Flatness and Completion for Infinitely Generated Modules over - PowerPoint PPT Presentation

1. Recalling the Completion of Finitely Generated Modules 1. Recalling the Completion of Finitely Generated Modules Let A be a commutative ring, with ideal a . For an A -module M , its a -adic completion is the A -module � ← i M / a i M . M : = lim The operation M �→ � M is an additive functor from the category Mod A of A -modules to itself. There is a canonical homomorphism τ M : M → � M . The module M is said to be a -adically complete if τ M is an isomorphism. Some texts would say that “ M is separated and complete”. Amnon Yekutieli (BGU) Flatness and Completion 3 / 27

1. Recalling the Completion of Finitely Generated Modules The completion � A of A is an A -algebra; and for any A -module M its completion � M is an � A -module. Now assume that A is noetherian. As we know from the course on commutative algebra, a -adic completion enjoys these properties: ◮ The A -algebra � A is flat. ◮ The completion functor M �→ � M is exact on the category Mod f A of finitely generated modules. ◮ If M ∈ Mod f A then the canonical homomorphism A ⊗ A M → � � M is an isomorphism. In this talk we shall be interested in infinitely generated A -modules. We shall see that the last two properties are false in the infinite case. Amnon Yekutieli (BGU) Flatness and Completion 4 / 27

1. Recalling the Completion of Finitely Generated Modules The completion � A of A is an A -algebra; and for any A -module M its completion � M is an � A -module. Now assume that A is noetherian. As we know from the course on commutative algebra, a -adic completion enjoys these properties: ◮ The A -algebra � A is flat. ◮ The completion functor M �→ � M is exact on the category Mod f A of finitely generated modules. ◮ If M ∈ Mod f A then the canonical homomorphism A ⊗ A M → � � M is an isomorphism. In this talk we shall be interested in infinitely generated A -modules. We shall see that the last two properties are false in the infinite case. Amnon Yekutieli (BGU) Flatness and Completion 4 / 27

1. Recalling the Completion of Finitely Generated Modules The completion � A of A is an A -algebra; and for any A -module M its completion � M is an � A -module. Now assume that A is noetherian. As we know from the course on commutative algebra, a -adic completion enjoys these properties: ◮ The A -algebra � A is flat. ◮ The completion functor M �→ � M is exact on the category Mod f A of finitely generated modules. ◮ If M ∈ Mod f A then the canonical homomorphism A ⊗ A M → � � M is an isomorphism. In this talk we shall be interested in infinitely generated A -modules. We shall see that the last two properties are false in the infinite case. Amnon Yekutieli (BGU) Flatness and Completion 4 / 27

1. Recalling the Completion of Finitely Generated Modules The completion � A of A is an A -algebra; and for any A -module M its completion � M is an � A -module. Now assume that A is noetherian. As we know from the course on commutative algebra, a -adic completion enjoys these properties: ◮ The A -algebra � A is flat. ◮ The completion functor M �→ � M is exact on the category Mod f A of finitely generated modules. ◮ If M ∈ Mod f A then the canonical homomorphism A ⊗ A M → � � M is an isomorphism. In this talk we shall be interested in infinitely generated A -modules. We shall see that the last two properties are false in the infinite case. Amnon Yekutieli (BGU) Flatness and Completion 4 / 27

1. Recalling the Completion of Finitely Generated Modules The completion � A of A is an A -algebra; and for any A -module M its completion � M is an � A -module. Now assume that A is noetherian. As we know from the course on commutative algebra, a -adic completion enjoys these properties: ◮ The A -algebra � A is flat. ◮ The completion functor M �→ � M is exact on the category Mod f A of finitely generated modules. ◮ If M ∈ Mod f A then the canonical homomorphism A ⊗ A M → � � M is an isomorphism. In this talk we shall be interested in infinitely generated A -modules. We shall see that the last two properties are false in the infinite case. Amnon Yekutieli (BGU) Flatness and Completion 4 / 27

1. Recalling the Completion of Finitely Generated Modules The completion � A of A is an A -algebra; and for any A -module M its completion � M is an � A -module. Now assume that A is noetherian. As we know from the course on commutative algebra, a -adic completion enjoys these properties: ◮ The A -algebra � A is flat. ◮ The completion functor M �→ � M is exact on the category Mod f A of finitely generated modules. ◮ If M ∈ Mod f A then the canonical homomorphism A ⊗ A M → � � M is an isomorphism. In this talk we shall be interested in infinitely generated A -modules. We shall see that the last two properties are false in the infinite case. Amnon Yekutieli (BGU) Flatness and Completion 4 / 27

2. Examples of Strange Behavior 2. Examples of Strange Behavior The first example will show that the a -adic completion � M of an A -module M is not always a -adically complete. This certainly sounds odd, and even “wrong”! I will give an explanation afterwards. Amnon Yekutieli (BGU) Flatness and Completion 5 / 27

2. Examples of Strange Behavior 2. Examples of Strange Behavior The first example will show that the a -adic completion � M of an A -module M is not always a -adically complete. This certainly sounds odd, and even “wrong”! I will give an explanation afterwards. Amnon Yekutieli (BGU) Flatness and Completion 5 / 27

2. Examples of Strange Behavior 2. Examples of Strange Behavior The first example will show that the a -adic completion � M of an A -module M is not always a -adically complete. This certainly sounds odd, and even “wrong”! I will give an explanation afterwards. Amnon Yekutieli (BGU) Flatness and Completion 5 / 27

2. Examples of Strange Behavior 2. Examples of Strange Behavior The first example will show that the a -adic completion � M of an A -module M is not always a -adically complete. This certainly sounds odd, and even “wrong”! I will give an explanation afterwards. Amnon Yekutieli (BGU) Flatness and Completion 5 / 27

2. Examples of Strange Behavior Example 2.1. Let K be a field, and let A : = K [ t 1 , t 2 , . . . ] , the ring of polynomials in countably many variables over K . Consider the maximal ideal a : = ( t 1 , t 2 , . . . ) in A . Take the module M : = A . Then the completion � M is not an a -adically complete A -module. Namely the canonical homomorphism M / a i � M : � � M → lim M τ � ← i is not bijective. Amnon Yekutieli (BGU) Flatness and Completion 6 / 27

2. Examples of Strange Behavior Example 2.1. Let K be a field, and let A : = K [ t 1 , t 2 , . . . ] , the ring of polynomials in countably many variables over K . Consider the maximal ideal a : = ( t 1 , t 2 , . . . ) in A . Take the module M : = A . Then the completion � M is not an a -adically complete A -module. Namely the canonical homomorphism M / a i � M : � � M → lim M τ � ← i is not bijective. Amnon Yekutieli (BGU) Flatness and Completion 6 / 27

2. Examples of Strange Behavior Example 2.1. Let K be a field, and let A : = K [ t 1 , t 2 , . . . ] , the ring of polynomials in countably many variables over K . Consider the maximal ideal a : = ( t 1 , t 2 , . . . ) in A . Take the module M : = A . Then the completion � M is not an a -adically complete A -module. Namely the canonical homomorphism M / a i � M : � � M → lim M τ � ← i is not bijective. Amnon Yekutieli (BGU) Flatness and Completion 6 / 27

2. Examples of Strange Behavior Example 2.1. Let K be a field, and let A : = K [ t 1 , t 2 , . . . ] , the ring of polynomials in countably many variables over K . Consider the maximal ideal a : = ( t 1 , t 2 , . . . ) in A . Take the module M : = A . Then the completion � M is not an a -adically complete A -module. Namely the canonical homomorphism M / a i � M : � � M → lim M τ � ← i is not bijective. Amnon Yekutieli (BGU) Flatness and Completion 6 / 27

2. Examples of Strange Behavior Example 2.1. Let K be a field, and let A : = K [ t 1 , t 2 , . . . ] , the ring of polynomials in countably many variables over K . Consider the maximal ideal a : = ( t 1 , t 2 , . . . ) in A . Take the module M : = A . Then the completion � M is not an a -adically complete A -module. Namely the canonical homomorphism M / a i � M : � � M → lim M τ � ← i is not bijective. Amnon Yekutieli (BGU) Flatness and Completion 6 / 27

2. Examples of Strange Behavior Here is an explanation. First a criterion for completeness. For every i ∈ N let A i : = A / a i + 1 . Theorem 2.2. Let M be an A -module, with a -adic completion � M . The following conditions are equivalent: (i) � M is a -adically complete. (ii) For every i the homomorphism id A i ⊗ τ M : A i ⊗ A M → A i ⊗ A � M is bijective. This is [Ye1, Theorem 1.5]; but it already appears implicitly in earlier work (e.g. [St]). It is not hard to apply this criterion to the module in the example (it is a calculation). Amnon Yekutieli (BGU) Flatness and Completion 7 / 27

2. Examples of Strange Behavior Here is an explanation. First a criterion for completeness. For every i ∈ N let A i : = A / a i + 1 . Theorem 2.2. Let M be an A -module, with a -adic completion � M . The following conditions are equivalent: (i) � M is a -adically complete. (ii) For every i the homomorphism id A i ⊗ τ M : A i ⊗ A M → A i ⊗ A � M is bijective. This is [Ye1, Theorem 1.5]; but it already appears implicitly in earlier work (e.g. [St]). It is not hard to apply this criterion to the module in the example (it is a calculation). Amnon Yekutieli (BGU) Flatness and Completion 7 / 27

2. Examples of Strange Behavior Here is an explanation. First a criterion for completeness. For every i ∈ N let A i : = A / a i + 1 . Theorem 2.2. Let M be an A -module, with a -adic completion � M . The following conditions are equivalent: (i) � M is a -adically complete. (ii) For every i the homomorphism id A i ⊗ τ M : A i ⊗ A M → A i ⊗ A � M is bijective. This is [Ye1, Theorem 1.5]; but it already appears implicitly in earlier work (e.g. [St]). It is not hard to apply this criterion to the module in the example (it is a calculation). Amnon Yekutieli (BGU) Flatness and Completion 7 / 27

2. Examples of Strange Behavior Here is an explanation. First a criterion for completeness. For every i ∈ N let A i : = A / a i + 1 . Theorem 2.2. Let M be an A -module, with a -adic completion � M . The following conditions are equivalent: (i) � M is a -adically complete. (ii) For every i the homomorphism id A i ⊗ τ M : A i ⊗ A M → A i ⊗ A � M is bijective. This is [Ye1, Theorem 1.5]; but it already appears implicitly in earlier work (e.g. [St]). It is not hard to apply this criterion to the module in the example (it is a calculation). Amnon Yekutieli (BGU) Flatness and Completion 7 / 27

2. Examples of Strange Behavior Here is an explanation. First a criterion for completeness. For every i ∈ N let A i : = A / a i + 1 . Theorem 2.2. Let M be an A -module, with a -adic completion � M . The following conditions are equivalent: (i) � M is a -adically complete. (ii) For every i the homomorphism id A i ⊗ τ M : A i ⊗ A M → A i ⊗ A � M is bijective. This is [Ye1, Theorem 1.5]; but it already appears implicitly in earlier work (e.g. [St]). It is not hard to apply this criterion to the module in the example (it is a calculation). Amnon Yekutieli (BGU) Flatness and Completion 7 / 27

