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

SLIDE 1

Flatness and Completion for Infinitely Generated Modules over Noetherian Rings

Amnon Yekutieli

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

Written 25 Oct 2010 Amnon Yekutieli (BGU) Flatness and Completion 1 / 27

SLIDE 2

Introduction

Introduction In this lecture I will discuss certain properties of the adic completion of infinitely generated modules over a noetherian commutative ring. Some of the results that will be mentioned are new, whereas others can be found in the literature. But I believe all are quite interesting. Here is an outline of the lecture:

1. Recalling the Completion of Finitely Generated Modules
2. Examples of Strange Behavior
3. The Module of Decaying Functions
4. Adically Free Modules
5. Sheaves of Complete Flat Modules

Amnon Yekutieli (BGU) Flatness and Completion 2 / 27

SLIDE 3

Introduction

Introduction In this lecture I will discuss certain properties of the adic completion of infinitely generated modules over a noetherian commutative ring. Some of the results that will be mentioned are new, whereas others can be found in the literature. But I believe all are quite interesting. Here is an outline of the lecture:

1. Recalling the Completion of Finitely Generated Modules
2. Examples of Strange Behavior
3. The Module of Decaying Functions
4. Adically Free Modules
5. Sheaves of Complete Flat Modules

Amnon Yekutieli (BGU) Flatness and Completion 2 / 27

SLIDE 4

Introduction

Introduction In this lecture I will discuss certain properties of the adic completion of infinitely generated modules over a noetherian commutative ring. Some of the results that will be mentioned are new, whereas others can be found in the literature. But I believe all are quite interesting. Here is an outline of the lecture:

1. Recalling the Completion of Finitely Generated Modules
2. Examples of Strange Behavior
3. The Module of Decaying Functions
4. Adically Free Modules
5. Sheaves of Complete Flat Modules

Amnon Yekutieli (BGU) Flatness and Completion 2 / 27

SLIDE 5

Introduction

Introduction In this lecture I will discuss certain properties of the adic completion of infinitely generated modules over a noetherian commutative ring. Some of the results that will be mentioned are new, whereas others can be found in the literature. But I believe all are quite interesting. Here is an outline of the lecture:

1. Recalling the Completion of Finitely Generated Modules
2. Examples of Strange Behavior
3. The Module of Decaying Functions
4. Adically Free Modules
5. Sheaves of Complete Flat Modules

Amnon Yekutieli (BGU) Flatness and Completion 2 / 27

SLIDE 6

Introduction

Introduction In this lecture I will discuss certain properties of the adic completion of infinitely generated modules over a noetherian commutative ring. Some of the results that will be mentioned are new, whereas others can be found in the literature. But I believe all are quite interesting. Here is an outline of the lecture:

1. Recalling the Completion of Finitely Generated Modules
2. Examples of Strange Behavior
3. The Module of Decaying Functions
4. Adically Free Modules
5. Sheaves of Complete Flat Modules

Amnon Yekutieli (BGU) Flatness and Completion 2 / 27

SLIDE 7

Introduction

Introduction In this lecture I will discuss certain properties of the adic completion of infinitely generated modules over a noetherian commutative ring. Some of the results that will be mentioned are new, whereas others can be found in the literature. But I believe all are quite interesting. Here is an outline of the lecture:

1. Recalling the Completion of Finitely Generated Modules
2. Examples of Strange Behavior
3. The Module of Decaying Functions
4. Adically Free Modules
5. Sheaves of Complete Flat Modules

Amnon Yekutieli (BGU) Flatness and Completion 2 / 27

SLIDE 8

Introduction

Introduction In this lecture I will discuss certain properties of the adic completion of infinitely generated modules over a noetherian commutative ring. Some of the results that will be mentioned are new, whereas others can be found in the literature. But I believe all are quite interesting. Here is an outline of the lecture:

1. Recalling the Completion of Finitely Generated Modules
2. Examples of Strange Behavior
3. The Module of Decaying Functions
4. Adically Free Modules
5. Sheaves of Complete Flat Modules

Amnon Yekutieli (BGU) Flatness and Completion 2 / 27

SLIDE 9

Introduction

Introduction In this lecture I will discuss certain properties of the adic completion of infinitely generated modules over a noetherian commutative ring. Some of the results that will be mentioned are new, whereas others can be found in the literature. But I believe all are quite interesting. Here is an outline of the lecture:

1. Recalling the Completion of Finitely Generated Modules
2. Examples of Strange Behavior
3. The Module of Decaying Functions
4. Adically Free Modules
5. Sheaves of Complete Flat Modules

Amnon Yekutieli (BGU) Flatness and Completion 2 / 27

SLIDE 10

Introduction

Introduction In this lecture I will discuss certain properties of the adic completion of infinitely generated modules over a noetherian commutative ring. Some of the results that will be mentioned are new, whereas others can be found in the literature. But I believe all are quite interesting. Here is an outline of the lecture:

1. Recalling the Completion of Finitely Generated Modules
2. Examples of Strange Behavior
3. The Module of Decaying Functions
4. Adically Free Modules
5. Sheaves of Complete Flat Modules

Amnon Yekutieli (BGU) Flatness and Completion 2 / 27

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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

M := lim

←i M/aiM.

The operation M → M is an additive functor from the category Mod A

f A-modules to itself.

There is a canonical homomorphism τM : M → M. The module M is said to be a-adically complete if τM is an isomorphism. Some texts would say that “M is separated and complete”.

Amnon Yekutieli (BGU) Flatness and Completion 3 / 27

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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

M := lim

←i M/aiM.

The operation M → M is an additive functor from the category Mod A

f A-modules to itself.

There is a canonical homomorphism τM : M → M. The module M is said to be a-adically complete if τM is an isomorphism. Some texts would say that “M is separated and complete”.

Amnon Yekutieli (BGU) Flatness and Completion 3 / 27

SLIDE 13

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

M := lim

←i M/aiM.

The operation M → M is an additive functor from the category Mod A

f A-modules to itself.

There is a canonical homomorphism τM : M → M. The module M is said to be a-adically complete if τM is an isomorphism. Some texts would say that “M is separated and complete”.

Amnon Yekutieli (BGU) Flatness and Completion 3 / 27

SLIDE 14

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

M := lim

←i M/aiM.

The operation M → M is an additive functor from the category Mod A

f A-modules to itself.

There is a canonical homomorphism τM : M → M. The module M is said to be a-adically complete if τM is an isomorphism. Some texts would say that “M is separated and complete”.

Amnon Yekutieli (BGU) Flatness and Completion 3 / 27

SLIDE 15

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

M := lim

←i M/aiM.

The operation M → M is an additive functor from the category Mod A

f A-modules to itself.

There is a canonical homomorphism τM : M → M. The module M is said to be a-adically complete if τM is an isomorphism. Some texts would say that “M is separated and complete”.

Amnon Yekutieli (BGU) Flatness and Completion 3 / 27

SLIDE 16

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

M := lim

←i M/aiM.

The operation M → M is an additive functor from the category Mod A

f A-modules to itself.

There is a canonical homomorphism τM : M → M. The module M is said to be a-adically complete if τM is an isomorphism. Some texts would say that “M is separated and complete”.

Amnon Yekutieli (BGU) Flatness and Completion 3 / 27

SLIDE 17

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

M := lim

←i M/aiM.

The operation M → M is an additive functor from the category Mod A

f A-modules to itself.

There is a canonical homomorphism τM : M → M. The module M is said to be a-adically complete if τM is an isomorphism. Some texts would say that “M is separated and complete”.

Amnon Yekutieli (BGU) Flatness and Completion 3 / 27

SLIDE 18

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 Modf A

f finitely generated modules.

◮ If M ∈ Modf 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

SLIDE 19

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 Modf A

f finitely generated modules.

◮ If M ∈ Modf 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

SLIDE 20

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 Modf A

f finitely generated modules.

◮ If M ∈ Modf 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

SLIDE 21

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 Modf A

f finitely generated modules.

◮ If M ∈ Modf 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

SLIDE 22

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 Modf A

f finitely generated modules.

◮ If M ∈ Modf 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

SLIDE 23

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 Modf A

f finitely generated modules.

◮ If M ∈ Modf 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

SLIDE 24

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

SLIDE 25

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

SLIDE 26

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

SLIDE 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

SLIDE 28

2. Examples of Strange Behavior

Example 2.1. Let K be a field, and let A := K[t1, t2, . . .], the ring of polynomials in countably many variables over K. Consider the maximal ideal a := (t1, t2, . . .) in A. Take the module M := A. Then the completion M is not an a-adically complete A-module. Namely the canonical homomorphism τ

M :

M → lim

←i

M/ai

M is not bijective.

Amnon Yekutieli (BGU) Flatness and Completion 6 / 27

SLIDE 29

2. Examples of Strange Behavior

Example 2.1. Let K be a field, and let A := K[t1, t2, . . .], the ring of polynomials in countably many variables over K. Consider the maximal ideal a := (t1, t2, . . .) in A. Take the module M := A. Then the completion M is not an a-adically complete A-module. Namely the canonical homomorphism τ

M :

M → lim

←i

M/ai

M is not bijective.

Amnon Yekutieli (BGU) Flatness and Completion 6 / 27

SLIDE 30

2. Examples of Strange Behavior

Example 2.1. Let K be a field, and let A := K[t1, t2, . . .], the ring of polynomials in countably many variables over K. Consider the maximal ideal a := (t1, t2, . . .) in A. Take the module M := A. Then the completion M is not an a-adically complete A-module. Namely the canonical homomorphism τ

M :

M → lim

←i

M/ai

M is not bijective.

Amnon Yekutieli (BGU) Flatness and Completion 6 / 27

SLIDE 31

2. Examples of Strange Behavior

Example 2.1. Let K be a field, and let A := K[t1, t2, . . .], the ring of polynomials in countably many variables over K. Consider the maximal ideal a := (t1, t2, . . .) in A. Take the module M := A. Then the completion M is not an a-adically complete A-module. Namely the canonical homomorphism τ

M :

M → lim

←i

M/ai

M is not bijective.

Amnon Yekutieli (BGU) Flatness and Completion 6 / 27

SLIDE 32

2. Examples of Strange Behavior

Example 2.1. Let K be a field, and let A := K[t1, t2, . . .], the ring of polynomials in countably many variables over K. Consider the maximal ideal a := (t1, t2, . . .) in A. Take the module M := A. Then the completion M is not an a-adically complete A-module. Namely the canonical homomorphism τ

M :

M → lim

←i

M/ai

M is not bijective.

Amnon Yekutieli (BGU) Flatness and Completion 6 / 27

SLIDE 33

2. Examples of Strange Behavior

Here is an explanation. First a criterion for completeness. For every i ∈ N let Ai := A/ai+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 idAi ⊗τM : Ai ⊗A M → Ai ⊗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

SLIDE 34

2. Examples of Strange Behavior

Here is an explanation. First a criterion for completeness. For every i ∈ N let Ai := A/ai+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 idAi ⊗τM : Ai ⊗A M → Ai ⊗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

SLIDE 35

2. Examples of Strange Behavior

Here is an explanation. First a criterion for completeness. For every i ∈ N let Ai := A/ai+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 idAi ⊗τM : Ai ⊗A M → Ai ⊗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

SLIDE 36

2. Examples of Strange Behavior

Here is an explanation. First a criterion for completeness. For every i ∈ N let Ai := A/ai+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 idAi ⊗τM : Ai ⊗A M → Ai ⊗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

SLIDE 37

2. Examples of Strange Behavior

Here is an explanation. First a criterion for completeness. For every i ∈ N let Ai := A/ai+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 idAi ⊗τM : Ai ⊗A M → Ai ⊗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

SLIDE 38

2. Examples of Strange Behavior

Here is an explanation. First a criterion for completeness. For every i ∈ N let Ai := A/ai+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 idAi ⊗τM : Ai ⊗A M → Ai ⊗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

SLIDE 39

2. Examples of Strange Behavior

Here is an explanation. First a criterion for completeness. For every i ∈ N let Ai := A/ai+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 idAi ⊗τM : Ai ⊗A M → Ai ⊗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

SLIDE 40

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

SLIDE 41

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

SLIDE 42

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

SLIDE 43

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

SLIDE 44

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 Modf 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

SLIDE 45

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 Modf 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

SLIDE 46

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 Modf 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

SLIDE 47

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 Modf 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

SLIDE 48

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 {δi}i∈N, and let Q be another copy of P. Define a homomorphism φ : P → Q by the formula φ(δi) := tiδi. It is easy to see that φ is injective, and hence there is an exact sequence 0 → P

φ

− → Q

ψ

− → M → 0, where M is the cokernel of φ. A calculation (see [Ye1, Example 3.20]) shows that the sequence 0 → P

φ

− → Q

ψ

− → M → 0 is not exact at Q.

Amnon Yekutieli (BGU) Flatness and Completion 10 / 27

SLIDE 49

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 {δi}i∈N, and let Q be another copy of P. Define a homomorphism φ : P → Q by the formula φ(δi) := tiδi. It is easy to see that φ is injective, and hence there is an exact sequence 0 → P

φ

− → Q

ψ

− → M → 0, where M is the cokernel of φ. A calculation (see [Ye1, Example 3.20]) shows that the sequence 0 → P

φ

− → Q

ψ

− → M → 0 is not exact at Q.

Amnon Yekutieli (BGU) Flatness and Completion 10 / 27

SLIDE 50

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 {δi}i∈N, and let Q be another copy of P. Define a homomorphism φ : P → Q by the formula φ(δi) := tiδi. It is easy to see that φ is injective, and hence there is an exact sequence 0 → P

φ

− → Q

ψ

− → M → 0, where M is the cokernel of φ. A calculation (see [Ye1, Example 3.20]) shows that the sequence 0 → P

φ

− → Q

ψ

− → M → 0 is not exact at Q.

Amnon Yekutieli (BGU) Flatness and Completion 10 / 27

SLIDE 51

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 {δi}i∈N, and let Q be another copy of P. Define a homomorphism φ : P → Q by the formula φ(δi) := tiδi. It is easy to see that φ is injective, and hence there is an exact sequence 0 → P

φ

− → Q

ψ

− → M → 0, where M is the cokernel of φ. A calculation (see [Ye1, Example 3.20]) shows that the sequence 0 → P

φ

− → Q

ψ

− → M → 0 is not exact at Q.

Amnon Yekutieli (BGU) Flatness and Completion 10 / 27

SLIDE 52

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 {δi}i∈N, and let Q be another copy of P. Define a homomorphism φ : P → Q by the formula φ(δi) := tiδi. It is easy to see that φ is injective, and hence there is an exact sequence 0 → P

φ

− → Q

ψ

− → M → 0, where M is the cokernel of φ. A calculation (see [Ye1, Example 3.20]) shows that the sequence 0 → P

φ

− → Q

ψ

− → M → 0 is not exact at Q.

Amnon Yekutieli (BGU) Flatness and Completion 10 / 27

SLIDE 53

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 {δi}i∈N, and let Q be another copy of P. Define a homomorphism φ : P → Q by the formula φ(δi) := tiδi. It is easy to see that φ is injective, and hence there is an exact sequence 0 → P

φ

− → Q

ψ

− → M → 0, where M is the cokernel of φ. A calculation (see [Ye1, Example 3.20]) shows that the sequence 0 → P

φ

− → Q

ψ

− → M → 0 is not exact at Q.

Amnon Yekutieli (BGU) Flatness and Completion 10 / 27

SLIDE 54

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

SLIDE 55

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

SLIDE 56

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. If P has finite rank, say P ∼ = Ar for some natural number r, then clearly

P ∼

= Ar.

Amnon Yekutieli (BGU) Flatness and Completion 12 / 27

SLIDE 57

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. If P has finite rank, say P ∼ = Ar for some natural number r, then clearly

P ∼

= Ar.

Amnon Yekutieli (BGU) Flatness and Completion 12 / 27

SLIDE 58

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. If P has finite rank, say P ∼ = Ar for some natural number r, then clearly

P ∼

= Ar.

Amnon Yekutieli (BGU) Flatness and Completion 12 / 27

SLIDE 59

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. If P has finite rank, say P ∼ = Ar for some natural number r, then clearly

P ∼

= Ar.

Amnon Yekutieli (BGU) Flatness and Completion 12 / 27

SLIDE 60

3. The Module of Decaying Functions

Now suppose P has infinite rank. Then P ∼ = Ffin(Z, A), where Z is some set (its cardinality being the rank of P), and Ffin(Z, A) denotes the set of finitely supported functions f : Z → A. Note that the free module Ffin(Z, A) comes equipped with a canonical basis: the collection of delta functions {δz}z∈Z. We are going to describe the completion P.

Amnon Yekutieli (BGU) Flatness and Completion 13 / 27

SLIDE 61

3. The Module of Decaying Functions

Now suppose P has infinite rank. Then P ∼ = Ffin(Z, A), where Z is some set (its cardinality being the rank of P), and Ffin(Z, A) denotes the set of finitely supported functions f : Z → A. Note that the free module Ffin(Z, A) comes equipped with a canonical basis: the collection of delta functions {δz}z∈Z. We are going to describe the completion P.

Amnon Yekutieli (BGU) Flatness and Completion 13 / 27

SLIDE 62

3. The Module of Decaying Functions

Now suppose P has infinite rank. Then P ∼ = Ffin(Z, A), where Z is some set (its cardinality being the rank of P), and Ffin(Z, A) denotes the set of finitely supported functions f : Z → A. Note that the free module Ffin(Z, A) comes equipped with a canonical basis: the collection of delta functions {δz}z∈Z. We are going to describe the completion P.

Amnon Yekutieli (BGU) Flatness and Completion 13 / 27

SLIDE 63

3. The Module of Decaying Functions

For any a ∈ A we denote by ord(a) its a-adic order; namely

rd(a) := sup {i ∈ N | a ∈ ai}.

Note that

rd(a) ∈ N ∪ {∞},

and ord(a) = ∞ iff a = 0. The a-adic metric on A is given by the formula dist(a1, a2) := ( 1

2)ord(a1−a2).

(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

SLIDE 64

3. The Module of Decaying Functions

For any a ∈ A we denote by ord(a) its a-adic order; namely

rd(a) := sup {i ∈ N | a ∈ ai}.

Note that

rd(a) ∈ N ∪ {∞},

and ord(a) = ∞ iff a = 0. The a-adic metric on A is given by the formula dist(a1, a2) := ( 1

2)ord(a1−a2).

(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

SLIDE 65

3. The Module of Decaying Functions

For any a ∈ A we denote by ord(a) its a-adic order; namely

rd(a) := sup {i ∈ N | a ∈ ai}.

Note that

rd(a) ∈ N ∪ {∞},

and ord(a) = ∞ iff a = 0. The a-adic metric on A is given by the formula dist(a1, a2) := ( 1

2)ord(a1−a2).

(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

SLIDE 66

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 Fdec(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 Fdec(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 Ffin(Z, A) ⊂ Fdec(Z, A) ⊂ F(Z, A).

Amnon Yekutieli (BGU) Flatness and Completion 15 / 27

SLIDE 67

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 Fdec(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 Fdec(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 Ffin(Z, A) ⊂ Fdec(Z, A) ⊂ F(Z, A).

Amnon Yekutieli (BGU) Flatness and Completion 15 / 27

SLIDE 68

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 Fdec(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 Fdec(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 Ffin(Z, A) ⊂ Fdec(Z, A) ⊂ F(Z, A).

Amnon Yekutieli (BGU) Flatness and Completion 15 / 27

SLIDE 69

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 Fdec(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 Fdec(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 Ffin(Z, A) ⊂ Fdec(Z, A) ⊂ F(Z, A).

Amnon Yekutieli (BGU) Flatness and Completion 15 / 27

SLIDE 70

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 Fdec(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 Fdec(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 Ffin(Z, A) ⊂ Fdec(Z, A) ⊂ F(Z, A).

Amnon Yekutieli (BGU) Flatness and Completion 15 / 27

SLIDE 71

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 Fdec(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 Fdec(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 Ffin(Z, A) ⊂ Fdec(Z, A) ⊂ F(Z, A).

Amnon Yekutieli (BGU) Flatness and Completion 15 / 27

SLIDE 72

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 Fdec(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 Fdec(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 Ffin(Z, A) ⊂ Fdec(Z, A) ⊂ F(Z, A).

Amnon Yekutieli (BGU) Flatness and Completion 15 / 27

SLIDE 73

3. The Module of Decaying Functions

The canonical homomorphism A → A induces a homomorphism Ffin(Z, A) → Fdec(Z, A). Theorem 3.2. Let Z be any set.

1. The A-module Fdec(Z,

A) is isomorphic to the a-adic completion of the A-module Ffin(Z, A).

2. The A-module Fdec(Z,

A) is a-adically complete.

3. The A-module Fdec(Z,

A) is flat.

Amnon Yekutieli (BGU) Flatness and Completion 16 / 27

SLIDE 74

3. The Module of Decaying Functions

The canonical homomorphism A → A induces a homomorphism Ffin(Z, A) → Fdec(Z, A). Theorem 3.2. Let Z be any set.

1. The A-module Fdec(Z,

A) is isomorphic to the a-adic completion of the A-module Ffin(Z, A).

2. The A-module Fdec(Z,

A) is a-adically complete.

3. The A-module Fdec(Z,

A) is flat.

Amnon Yekutieli (BGU) Flatness and Completion 16 / 27

SLIDE 75

3. The Module of Decaying Functions

The canonical homomorphism A → A induces a homomorphism Ffin(Z, A) → Fdec(Z, A). Theorem 3.2. Let Z be any set.

1. The A-module Fdec(Z,

A) is isomorphic to the a-adic completion of the A-module Ffin(Z, A).

2. The A-module Fdec(Z,

A) is a-adically complete.

3. The A-module Fdec(Z,

A) is flat.

Amnon Yekutieli (BGU) Flatness and Completion 16 / 27

SLIDE 76

3. The Module of Decaying Functions

The canonical homomorphism A → A induces a homomorphism Ffin(Z, A) → Fdec(Z, A). Theorem 3.2. Let Z be any set.

1. The A-module Fdec(Z,

A) is isomorphic to the a-adic completion of the A-module Ffin(Z, A).

2. The A-module Fdec(Z,

A) is a-adically complete.

3. The A-module Fdec(Z,

A) is flat.

Amnon Yekutieli (BGU) