⊗ � A ) → F dec ( Z, M ) is bijective. Amnon Yekutieli (BGU) Cohomologically Complete Complexes 8 / 36

2. Adically Projective Modules Let me say a few words on the structure of a -adically free A -modules. Take a set Z , and consider the set F fin ( Z, A ) of functions φ : Z → A with finite support. It is a free A -module with basis the set { δ z } z ∈ Z of delta functions. Of course any free A -module looks like this. The completion Λ a F fin ( Z, A ) is canonically isomorphic to the A -module F dec ( Z, � A ) of a -adically decaying functions φ : Z → � A , where � A := Λ a A . Theorem 2.2 is proved in [Ye] by showing that the functor M �→ F dec ( Z, M ) is exact on the category Mod f � A of finitely generated � A -modules, and that for any such M the canonical homomorphism A F dec ( Z, � M ⊗ � A ) → F dec ( Z, M ) is bijective. Amnon Yekutieli (BGU) Cohomologically Complete Complexes 8 / 36

2. Adically Projective Modules Let me say a few words on the structure of a -adically free A -modules. Take a set Z , and consider the set F fin ( Z, A ) of functions φ : Z → A with finite support. It is a free A -module with basis the set { δ z } z ∈ Z of delta functions. Of course any free A -module looks like this. The completion Λ a F fin ( Z, A ) is canonically isomorphic to the A -module F dec ( Z, � A ) of a -adically decaying functions φ : Z → � A , where � A := Λ a A . Theorem 2.2 is proved in [Ye] by showing that the functor M �→ F dec ( Z, M ) is exact on the category Mod f � A of finitely generated � A -modules, and that for any such M the canonical homomorphism A F dec ( Z, � M ⊗ � A ) → F dec ( Z, M ) is bijective. Amnon Yekutieli (BGU) Cohomologically Complete Complexes 8 / 36

2. Adically Projective Modules Let me say a few words on the structure of a -adically free A -modules. Take a set Z , and consider the set F fin ( Z, A ) of functions φ : Z → A with finite support. It is a free A -module with basis the set { δ z } z ∈ Z of delta functions. Of course any free A -module looks like this. The completion Λ a F fin ( Z, A ) is canonically isomorphic to the A -module F dec ( Z, � A ) of a -adically decaying functions φ : Z → � A , where � A := Λ a A . Theorem 2.2 is proved in [Ye] by showing that the functor M �→ F dec ( Z, M ) is exact on the category Mod f � A of finitely generated � A -modules, and that for any such M the canonical homomorphism A F dec ( Z, � M ⊗ � A ) → F dec ( Z, M ) is bijective. Amnon Yekutieli (BGU) Cohomologically Complete Complexes 8 / 36

� � � 2. Adically Projective Modules Definition 2.3. An A module P is called a -adically projective if it is a -adically complete, and any diagram P � N M φ where M and N are a -adically complete and φ is surjective, extends to a diagram P � N M φ Amnon Yekutieli (BGU) Cohomologically Complete Complexes 9 / 36

� � � 2. Adically Projective Modules Definition 2.3. An A module P is called a -adically projective if it is a -adically complete, and any diagram P � N M φ where M and N are a -adically complete and φ is surjective, extends to a diagram P � N M φ Amnon Yekutieli (BGU) Cohomologically Complete Complexes 9 / 36

2. Adically Projective Modules Here are some results from [Ye], which show that the concept of a -adically projective module is useful. Proposition 2.4. 1. Let M be an a -adically complete module. Then M is a quotient of an a -adically free module P . 2. A module P is a -adically projective iff it is a direct summand of an a -adically free module. 3. An a -adically projective module P is flat. 4. If Q is a projective A -module, then its completion P := Λ a Q is a -adically projective. Amnon Yekutieli (BGU) Cohomologically Complete Complexes 10 / 36

2. Adically Projective Modules Here are some results from [Ye], which show that the concept of a -adically projective module is useful. Proposition 2.4. 1. Let M be an a -adically complete module. Then M is a quotient of an a -adically free module P . 2. A module P is a -adically projective iff it is a direct summand of an a -adically free module. 3. An a -adically projective module P is flat. 4. If Q is a projective A -module, then its completion P := Λ a Q is a -adically projective. Amnon Yekutieli (BGU) Cohomologically Complete Complexes 10 / 36

2. Adically Projective Modules Here are some results from [Ye], which show that the concept of a -adically projective module is useful. Proposition 2.4. 1. Let M be an a -adically complete module. Then M is a quotient of an a -adically free module P . 2. A module P is a -adically projective iff it is a direct summand of an a -adically free module. 3. An a -adically projective module P is flat. 4. If Q is a projective A -module, then its completion P := Λ a Q is a -adically projective. Amnon Yekutieli (BGU) Cohomologically Complete Complexes 10 / 36

2. Adically Projective Modules Here are some results from [Ye], which show that the concept of a -adically projective module is useful. Proposition 2.4. 1. Let M be an a -adically complete module. Then M is a quotient of an a -adically free module P . 2. A module P is a -adically projective iff it is a direct summand of an a -adically free module. 3. An a -adically projective module P is flat. 4. If Q is a projective A -module, then its completion P := Λ a Q is a -adically projective. Amnon Yekutieli (BGU) Cohomologically Complete Complexes 10 / 36

2. Adically Projective Modules Here are some results from [Ye], which show that the concept of a -adically projective module is useful. Proposition 2.4. 1. Let M be an a -adically complete module. Then M is a quotient of an a -adically free module P . 2. A module P is a -adically projective iff it is a direct summand of an a -adically free module. 3. An a -adically projective module P is flat. 4. If Q is a projective A -module, then its completion P := Λ a Q is a -adically projective. Amnon Yekutieli (BGU) Cohomologically Complete Complexes 10 / 36

2. Adically Projective Modules Here are some results from [Ye], which show that the concept of a -adically projective module is useful. Proposition 2.4. 1. Let M be an a -adically complete module. Then M is a quotient of an a -adically free module P . 2. A module P is a -adically projective iff it is a direct summand of an a -adically free module. 3. An a -adically projective module P is flat. 4. If Q is a projective A -module, then its completion P := Λ a Q is a -adically projective. Amnon Yekutieli (BGU) Cohomologically Complete Complexes 10 / 36

2. Adically Projective Modules When A is a local ring, more can be said. Theorem 2.5. ([Ye, Corollary 4.5]) Assume A is a complete local ring, with maximal ideal m . The following conditions are equivalent for an A -module M : (i) M is flat and m -adically complete. (ii) M is m -adically free. Amnon Yekutieli (BGU) Cohomologically Complete Complexes 11 / 36

2. Adically Projective Modules When A is a local ring, more can be said. Theorem 2.5. ([Ye, Corollary 4.5]) Assume A is a complete local ring, with maximal ideal m . The following conditions are equivalent for an A -module M : (i) M is flat and m -adically complete. (ii) M is m -adically free. Amnon Yekutieli (BGU) Cohomologically Complete Complexes 11 / 36

2. Adically Projective Modules When A is a local ring, more can be said. Theorem 2.5. ([Ye, Corollary 4.5]) Assume A is a complete local ring, with maximal ideal m . The following conditions are equivalent for an A -module M : (i) M is flat and m -adically complete. (ii) M is m -adically free. Amnon Yekutieli (BGU) Cohomologically Complete Complexes 11 / 36

2. Adically Projective Modules When A is a local ring, more can be said. Theorem 2.5. ([Ye, Corollary 4.5]) Assume A is a complete local ring, with maximal ideal m . The following conditions are equivalent for an A -module M : (i) M is flat and m -adically complete. (ii) M is m -adically free. Amnon Yekutieli (BGU) Cohomologically Complete Complexes 11 / 36

2. Adically Projective Modules When A is a local ring, more can be said. Theorem 2.5. ([Ye, Corollary 4.5]) Assume A is a complete local ring, with maximal ideal m . The following conditions are equivalent for an A -module M : (i) M is flat and m -adically complete. (ii) M is m -adically free. Amnon Yekutieli (BGU) Cohomologically Complete Complexes 11 / 36

2. Adically Projective Modules When A is a local ring, more can be said. Theorem 2.5. ([Ye, Corollary 4.5]) Assume A is a complete local ring, with maximal ideal m . The following conditions are equivalent for an A -module M : (i) M is flat and m -adically complete. (ii) M is m -adically free. Amnon Yekutieli (BGU) Cohomologically Complete Complexes 11 / 36

3. Derived Completion and Torsion Functors 3. Derived Completion and Torsion Functors Let us denote by D ( Mod A ) the derived category of (unbounded) complexes of A -modules. We shall require the notions of K-flat resolution and K-injective resolution. A complex of A -modules P is called K-flat if for any acyclic complex N , the complex P ⊗ A N is acyclic. A K-flat resolution of a complex M is a quasi-isomorphism P → M , with P a K-flat complex. Such resolutions always exist. Example 3.1. Suppose M is a bounded above complex. Let P → M be a flat resolution (in the usual sense, i.e. P is a bounded above complex of flat modules). Then P → M is a K-flat resolution. Amnon Yekutieli (BGU) Cohomologically Complete Complexes 12 / 36

3. Derived Completion and Torsion Functors 3. Derived Completion and Torsion Functors Let us denote by D ( Mod A ) the derived category of (unbounded) complexes of A -modules. We shall require the notions of K-flat resolution and K-injective resolution. A complex of A -modules P is called K-flat if for any acyclic complex N , the complex P ⊗ A N is acyclic. A K-flat resolution of a complex M is a quasi-isomorphism P → M , with P a K-flat complex. Such resolutions always exist. Example 3.1. Suppose M is a bounded above complex. Let P → M be a flat resolution (in the usual sense, i.e. P is a bounded above complex of flat modules). Then P → M is a K-flat resolution. Amnon Yekutieli (BGU) Cohomologically Complete Complexes 12 / 36

3. Derived Completion and Torsion Functors 3. Derived Completion and Torsion Functors Let us denote by D ( Mod A ) the derived category of (unbounded) complexes of A -modules. We shall require the notions of K-flat resolution and K-injective resolution. A complex of A -modules P is called K-flat if for any acyclic complex N , the complex P ⊗ A N is acyclic. A K-flat resolution of a complex M is a quasi-isomorphism P → M , with P a K-flat complex. Such resolutions always exist. Example 3.1. Suppose M is a bounded above complex. Let P → M be a flat resolution (in the usual sense, i.e. P is a bounded above complex of flat modules). Then P → M is a K-flat resolution. Amnon Yekutieli (BGU) Cohomologically Complete Complexes 12 / 36

3. Derived Completion and Torsion Functors 3. Derived Completion and Torsion Functors Let us denote by D ( Mod A ) the derived category of (unbounded) complexes of A -modules. We shall require the notions of K-flat resolution and K-injective resolution. A complex of A -modules P is called K-flat if for any acyclic complex N , the complex P ⊗ A N is acyclic. A K-flat resolution of a complex M is a quasi-isomorphism P → M , with P a K-flat complex. Such resolutions always exist. Example 3.1. Suppose M is a bounded above complex. Let P → M be a flat resolution (in the usual sense, i.e. P is a bounded above complex of flat modules). Then P → M is a K-flat resolution. Amnon Yekutieli (BGU) Cohomologically Complete Complexes 12 / 36

3. Derived Completion and Torsion Functors 3. Derived Completion and Torsion Functors Let us denote by D ( Mod A ) the derived category of (unbounded) complexes of A -modules. We shall require the notions of K-flat resolution and K-injective resolution. A complex of A -modules P is called K-flat if for any acyclic complex N , the complex P ⊗ A N is acyclic. A K-flat resolution of a complex M is a quasi-isomorphism P → M , with P a K-flat complex. Such resolutions always exist. Example 3.1. Suppose M is a bounded above complex. Let P → M be a flat resolution (in the usual sense, i.e. P is a bounded above complex of flat modules). Then P → M is a K-flat resolution. Amnon Yekutieli (BGU) Cohomologically Complete Complexes 12 / 36

3. Derived Completion and Torsion Functors 3. Derived Completion and Torsion Functors Let us denote by D ( Mod A ) the derived category of (unbounded) complexes of A -modules. We shall require the notions of K-flat resolution and K-injective resolution. A complex of A -modules P is called K-flat if for any acyclic complex N , the complex P ⊗ A N is acyclic. A K-flat resolution of a complex M is a quasi-isomorphism P → M , with P a K-flat complex. Such resolutions always exist. Example 3.1. Suppose M is a bounded above complex. Let P → M be a flat resolution (in the usual sense, i.e. P is a bounded above complex of flat modules). Then P → M is a K-flat resolution. Amnon Yekutieli (BGU) Cohomologically Complete Complexes 12 / 36

3. Derived Completion and Torsion Functors 3. Derived Completion and Torsion Functors Let us denote by D ( Mod A ) the derived category of (unbounded) complexes of A -modules. We shall require the notions of K-flat resolution and K-injective resolution. A complex of A -modules P is called K-flat if for any acyclic complex N , the complex P ⊗ A N is acyclic. A K-flat resolution of a complex M is a quasi-isomorphism P → M , with P a K-flat complex. Such resolutions always exist. Example 3.1. Suppose M is a bounded above complex. Let P → M be a flat resolution (in the usual sense, i.e. P is a bounded above complex of flat modules). Then P → M is a K-flat resolution. Amnon Yekutieli (BGU) Cohomologically Complete Complexes 12 / 36

3. Derived Completion and Torsion Functors 3. Derived Completion and Torsion Functors Let us denote by D ( Mod A ) the derived category of (unbounded) complexes of A -modules. We shall require the notions of K-flat resolution and K-injective resolution. A complex of A -modules P is called K-flat if for any acyclic complex N , the complex P ⊗ A N is acyclic. A K-flat resolution of a complex M is a quasi-isomorphism P → M , with P a K-flat complex. Such resolutions always exist. Example 3.1. Suppose M is a bounded above complex. Let P → M be a flat resolution (in the usual sense, i.e. P is a bounded above complex of flat modules). Then P → M is a K-flat resolution. Amnon Yekutieli (BGU) Cohomologically Complete Complexes 12 / 36

3. Derived Completion and Torsion Functors A complex I is called K-injective if for any acyclic complex N , the complex Hom A ( N, I ) is acyclic. A K-injective resolution of a complex M is a quasi-isomorphism M → I , with I a K-injective complex. Such resolutions always exist. Example 3.2. Suppose M is a bounded below complex. Let M → I be an injective resolution (in the usual sense, i.e. I is a bounded below complex of injective modules). Then M → I is a K-injective resolution. Amnon Yekutieli (BGU) Cohomologically Complete Complexes 13 / 36

3. Derived Completion and Torsion Functors A complex I is called K-injective if for any acyclic complex N , the complex Hom A ( N, I ) is acyclic. A K-injective resolution of a complex M is a quasi-isomorphism M → I , with I a K-injective complex. Such resolutions always exist. Example 3.2. Suppose M is a bounded below complex. Let M → I be an injective resolution (in the usual sense, i.e. I is a bounded below complex of injective modules). Then M → I is a K-injective resolution. Amnon Yekutieli (BGU) Cohomologically Complete Complexes 13 / 36

3. Derived Completion and Torsion Functors A complex I is called K-injective if for any acyclic complex N , the complex Hom A ( N, I ) is acyclic. A K-injective resolution of a complex M is a quasi-isomorphism M → I , with I a K-injective complex. Such resolutions always exist. Example 3.2. Suppose M is a bounded below complex. Let M → I be an injective resolution (in the usual sense, i.e. I is a bounded below complex of injective modules). Then M → I is a K-injective resolution. Amnon Yekutieli (BGU) Cohomologically Complete Complexes 13 / 36

3. Derived Completion and Torsion Functors A complex I is called K-injective if for any acyclic complex N , the complex Hom A ( N, I ) is acyclic. A K-injective resolution of a complex M is a quasi-isomorphism M → I , with I a K-injective complex. Such resolutions always exist. Example 3.2. Suppose M is a bounded below complex. Let M → I be an injective resolution (in the usual sense, i.e. I is a bounded below complex of injective modules). Then M → I is a K-injective resolution. Amnon Yekutieli (BGU) Cohomologically Complete Complexes 13 / 36

3. Derived Completion and Torsion Functors A complex I is called K-injective if for any acyclic complex N , the complex Hom A ( N, I ) is acyclic. A K-injective resolution of a complex M is a quasi-isomorphism M → I , with I a K-injective complex. Such resolutions always exist. Example 3.2. Suppose M is a bounded below complex. Let M → I be an injective resolution (in the usual sense, i.e. I is a bounded below complex of injective modules). Then M → I is a K-injective resolution. Amnon Yekutieli (BGU) Cohomologically Complete Complexes 13 / 36

3. Derived Completion and Torsion Functors The additive functors Λ a , Γ a : Mod A → Mod A have derived functors LΛ a , RΓ a : D ( Mod A ) → D ( Mod A ) . The left derived functor LΛ a is constructed like this: given a complex M of A -modules, we choose a K-flat resolution P → M , and we let LΛ a M := Λ a P. The right derived functor RΓ a is constructed like this: RΓ a M := Γ a I, where M → I is a K-injective resolution. Amnon Yekutieli (BGU) Cohomologically Complete Complexes 14 / 36

3. Derived Completion and Torsion Functors The additive functors Λ a , Γ a : Mod A → Mod A have derived functors LΛ a , RΓ a : D ( Mod A ) → D ( Mod A ) . The left derived functor LΛ a is constructed like this: given a complex M of A -modules, we choose a K-flat resolution P → M , and we let LΛ a M := Λ a P. The right derived functor RΓ a is constructed like this: RΓ a M := Γ a I, where M → I is a K-injective resolution. Amnon Yekutieli (BGU) Cohomologically Complete Complexes 14 / 36

3. Derived Completion and Torsion Functors The additive functors Λ a , Γ a : Mod A → Mod A have derived functors LΛ a , RΓ a : D ( Mod A ) → D ( Mod A ) . The left derived functor LΛ a is constructed like this: given a complex M of A -modules, we choose a K-flat resolution P → M , and we let LΛ a M := Λ a P. The right derived functor RΓ a is constructed like this: RΓ a M := Γ a I, where M → I is a K-injective resolution. Amnon Yekutieli (BGU) Cohomologically Complete Complexes 14 / 36

3. Derived Completion and Torsion Functors There are natural transformations τ L M : M → LΛ a M and σ R M : RΓ a M → M. Definition 3.3. A complex M ∈ D ( Mod A ) is called a cohomologically a -torsion complex if the morphism σ R M is an isomorphism. Cohomologically a -torsion complexes are not hard to identify, because of the next result. Theorem 3.4. [PSY, Corollary 5.4] A complex M ∈ D ( Mod A ) is cohomologically a -torsion iff for every i the A -module H i M is a -torsion. Amnon Yekutieli (BGU) Cohomologically Complete Complexes 15 / 36

3. Derived Completion and Torsion Functors There are natural transformations τ L M : M → LΛ a M and σ R M : RΓ a M → M. Definition 3.3. A complex M ∈ D ( Mod A ) is called a cohomologically a -torsion complex if the morphism σ R M is an isomorphism. Cohomologically a -torsion complexes are not hard to identify, because of the next result. Theorem 3.4. [PSY, Corollary 5.4] A complex M ∈ D ( Mod A ) is cohomologically a -torsion iff for every i the A -module H i M is a -torsion. Amnon Yekutieli (BGU) Cohomologically Complete Complexes 15 / 36

3. Derived Completion and Torsion Functors There are natural transformations τ L M : M → LΛ a M and σ R M : RΓ a M → M. Definition 3.3. A complex M ∈ D ( Mod A ) is called a cohomologically a -torsion complex if the morphism σ R M is an isomorphism. Cohomologically a -torsion complexes are not hard to identify, because of the next result. Theorem 3.4. [PSY, Corollary 5.4] A complex M ∈ D ( Mod A ) is cohomologically a -torsion iff for every i the A -module H i M is a -torsion. Amnon Yekutieli (BGU) Cohomologically Complete Complexes 15 / 36

3. Derived Completion and Torsion Functors There are natural transformations τ L M : M → LΛ a M and σ R M : RΓ a M → M. Definition 3.3. A complex M ∈ D ( Mod A ) is called a cohomologically a -torsion complex if the morphism σ R M is an isomorphism. Cohomologically a -torsion complexes are not hard to identify, because of the next result. Theorem 3.4. [PSY, Corollary 5.4] A complex M ∈ D ( Mod A ) is cohomologically a -torsion iff for every i the A -module H i M is a -torsion. Amnon Yekutieli (BGU) Cohomologically Complete Complexes 15 / 36

3. Derived Completion and Torsion Functors There are natural transformations τ L M : M → LΛ a M and σ R M : RΓ a M → M. Definition 3.3. A complex M ∈ D ( Mod A ) is called a cohomologically a -torsion complex if the morphism σ R M is an isomorphism. Cohomologically a -torsion complexes are not hard to identify, because of the next result. Theorem 3.4. [PSY, Corollary 5.4] A complex M ∈ D ( Mod A ) is cohomologically a -torsion iff for every i the A -module H i M is a -torsion. Amnon Yekutieli (BGU) Cohomologically Complete Complexes 15 / 36

3. Derived Completion and Torsion Functors Definition 3.5. A complex M ∈ D ( Mod A ) is called a cohomologically a -adically complete complex if the morphism τ L M : M → LΛ a M is an isomorphism. We first saw the notion of cohomologically complete complex in the paper [KS] of Kashiwara and Schapira. They considered the ring A := K [ t ] and the ideal a := ( t ) ; we refer to this as the principal case. The definition above is different from the one in [KS]; but we will prove later that they are equivalent (in Section 5). Amnon Yekutieli (BGU) Cohomologically Complete Complexes 16 / 36

3. Derived Completion and Torsion Functors Definition 3.5. A complex M ∈ D ( Mod A ) is called a cohomologically a -adically complete complex if the morphism τ L M : M → LΛ a M is an isomorphism. We first saw the notion of cohomologically complete complex in the paper [KS] of Kashiwara and Schapira. They considered the ring A := K [ t ] and the ideal a := ( t ) ; we refer to this as the principal case. The definition above is different from the one in [KS]; but we will prove later that they are equivalent (in Section 5). Amnon Yekutieli (BGU) Cohomologically Complete Complexes 16 / 36

3. Derived Completion and Torsion Functors Definition 3.5. A complex M ∈ D ( Mod A ) is called a cohomologically a -adically complete complex if the morphism τ L M : M → LΛ a M is an isomorphism. We first saw the notion of cohomologically complete complex in the paper [KS] of Kashiwara and Schapira. They considered the ring A := K [ t ] and the ideal a := ( t ) ; we refer to this as the principal case. The definition above is different from the one in [KS]; but we will prove later that they are equivalent (in Section 5). Amnon Yekutieli (BGU) Cohomologically Complete Complexes 16 / 36

3. Derived Completion and Torsion Functors Definition 3.5. A complex M ∈ D ( Mod A ) is called a cohomologically a -adically complete complex if the morphism τ L M : M → LΛ a M is an isomorphism. We first saw the notion of cohomologically complete complex in the paper [KS] of Kashiwara and Schapira. They considered the ring A := K [ t ] and the ideal a := ( t ) ; we refer to this as the principal case. The definition above is different from the one in [KS]; but we will prove later that they are equivalent (in Section 5). Amnon Yekutieli (BGU) Cohomologically Complete Complexes 16 / 36

3. Derived Completion and Torsion Functors Cohomologically complete complexes are somewhat mysterious, as the next example shows. Example 3.6. [PSY, Example 2.27] Let A := K [[ t ]] and a := ( t ) . There is a complex � � · · · → 0 → P − 1 → P 0 → 0 → · · · P = with these properties: ◮ P − 1 and P 0 are a -adically projective modules. ◮ The complex P is cohomologically a -adically complete. ◮ H − 1 P = 0 , but the A -module H 0 P is not a -adically complete. Amnon Yekutieli (BGU) Cohomologically Complete Complexes 17 / 36

3. Derived Completion and Torsion Functors Cohomologically complete complexes are somewhat mysterious, as the next example shows. Example 3.6. [PSY, Example 2.27] Let A := K [[ t ]] and a := ( t ) . There is a complex � � · · · → 0 → P − 1 → P 0 → 0 → · · · P = with these properties: ◮ P − 1 and P 0 are a -adically projective modules. ◮ The complex P is cohomologically a -adically complete. ◮ H − 1 P = 0 , but the A -module H 0 P is not a -adically complete. Amnon Yekutieli (BGU) Cohomologically Complete Complexes 17 / 36

3. Derived Completion and Torsion Functors Cohomologically complete complexes are somewhat mysterious, as the next example shows. Example 3.6. [PSY, Example 2.27] Let A := K [[ t ]] and a := ( t ) . There is a complex � � · · · → 0 → P − 1 → P 0 → 0 → · · · P = with these properties: ◮ P − 1 and P 0 are a -adically projective modules. ◮ The complex P is cohomologically a -adically complete. ◮ H − 1 P = 0 , but the A -module H 0 P is not a -adically complete. Amnon Yekutieli (BGU) Cohomologically Complete Complexes 17 / 36

3. Derived Completion and Torsion Functors Cohomologically complete complexes are somewhat mysterious, as the next example shows. Example 3.6. [PSY, Example 2.27] Let A := K [[ t ]] and a := ( t ) . There is a complex � � · · · → 0 → P − 1 → P 0 → 0 → · · · P = with these properties: ◮ P − 1 and P 0 are a -adically projective modules. ◮ The complex P is cohomologically a -adically complete. ◮ H − 1 P = 0 , but the A -module H 0 P is not a -adically complete. Amnon Yekutieli (BGU) Cohomologically Complete Complexes 17 / 36

3. Derived Completion and Torsion Functors Cohomologically complete complexes are somewhat mysterious, as the next example shows. Example 3.6. [PSY, Example 2.27] Let A := K [[ t ]] and a := ( t ) . There is a complex � � · · · → 0 → P − 1 → P 0 → 0 → · · · P = with these properties: ◮ P − 1 and P 0 are a -adically projective modules. ◮ The complex P is cohomologically a -adically complete. ◮ H − 1 P = 0 , but the A -module H 0 P is not a -adically complete. Amnon Yekutieli (BGU) Cohomologically Complete Complexes 17 / 36

3. Derived Completion and Torsion Functors Cohomologically complete complexes are somewhat mysterious, as the next example shows. Example 3.6. [PSY, Example 2.27] Let A := K [[ t ]] and a := ( t ) . There is a complex � � · · · → 0 → P − 1 → P 0 → 0 → · · · P = with these properties: ◮ P − 1 and P 0 are a -adically projective modules. ◮ The complex P is cohomologically a -adically complete. ◮ H − 1 P = 0 , but the A -module H 0 P is not a -adically complete. Amnon Yekutieli (BGU) Cohomologically Complete Complexes 17 / 36

3. Derived Completion and Torsion Functors We do have this partial characterization: Theorem 3.7. [PSY, Theorem 2.24] A bounded above complex M is cohomologically a -adically complete iff it is isomorphic, in D ( Mod A ) , to a bounded above complex P of a -adically projective complexes. Let us denote by D − f ( Mod A ) the category of bounded above complexes with finitely generated cohomologies. Corollary 3.8. Assume that A is a -adically complete. Then D − f ( Mod A ) ⊂ D ( Mod A ) a -com . Proof. Let M ∈ D − f ( Mod A ) . There there is a resolution P → M , where P is a bounded above complex of finitely generated projective A -modules. Since A is complete, each P i is an a -adically projective module. � Amnon Yekutieli (BGU) Cohomologically Complete Complexes 18 / 36

3. Derived Completion and Torsion Functors We do have this partial characterization: Theorem 3.7. [PSY, Theorem 2.24] A bounded above complex M is cohomologically a -adically complete iff it is isomorphic, in D ( Mod A ) , to a bounded above complex P of a -adically projective complexes. Let us denote by D − f ( Mod A ) the category of bounded above complexes with finitely generated cohomologies. Corollary 3.8. Assume that A is a -adically complete. Then D − f ( Mod A ) ⊂ D ( Mod A ) a -com . Proof. Let M ∈ D − f ( Mod A ) . There there is a resolution P → M , where P is a bounded above complex of finitely generated projective A -modules. Since A is complete, each P i is an a -adically projective module. � Amnon Yekutieli (BGU) Cohomologically Complete Complexes 18 / 36

3. Derived Completion and Torsion Functors We do have this partial characterization: Theorem 3.7. [PSY, Theorem 2.24] A bounded above complex M is cohomologically a -adically complete iff it is isomorphic, in D ( Mod A ) , to a bounded above complex P of a -adically projective complexes. Let us denote by D − f ( Mod A ) the category of bounded above complexes with finitely generated cohomologies. Corollary 3.8. Assume that A is a -adically complete. Then D − f ( Mod A ) ⊂ D ( Mod A ) a -com . Proof. Let M ∈ D − f ( Mod A ) . There there is a resolution P → M , where P is a bounded above complex of finitely generated projective A -modules. Since A is complete, each P i is an a -adically projective module. � Amnon Yekutieli (BGU) Cohomologically Complete Complexes 18 / 36

3. Derived Completion and Torsion Functors We do have this partial characterization: Theorem 3.7. [PSY, Theorem 2.24] A bounded above complex M is cohomologically a -adically complete iff it is isomorphic, in D ( Mod A ) , to a bounded above complex P of a -adically projective complexes. Let us denote by D − f ( Mod A ) the category of bounded above complexes with finitely generated cohomologies. Corollary 3.8. Assume that A is a -adically complete. Then D − f ( Mod A ) ⊂ D ( Mod A ) a -com . Proof. Let M ∈ D − f ( Mod A ) . There there is a resolution P → M , where P is a bounded above complex of finitely generated projective A -modules. Since A is complete, each P i is an a -adically projective module. � Amnon Yekutieli (BGU) Cohomologically Complete Complexes 18 / 36

3. Derived Completion and Torsion Functors We do have this partial characterization: Theorem 3.7. [PSY, Theorem 2.24] A bounded above complex M is cohomologically a -adically complete iff it is isomorphic, in D ( Mod A ) , to a bounded above complex P of a -adically projective complexes. Let us denote by D − f ( Mod A ) the category of bounded above complexes with finitely generated cohomologies. Corollary 3.8. Assume that A is a -adically complete. Then D − f ( Mod A ) ⊂ D ( Mod A ) a -com . Proof. Let M ∈ D − f ( Mod A ) . There there is a resolution P → M , where P is a bounded above complex of finitely generated projective A -modules. Since A is complete, each P i is an a -adically projective module. � Amnon Yekutieli (BGU) Cohomologically Complete Complexes 18 / 36

3. Derived Completion and Torsion Functors We do have this partial characterization: Theorem 3.7. [PSY, Theorem 2.24] A bounded above complex M is cohomologically a -adically complete iff it is isomorphic, in D ( Mod A ) , to a bounded above complex P of a -adically projective complexes. Let us denote by D − f ( Mod A ) the category of bounded above complexes with finitely generated cohomologies. Corollary 3.8. Assume that A is a -adically complete. Then D − f ( Mod A ) ⊂ D ( Mod A ) a -com . Proof. Let M ∈ D − f ( Mod A ) . There there is a resolution P → M , where P is a bounded above complex of finitely generated projective A -modules. Since A is complete, each P i is an a -adically projective module. � Amnon Yekutieli (BGU) Cohomologically Complete Complexes 18 / 36

3. Derived Completion and Torsion Functors We do have this partial characterization: Theorem 3.7. [PSY, Theorem 2.24] A bounded above complex M is cohomologically a -adically complete iff it is isomorphic, in D ( Mod A ) , to a bounded above complex P of a -adically projective complexes. Let us denote by D − f ( Mod A ) the category of bounded above complexes with finitely generated cohomologies. Corollary 3.8. Assume that A is a -adically complete. Then D − f ( Mod A ) ⊂ D ( Mod A ) a -com . Proof. Let M ∈ D − f ( Mod A ) . There there is a resolution P → M , where P is a bounded above complex of finitely generated projective A -modules. Since A is complete, each P i is an a -adically projective module. � Amnon Yekutieli (BGU) Cohomologically Complete Complexes 18 / 36

3. Derived Completion and Torsion Functors We do have this partial characterization: Theorem 3.7. [PSY, Theorem 2.24] A bounded above complex M is cohomologically a -adically complete iff it is isomorphic, in D ( Mod A ) , to a bounded above complex P of a -adically projective complexes. Let us denote by D − f ( Mod A ) the category of bounded above complexes with finitely generated cohomologies. Corollary 3.8. Assume that A is a -adically complete. Then D − f ( Mod A ) ⊂ D ( Mod A ) a -com . Proof. Let M ∈ D − f ( Mod A ) . There there is a resolution P → M , where P is a bounded above complex of finitely generated projective A -modules. Since A is complete, each P i is an a -adically projective module. � Amnon Yekutieli (BGU) Cohomologically Complete Complexes 18 / 36

4. MGM Equivalence and GM Duality 4. MGM Equivalence and GM Duality Most of the results in this section were already proved by Alonso, Jeremias and Lipman in the paper [AJL1]; and some results even go back to Matlis [Ma], and Greenlees and May [GM]. The proofs given in [PSY] are somewhat different. For convenience I will indicate the references in [PSY]. Let us denote by D ( Mod A ) a -com full subcategory of D ( Mod A ) consisting of cohomologically a -adically complete complexes. Likewise let D ( Mod A ) a -tor be the category of cohomologically a -torsion complexes. These are triangulated subcategories. Amnon Yekutieli (BGU) Cohomologically Complete Complexes 19 / 36

4. MGM Equivalence and GM Duality 4. MGM Equivalence and GM Duality Most of the results in this section were already proved by Alonso, Jeremias and Lipman in the paper [AJL1]; and some results even go back to Matlis [Ma], and Greenlees and May [GM]. The proofs given in [PSY] are somewhat different. For convenience I will indicate the references in [PSY]. Let us denote by D ( Mod A ) a -com full subcategory of D ( Mod A ) consisting of cohomologically a -adically complete complexes. Likewise let D ( Mod A ) a -tor be the category of cohomologically a -torsion complexes. These are triangulated subcategories. Amnon Yekutieli (BGU) Cohomologically Complete Complexes 19 / 36

4. MGM Equivalence and GM Duality 4. MGM Equivalence and GM Duality Most of the results in this section were already proved by Alonso, Jeremias and Lipman in the paper [AJL1]; and some results even go back to Matlis [Ma], and Greenlees and May [GM]. The proofs given in [PSY] are somewhat different. For convenience I will indicate the references in [PSY]. Let us denote by D ( Mod A ) a -com full subcategory of D ( Mod A ) consisting of cohomologically a -adically complete complexes. Likewise let D ( Mod A ) a -tor be the category of cohomologically a -torsion complexes. These are triangulated subcategories. Amnon Yekutieli (BGU) Cohomologically Complete Complexes 19 / 36

4. MGM Equivalence and GM Duality 4. MGM Equivalence and GM Duality Most of the results in this section were already proved by Alonso, Jeremias and Lipman in the paper [AJL1]; and some results even go back to Matlis [Ma], and Greenlees and May [GM]. The proofs given in [PSY] are somewhat different. For convenience I will indicate the references in [PSY]. Let us denote by D ( Mod A ) a -com full subcategory of D ( Mod A ) consisting of cohomologically a -adically complete complexes. Likewise let D ( Mod A ) a -tor be the category of cohomologically a -torsion complexes. These are triangulated subcategories. Amnon Yekutieli (BGU) Cohomologically Complete Complexes 19 / 36

4. MGM Equivalence and GM Duality 4. MGM Equivalence and GM Duality Most of the results in this section were already proved by Alonso, Jeremias and Lipman in the paper [AJL1]; and some results even go back to Matlis [Ma], and Greenlees and May [GM]. The proofs given in [PSY] are somewhat different. For convenience I will indicate the references in [PSY]. Let us denote by D ( Mod A ) a -com full subcategory of D ( Mod A ) consisting of cohomologically a -adically complete complexes. Likewise let D ( Mod A ) a -tor be the category of cohomologically a -torsion complexes. These are triangulated subcategories. Amnon Yekutieli (BGU) Cohomologically Complete Complexes 19 / 36

4. MGM Equivalence and GM Duality 4. MGM Equivalence and GM Duality Most of the results in this section were already proved by Alonso, Jeremias and Lipman in the paper [AJL1]; and some results even go back to Matlis [Ma], and Greenlees and May [GM]. The proofs given in [PSY] are somewhat different. For convenience I will indicate the references in [PSY]. Let us denote by D ( Mod A ) a -com full subcategory of D ( Mod A ) consisting of cohomologically a -adically complete complexes. Likewise let D ( Mod A ) a -tor be the category of cohomologically a -torsion complexes. These are triangulated subcategories. Amnon Yekutieli (BGU) Cohomologically Complete Complexes 19 / 36

4. MGM Equivalence and GM Duality Theorem 4.1. [PSY, Theorems 5.1 and 7.1] Assume that the ideal a can be generated by n elements. Then the functors RΓ a and LΛ a have cohomological dimensions ≤ n . Theorem 4.2. (MGM Equivalence) [PSY, Theorem 7.3] 1. For any M ∈ D ( Mod A ) one has RΓ a M ∈ D ( Mod A ) a -tor and LΛ a M ∈ D ( Mod A ) a -com . 2. The functor RΓ a : D ( Mod A ) a -com → D ( Mod A ) a -tor is an equivalence, with quasi-inverse LΛ a . Amnon Yekutieli (BGU) Cohomologically Complete Complexes 20 / 36