2. Examples of Strange Behavior Here is an explanation. First a criterion for completeness. For every i ∈ N let A i : = A / a i + 1 . Theorem 2.2. Let M be an A -module, with a -adic completion � M . The following conditions are equivalent: (i) � M is a -adically complete. (ii) For every i the homomorphism id A i ⊗ τ M : A i ⊗ A M → A i ⊗ A � M is bijective. This is [Ye1, Theorem 1.5]; but it already appears implicitly in earlier work (e.g. [St]). It is not hard to apply this criterion to the module in the example (it is a calculation). Amnon Yekutieli (BGU) Flatness and Completion 7 / 27

2. Examples of Strange Behavior Here is an explanation. First a criterion for completeness. For every i ∈ N let A i : = A / a i + 1 . Theorem 2.2. Let M be an A -module, with a -adic completion � M . The following conditions are equivalent: (i) � M is a -adically complete. (ii) For every i the homomorphism id A i ⊗ τ M : A i ⊗ A M → A i ⊗ A � M is bijective. This is [Ye1, Theorem 1.5]; but it already appears implicitly in earlier work (e.g. [St]). It is not hard to apply this criterion to the module in the example (it is a calculation). Amnon Yekutieli (BGU) Flatness and Completion 7 / 27

2. Examples of Strange Behavior The confusion is due to the traditional concept that the a -adic completion of an A -module M coincides with the topological completion of M , with respect to the a -adic metric. (I will give a formula for this metric later.) In the example, the module � M is indeed the topological completion of M . So in particular it is a complete metric space, with the metric induced from the a -adic metric of M . However, in the example, the intrinsic a -adic metric of � M , coming from its a -adic filtration, is different from the induced metric! The module � M is not complete with respect its intrinsic a -adic metric. Amnon Yekutieli (BGU) Flatness and Completion 8 / 27

2. Examples of Strange Behavior The confusion is due to the traditional concept that the a -adic completion of an A -module M coincides with the topological completion of M , with respect to the a -adic metric. (I will give a formula for this metric later.) In the example, the module � M is indeed the topological completion of M . So in particular it is a complete metric space, with the metric induced from the a -adic metric of M . However, in the example, the intrinsic a -adic metric of � M , coming from its a -adic filtration, is different from the induced metric! The module � M is not complete with respect its intrinsic a -adic metric. Amnon Yekutieli (BGU) Flatness and Completion 8 / 27

2. Examples of Strange Behavior The confusion is due to the traditional concept that the a -adic completion of an A -module M coincides with the topological completion of M , with respect to the a -adic metric. (I will give a formula for this metric later.) In the example, the module � M is indeed the topological completion of M . So in particular it is a complete metric space, with the metric induced from the a -adic metric of M . However, in the example, the intrinsic a -adic metric of � M , coming from its a -adic filtration, is different from the induced metric! The module � M is not complete with respect its intrinsic a -adic metric. Amnon Yekutieli (BGU) Flatness and Completion 8 / 27

2. Examples of Strange Behavior The confusion is due to the traditional concept that the a -adic completion of an A -module M coincides with the topological completion of M , with respect to the a -adic metric. (I will give a formula for this metric later.) In the example, the module � M is indeed the topological completion of M . So in particular it is a complete metric space, with the metric induced from the a -adic metric of M . However, in the example, the intrinsic a -adic metric of � M , coming from its a -adic filtration, is different from the induced metric! The module � M is not complete with respect its intrinsic a -adic metric. Amnon Yekutieli (BGU) Flatness and Completion 8 / 27

2. Examples of Strange Behavior Fortunately, the anomaly above does not happen when A is a noetherian ring. I will return to this point later. From now on A will always be a noetherian ring. As noted before, the a -adic completion functor M �→ � M is exact on Mod f A . The next example shows that this functor is not left exact, nor right exact, on the whole category Mod A . Amnon Yekutieli (BGU) Flatness and Completion 9 / 27

2. Examples of Strange Behavior Fortunately, the anomaly above does not happen when A is a noetherian ring. I will return to this point later. From now on A will always be a noetherian ring. As noted before, the a -adic completion functor M �→ � M is exact on Mod f A . The next example shows that this functor is not left exact, nor right exact, on the whole category Mod A . Amnon Yekutieli (BGU) Flatness and Completion 9 / 27

2. Examples of Strange Behavior Fortunately, the anomaly above does not happen when A is a noetherian ring. I will return to this point later. From now on A will always be a noetherian ring. As noted before, the a -adic completion functor M �→ � M is exact on Mod f A . The next example shows that this functor is not left exact, nor right exact, on the whole category Mod A . Amnon Yekutieli (BGU) Flatness and Completion 9 / 27

2. Examples of Strange Behavior Fortunately, the anomaly above does not happen when A is a noetherian ring. I will return to this point later. From now on A will always be a noetherian ring. As noted before, the a -adic completion functor M �→ � M is exact on Mod f A . The next example shows that this functor is not left exact, nor right exact, on the whole category Mod A . Amnon Yekutieli (BGU) Flatness and Completion 9 / 27

2. Examples of Strange Behavior Example 2.3. Take A : = K [ t ] , the polynomial ring in one variable over the field K , and a : = ( t ) . Let P be the free A -module of countable rank with basis { δ i } i ∈ N , and let Q be another copy of P . Define a homomorphism φ : P → Q by the formula φ ( δ i ) : = t i δ i . It is easy to see that φ is injective, and hence there is an exact sequence φ ψ 0 → P − → Q − → M → 0, where M is the cokernel of φ . A calculation (see [Ye1, Example 3.20]) shows that the sequence � � φ ψ 0 → � → � → � − − M → 0 P Q is not exact at � Q . Amnon Yekutieli (BGU) Flatness and Completion 10 / 27

2. Examples of Strange Behavior Example 2.3. Take A : = K [ t ] , the polynomial ring in one variable over the field K , and a : = ( t ) . Let P be the free A -module of countable rank with basis { δ i } i ∈ N , and let Q be another copy of P . Define a homomorphism φ : P → Q by the formula φ ( δ i ) : = t i δ i . It is easy to see that φ is injective, and hence there is an exact sequence φ ψ 0 → P − → Q − → M → 0, where M is the cokernel of φ . A calculation (see [Ye1, Example 3.20]) shows that the sequence � � φ ψ 0 → � → � → � − − M → 0 P Q is not exact at � Q . Amnon Yekutieli (BGU) Flatness and Completion 10 / 27

2. Examples of Strange Behavior Example 2.3. Take A : = K [ t ] , the polynomial ring in one variable over the field K , and a : = ( t ) . Let P be the free A -module of countable rank with basis { δ i } i ∈ N , and let Q be another copy of P . Define a homomorphism φ : P → Q by the formula φ ( δ i ) : = t i δ i . It is easy to see that φ is injective, and hence there is an exact sequence φ ψ 0 → P − → Q − → M → 0, where M is the cokernel of φ . A calculation (see [Ye1, Example 3.20]) shows that the sequence � � φ ψ 0 → � → � → � − − M → 0 P Q is not exact at � Q . Amnon Yekutieli (BGU) Flatness and Completion 10 / 27

2. Examples of Strange Behavior Example 2.3. Take A : = K [ t ] , the polynomial ring in one variable over the field K , and a : = ( t ) . Let P be the free A -module of countable rank with basis { δ i } i ∈ N , and let Q be another copy of P . Define a homomorphism φ : P → Q by the formula φ ( δ i ) : = t i δ i . It is easy to see that φ is injective, and hence there is an exact sequence φ ψ 0 → P − → Q − → M → 0, where M is the cokernel of φ . A calculation (see [Ye1, Example 3.20]) shows that the sequence � � φ ψ 0 → � → � → � − − M → 0 P Q is not exact at � Q . Amnon Yekutieli (BGU) Flatness and Completion 10 / 27

2. Examples of Strange Behavior Example 2.3. Take A : = K [ t ] , the polynomial ring in one variable over the field K , and a : = ( t ) . Let P be the free A -module of countable rank with basis { δ i } i ∈ N , and let Q be another copy of P . Define a homomorphism φ : P → Q by the formula φ ( δ i ) : = t i δ i . It is easy to see that φ is injective, and hence there is an exact sequence φ ψ 0 → P − → Q − → M → 0, where M is the cokernel of φ . A calculation (see [Ye1, Example 3.20]) shows that the sequence � � φ ψ 0 → � → � → � − − M → 0 P Q is not exact at � Q . Amnon Yekutieli (BGU) Flatness and Completion 10 / 27

2. Examples of Strange Behavior Example 2.3. Take A : = K [ t ] , the polynomial ring in one variable over the field K , and a : = ( t ) . Let P be the free A -module of countable rank with basis { δ i } i ∈ N , and let Q be another copy of P . Define a homomorphism φ : P → Q by the formula φ ( δ i ) : = t i δ i . It is easy to see that φ is injective, and hence there is an exact sequence φ ψ 0 → P − → Q − → M → 0, where M is the cokernel of φ . A calculation (see [Ye1, Example 3.20]) shows that the sequence � � φ ψ 0 → � → � → � − − M → 0 P Q is not exact at � Q . Amnon Yekutieli (BGU) Flatness and Completion 10 / 27

2. Examples of Strange Behavior The fact that the completion functor is not exact is important. This is the subject of the very recent paper [PSY] by Marco Porta, Liran Shaul and myself. Amnon Yekutieli (BGU) Flatness and Completion 11 / 27

2. Examples of Strange Behavior The fact that the completion functor is not exact is important. This is the subject of the very recent paper [PSY] by Marco Porta, Liran Shaul and myself. Amnon Yekutieli (BGU) Flatness and Completion 11 / 27

3. The Module of Decaying Functions 3. The Module of Decaying Functions We continue with the assumption that A is a noetherian ring, and a is an ideal in it. The a -adic completion of A is � A . Let P be a free A -module. We want to understand the structure of its a -adic completion � P . = A r for some natural number r , then clearly If P has finite rank, say P ∼ P ∼ � = � A r . Amnon Yekutieli (BGU) Flatness and Completion 12 / 27

3. The Module of Decaying Functions 3. The Module of Decaying Functions We continue with the assumption that A is a noetherian ring, and a is an ideal in it. The a -adic completion of A is � A . Let P be a free A -module. We want to understand the structure of its a -adic completion � P . = A r for some natural number r , then clearly If P has finite rank, say P ∼ P ∼ � = � A r . Amnon Yekutieli (BGU) Flatness and Completion 12 / 27

3. The Module of Decaying Functions 3. The Module of Decaying Functions We continue with the assumption that A is a noetherian ring, and a is an ideal in it. The a -adic completion of A is � A . Let P be a free A -module. We want to understand the structure of its a -adic completion � P . = A r for some natural number r , then clearly If P has finite rank, say P ∼ P ∼ � = � A r . Amnon Yekutieli (BGU) Flatness and Completion 12 / 27

3. The Module of Decaying Functions 3. The Module of Decaying Functions We continue with the assumption that A is a noetherian ring, and a is an ideal in it. The a -adic completion of A is � A . Let P be a free A -module. We want to understand the structure of its a -adic completion � P . = A r for some natural number r , then clearly If P has finite rank, say P ∼ P ∼ � = � A r . Amnon Yekutieli (BGU) Flatness and Completion 12 / 27

3. The Module of Decaying Functions Now suppose P has infinite rank. Then P ∼ = F fin ( Z , A ) , where Z is some set (its cardinality being the rank of P ), and F fin ( Z , A ) denotes the set of finitely supported functions f : Z → A . Note that the free module F fin ( Z , A ) comes equipped with a canonical basis: the collection of delta functions { δ z } z ∈ Z . We are going to describe the completion � P . Amnon Yekutieli (BGU) Flatness and Completion 13 / 27

3. The Module of Decaying Functions Now suppose P has infinite rank. Then P ∼ = F fin ( Z , A ) , where Z is some set (its cardinality being the rank of P ), and F fin ( Z , A ) denotes the set of finitely supported functions f : Z → A . Note that the free module F fin ( Z , A ) comes equipped with a canonical basis: the collection of delta functions { δ z } z ∈ Z . We are going to describe the completion � P . Amnon Yekutieli (BGU) Flatness and Completion 13 / 27

3. The Module of Decaying Functions Now suppose P has infinite rank. Then P ∼ = F fin ( Z , A ) , where Z is some set (its cardinality being the rank of P ), and F fin ( Z , A ) denotes the set of finitely supported functions f : Z → A . Note that the free module F fin ( Z , A ) comes equipped with a canonical basis: the collection of delta functions { δ z } z ∈ Z . We are going to describe the completion � P . Amnon Yekutieli (BGU) Flatness and Completion 13 / 27

3. The Module of Decaying Functions For any a ∈ � A we denote by ord ( a ) its a -adic order; namely ord ( a ) : = sup { i ∈ N | a ∈ a i } . Note that ord ( a ) ∈ N ∪ { ∞ } , and ord ( a ) = ∞ iff a = 0. The a -adic metric on � A is given by the formula dist ( a 1 , a 2 ) : = ( 1 2 ) ord ( a 1 − a 2 ) . (We will not need this metric – this is just to relate to the discussion in Section 2.) Amnon Yekutieli (BGU) Flatness and Completion 14 / 27

3. The Module of Decaying Functions For any a ∈ � A we denote by ord ( a ) its a -adic order; namely ord ( a ) : = sup { i ∈ N | a ∈ a i } . Note that ord ( a ) ∈ N ∪ { ∞ } , and ord ( a ) = ∞ iff a = 0. The a -adic metric on � A is given by the formula dist ( a 1 , a 2 ) : = ( 1 2 ) ord ( a 1 − a 2 ) . (We will not need this metric – this is just to relate to the discussion in Section 2.) Amnon Yekutieli (BGU) Flatness and Completion 14 / 27

3. The Module of Decaying Functions For any a ∈ � A we denote by ord ( a ) its a -adic order; namely ord ( a ) : = sup { i ∈ N | a ∈ a i } . Note that ord ( a ) ∈ N ∪ { ∞ } , and ord ( a ) = ∞ iff a = 0. The a -adic metric on � A is given by the formula dist ( a 1 , a 2 ) : = ( 1 2 ) ord ( a 1 − a 2 ) . (We will not need this metric – this is just to relate to the discussion in Section 2.) Amnon Yekutieli (BGU) Flatness and Completion 14 / 27

3. The Module of Decaying Functions Definition 3.1. Let Z be a set. 1. A function f : Z → � A is called decaying if for every natural number i , the set { z ∈ Z | ord ( f ( z )) ≤ i } is finite. 2. The set of all decaying functions f : Z → � A is denoted by F dec ( Z , � A ) , and is called the module of decaying functions . It is easy to see that the support of any decaying function is countable. Also F dec ( Z , � A ) is an A -submodule of the module F ( Z , � A ) of all functions f : Z → � A . Any finitely supported function is decaying. Thus we get inclusions of A -modules F fin ( Z , � A ) ⊂ F dec ( Z , � A ) ⊂ F ( Z , � A ) . Amnon Yekutieli (BGU) Flatness and Completion 15 / 27

3. The Module of Decaying Functions Definition 3.1. Let Z be a set. 1. A function f : Z → � A is called decaying if for every natural number i , the set { z ∈ Z | ord ( f ( z )) ≤ i } is finite. 2. The set of all decaying functions f : Z → � A is denoted by F dec ( Z , � A ) , and is called the module of decaying functions . It is easy to see that the support of any decaying function is countable. Also F dec ( Z , � A ) is an A -submodule of the module F ( Z , � A ) of all functions f : Z → � A . Any finitely supported function is decaying. Thus we get inclusions of A -modules F fin ( Z , � A ) ⊂ F dec ( Z , � A ) ⊂ F ( Z , � A ) . Amnon Yekutieli (BGU) Flatness and Completion 15 / 27

3. The Module of Decaying Functions Definition 3.1. Let Z be a set. 1. A function f : Z → � A is called decaying if for every natural number i , the set { z ∈ Z | ord ( f ( z )) ≤ i } is finite. 2. The set of all decaying functions f : Z → � A is denoted by F dec ( Z , � A ) , and is called the module of decaying functions . It is easy to see that the support of any decaying function is countable. Also F dec ( Z , � A ) is an A -submodule of the module F ( Z , � A ) of all functions f : Z → � A . Any finitely supported function is decaying. Thus we get inclusions of A -modules F fin ( Z , � A ) ⊂ F dec ( Z , � A ) ⊂ F ( Z , � A ) . Amnon Yekutieli (BGU) Flatness and Completion 15 / 27

3. The Module of Decaying Functions Definition 3.1. Let Z be a set. 1. A function f : Z → � A is called decaying if for every natural number i , the set { z ∈ Z | ord ( f ( z )) ≤ i } is finite. 2. The set of all decaying functions f : Z → � A is denoted by F dec ( Z , � A ) , and is called the module of decaying functions . It is easy to see that the support of any decaying function is countable. Also F dec ( Z , � A ) is an A -submodule of the module F ( Z , � A ) of all functions f : Z → � A . Any finitely supported function is decaying. Thus we get inclusions of A -modules F fin ( Z , � A ) ⊂ F dec ( Z , � A ) ⊂ F ( Z , � A ) . Amnon Yekutieli (BGU) Flatness and Completion 15 / 27

3. The Module of Decaying Functions Definition 3.1. Let Z be a set. 1. A function f : Z → � A is called decaying if for every natural number i , the set { z ∈ Z | ord ( f ( z )) ≤ i } is finite. 2. The set of all decaying functions f : Z → � A is denoted by F dec ( Z , � A ) , and is called the module of decaying functions . It is easy to see that the support of any decaying function is countable. Also F dec ( Z , � A ) is an A -submodule of the module F ( Z , � A ) of all functions f : Z → � A . Any finitely supported function is decaying. Thus we get inclusions of A -modules F fin ( Z , � A ) ⊂ F dec ( Z , � A ) ⊂ F ( Z , � A ) . Amnon Yekutieli (BGU) Flatness and Completion 15 / 27

3. The Module of Decaying Functions Definition 3.1. Let Z be a set. 1. A function f : Z → � A is called decaying if for every natural number i , the set { z ∈ Z | ord ( f ( z )) ≤ i } is finite. 2. The set of all decaying functions f : Z → � A is denoted by F dec ( Z , � A ) , and is called the module of decaying functions . It is easy to see that the support of any decaying function is countable. Also F dec ( Z , � A ) is an A -submodule of the module F ( Z , � A ) of all functions f : Z → � A . Any finitely supported function is decaying. Thus we get inclusions of A -modules F fin ( Z , � A ) ⊂ F dec ( Z , � A ) ⊂ F ( Z , � A ) . Amnon Yekutieli (BGU) Flatness and Completion 15 / 27

3. The Module of Decaying Functions Definition 3.1. Let Z be a set. 1. A function f : Z → � A is called decaying if for every natural number i , the set { z ∈ Z | ord ( f ( z )) ≤ i } is finite. 2. The set of all decaying functions f : Z → � A is denoted by F dec ( Z , � A ) , and is called the module of decaying functions . It is easy to see that the support of any decaying function is countable. Also F dec ( Z , � A ) is an A -submodule of the module F ( Z , � A ) of all functions f : Z → � A . Any finitely supported function is decaying. Thus we get inclusions of A -modules F fin ( Z , � A ) ⊂ F dec ( Z , � A ) ⊂ F ( Z , � A ) . Amnon Yekutieli (BGU) Flatness and Completion 15 / 27

3. The Module of Decaying Functions The canonical homomorphism A → � A induces a homomorphism F fin ( Z , A ) → F dec ( Z , � A ) . Theorem 3.2. Let Z be any set. 1. The A -module F dec ( Z , � A ) is isomorphic to the a -adic completion of the A -module F fin ( Z , A ) . 2. The A -module F dec ( Z , � A ) is a -adically complete. 3. The A -module F dec ( Z , � A ) is flat. Amnon Yekutieli (BGU) Flatness and Completion 16 / 27

3. The Module of Decaying Functions The canonical homomorphism A → � A induces a homomorphism F fin ( Z , A ) → F dec ( Z , � A ) . Theorem 3.2. Let Z be any set. 1. The A -module F dec ( Z , � A ) is isomorphic to the a -adic completion of the A -module F fin ( Z , A ) . 2. The A -module F dec ( Z , � A ) is a -adically complete. 3. The A -module F dec ( Z , � A ) is flat. Amnon Yekutieli (BGU) Flatness and Completion 16 / 27

3. The Module of Decaying Functions The canonical homomorphism A → � A induces a homomorphism F fin ( Z , A ) → F dec ( Z , � A ) . Theorem 3.2. Let Z be any set. 1. The A -module F dec ( Z , � A ) is isomorphic to the a -adic completion of the A -module F fin ( Z , A ) . 2. The A -module F dec ( Z , � A ) is a -adically complete. 3. The A -module F dec ( Z , � A ) is flat. Amnon Yekutieli (BGU) Flatness and Completion 16 / 27

3. The Module of Decaying Functions The canonical homomorphism A → � A induces a homomorphism F fin ( Z , A ) → F dec ( Z , � A ) . Theorem 3.2. Let Z be any set. 1. The A -module F dec ( Z , � A ) is isomorphic to the a -adic completion of the A -module F fin ( Z , A ) . 2. The A -module F dec ( Z , � A ) is a -adically complete. 3. The A -module F dec ( Z , � A ) is flat. Amnon Yekutieli (BGU) Flatness and Completion 16 / 27

3. The Module of Decaying Functions The canonical homomorphism A → � A induces a homomorphism F fin ( Z , A ) → F dec ( Z , � A ) . Theorem 3.2. Let Z be any set. 1. The A -module F dec ( Z , � A ) is isomorphic to the a -adic completion of the A -module F fin ( Z , A ) . 2. The A -module F dec ( Z , � A ) is a -adically complete. 3. The A -module F dec ( Z , � A ) is flat. Amnon Yekutieli (BGU) Flatness and Completion 16 / 27

3. The Module of Decaying Functions The canonical homomorphism A → � A induces a homomorphism F fin ( Z , A ) → F dec ( Z , � A ) . Theorem 3.2. Let Z be any set. 1. The A -module F dec ( Z , � A ) is isomorphic to the a -adic completion of the A -module F fin ( Z , A ) . 2. The A -module F dec ( Z , � A ) is a -adically complete. 3. The A -module F dec ( Z , � A ) is flat. Amnon Yekutieli (BGU) Flatness and Completion 16 / 27

3. The Module of Decaying Functions Here is a sketch of the proof. (Details are in [Ye1, Corollary 2.9 and Theorem 3.4].) Given any finitely generated � A -module � M , we can consider the module of decaying functions F dec ( Z , � M ) . By direct calculation I prove the following facts: ◮ The functor M �→ F dec ( Z , � M ) is exact on Mod f A . ◮ For any M ∈ Mod f A the canonical homomorphism M ⊗ A F dec ( Z , � A ) → F dec ( Z , � M ) is bijective. Amnon Yekutieli (BGU) Flatness and Completion 17 / 27

3. The Module of Decaying Functions Here is a sketch of the proof. (Details are in [Ye1, Corollary 2.9 and Theorem 3.4].) Given any finitely generated � A -module � M , we can consider the module of decaying functions F dec ( Z , � M ) . By direct calculation I prove the following facts: ◮ The functor M �→ F dec ( Z , � M ) is exact on Mod f A . ◮ For any M ∈ Mod f A the canonical homomorphism M ⊗ A F dec ( Z , � A ) → F dec ( Z , � M ) is bijective. Amnon Yekutieli (BGU) Flatness and Completion 17 / 27

3. The Module of Decaying Functions Here is a sketch of the proof. (Details are in [Ye1, Corollary 2.9 and Theorem 3.4].) Given any finitely generated � A -module � M , we can consider the module of decaying functions F dec ( Z , � M ) . By direct calculation I prove the following facts: ◮ The functor M �→ F dec ( Z , � M ) is exact on Mod f A . ◮ For any M ∈ Mod f A the canonical homomorphism M ⊗ A F dec ( Z , � A ) → F dec ( Z , � M ) is bijective. Amnon Yekutieli (BGU) Flatness and Completion 17 / 27

3. The Module of Decaying Functions Here is a sketch of the proof. (Details are in [Ye1, Corollary 2.9 and Theorem 3.4].) Given any finitely generated � A -module � M , we can consider the module of decaying functions F dec ( Z , � M ) . By direct calculation I prove the following facts: ◮ The functor M �→ F dec ( Z , � M ) is exact on Mod f A . ◮ For any M ∈ Mod f A the canonical homomorphism M ⊗ A F dec ( Z , � A ) → F dec ( Z , � M ) is bijective. Amnon Yekutieli (BGU) Flatness and Completion 17 / 27

3. The Module of Decaying Functions Here is a sketch of the proof. (Details are in [Ye1, Corollary 2.9 and Theorem 3.4].) Given any finitely generated � A -module � M , we can consider the module of decaying functions F dec ( Z , � M ) . By direct calculation I prove the following facts: ◮ The functor M �→ F dec ( Z , � M ) is exact on Mod f A . ◮ For any M ∈ Mod f A the canonical homomorphism M ⊗ A F dec ( Z , � A ) → F dec ( Z , � M ) is bijective. Amnon Yekutieli (BGU) Flatness and Completion 17 / 27

3. The Module of Decaying Functions We see that the functor M �→ M ⊗ A F dec ( Z , � A ) is exact on Mod f A . Hence the A -module F dec ( Z , � A ) is flat. Another consequence is that for every i ∈ N the canonical homomorphism A i ⊗ A F dec ( Z , � A ) → F fin ( Z , A i ) is bijective. By Theorem 2.2 it follows that F dec ( Z , � A ) is a -adically complete. Amnon Yekutieli (BGU) Flatness and Completion 18 / 27

3. The Module of Decaying Functions We see that the functor M �→ M ⊗ A F dec ( Z , � A ) is exact on Mod f A . Hence the A -module F dec ( Z , � A ) is flat. Another consequence is that for every i ∈ N the canonical homomorphism A i ⊗ A F dec ( Z , � A ) → F fin ( Z , A i ) is bijective. By Theorem 2.2 it follows that F dec ( Z , � A ) is a -adically complete. Amnon Yekutieli (BGU) Flatness and Completion 18 / 27

3. The Module of Decaying Functions We see that the functor M �→ M ⊗ A F dec ( Z , � A ) is exact on Mod f A . Hence the A -module F dec ( Z , � A ) is flat. Another consequence is that for every i ∈ N the canonical homomorphism A i ⊗ A F dec ( Z , � A ) → F fin ( Z , A i ) is bijective. By Theorem 2.2 it follows that F dec ( Z , � A ) is a -adically complete. Amnon Yekutieli (BGU) Flatness and Completion 18 / 27

3. The Module of Decaying Functions We see that the functor M �→ M ⊗ A F dec ( Z , � A ) is exact on Mod f A . Hence the A -module F dec ( Z , � A ) is flat. Another consequence is that for every i ∈ N the canonical homomorphism A i ⊗ A F dec ( Z , � A ) → F fin ( Z , A i ) is bijective. By Theorem 2.2 it follows that F dec ( Z , � A ) is a -adically complete. Amnon Yekutieli (BGU) Flatness and Completion 18 / 27

3. The Module of Decaying Functions By combining Theorem 3.2 with Theorem 2.2 we can also deduce: Corollary 3.3. Let M be any A -module. Then its a -adic completion � M is a -adically complete. This important fact is actually not new; see the paper [Ma] or the book [St]. Amnon Yekutieli (BGU) Flatness and Completion 19 / 27

3. The Module of Decaying Functions By combining Theorem 3.2 with Theorem 2.2 we can also deduce: Corollary 3.3. Let M be any A -module. Then its a -adic completion � M is a -adically complete. This important fact is actually not new; see the paper [Ma] or the book [St]. Amnon Yekutieli (BGU) Flatness and Completion 19 / 27

3. The Module of Decaying Functions By combining Theorem 3.2 with Theorem 2.2 we can also deduce: Corollary 3.3. Let M be any A -module. Then its a -adic completion � M is a -adically complete. This important fact is actually not new; see the paper [Ma] or the book [St]. Amnon Yekutieli (BGU) Flatness and Completion 19 / 27

4. Adically Free Modules 4. Adically Free Modules Let Z be a set, M an a -adicaly complete A -module, and f : Z → M a function. Then there is a unique A -linear homomorphism φ : F dec ( Z , � A ) → M such that φ ( δ z ) = f ( z ) for every z ∈ Z . The formula is φ ( g ) = ∑ g ( z ) f ( z ) ∈ M z ∈ Z for g ∈ F dec ( Z , � A ) . Amnon Yekutieli (BGU) Flatness and Completion 20 / 27

4. Adically Free Modules 4. Adically Free Modules Let Z be a set, M an a -adicaly complete A -module, and f : Z → M a function. Then there is a unique A -linear homomorphism φ : F dec ( Z , � A ) → M such that φ ( δ z ) = f ( z ) for every z ∈ Z . The formula is φ ( g ) = ∑ g ( z ) f ( z ) ∈ M z ∈ Z for g ∈ F dec ( Z , � A ) . Amnon Yekutieli (BGU) Flatness and Completion 20 / 27

4. Adically Free Modules 4. Adically Free Modules Let Z be a set, M an a -adicaly complete A -module, and f : Z → M a function. Then there is a unique A -linear homomorphism φ : F dec ( Z , � A ) → M such that φ ( δ z ) = f ( z ) for every z ∈ Z . The formula is φ ( g ) = ∑ g ( z ) f ( z ) ∈ M z ∈ Z for g ∈ F dec ( Z , � A ) . Amnon Yekutieli (BGU) Flatness and Completion 20 / 27

4. Adically Free Modules 4. Adically Free Modules Let Z be a set, M an a -adicaly complete A -module, and f : Z → M a function. Then there is a unique A -linear homomorphism φ : F dec ( Z , � A ) → M such that φ ( δ z ) = f ( z ) for every z ∈ Z . The formula is φ ( g ) = ∑ g ( z ) f ( z ) ∈ M z ∈ Z for g ∈ F dec ( Z , � A ) . Amnon Yekutieli (BGU) Flatness and Completion 20 / 27

4. Adically Free Modules The observation above justifies the next definition: Definition 4.1. An a -adically free module is an A -module P that is isomorphic to F dec ( Z , � A ) for some set Z . It is not hard to see that the cardinality of the set Z equals the rank of the free A 0 -module A 0 ⊗ A P . This cardinality is called the a -adic rank of P . Amnon Yekutieli (BGU) Flatness and Completion 21 / 27

4. Adically Free Modules The observation above justifies the next definition: Definition 4.1. An a -adically free module is an A -module P that is isomorphic to F dec ( Z , � A ) for some set Z . It is not hard to see that the cardinality of the set Z equals the rank of the free A 0 -module A 0 ⊗ A P . This cardinality is called the a -adic rank of P . Amnon Yekutieli (BGU) Flatness and Completion 21 / 27

4. Adically Free Modules The observation above justifies the next definition: Definition 4.1. An a -adically free module is an A -module P that is isomorphic to F dec ( Z , � A ) for some set Z . It is not hard to see that the cardinality of the set Z equals the rank of the free A 0 -module A 0 ⊗ A P . This cardinality is called the a -adic rank of P . Amnon Yekutieli (BGU) Flatness and Completion 21 / 27

5. Sheaves of Complete Flat Modules 5. Sheaves of Complete Flat Modules In this last section we assume that A is a complete noetherian local ring, with maximal ideal m . It is well known that a finitely generated A -module is flat iff it is free. Here is the corresponding infinite version (see [Ye1, Corollary 4.5]). Theorem 5.1. The following conditions are equivalent for an A -module M : (i) M is flat and m -adically complete. (ii) M is m -adically free. The principal case (when the ideal m is generated by a single regular element) was known before (see [CFT]). Amnon Yekutieli (BGU) Flatness and Completion 22 / 27

5. Sheaves of Complete Flat Modules 5. Sheaves of Complete Flat Modules In this last section we assume that A is a complete noetherian local ring, with maximal ideal m . It is well known that a finitely generated A -module is flat iff it is free. Here is the corresponding infinite version (see [Ye1, Corollary 4.5]). Theorem 5.1. The following conditions are equivalent for an A -module M : (i) M is flat and m -adically complete. (ii) M is m -adically free. The principal case (when the ideal m is generated by a single regular element) was known before (see [CFT]). Amnon Yekutieli (BGU) Flatness and Completion 22 / 27

5. Sheaves of Complete Flat Modules 5. Sheaves of Complete Flat Modules In this last section we assume that A is a complete noetherian local ring, with maximal ideal m . It is well known that a finitely generated A -module is flat iff it is free. Here is the corresponding infinite version (see [Ye1, Corollary 4.5]). Theorem 5.1. The following conditions are equivalent for an A -module M : (i) M is flat and m -adically complete. (ii) M is m -adically free. The principal case (when the ideal m is generated by a single regular element) was known before (see [CFT]). Amnon Yekutieli (BGU) Flatness and Completion 22 / 27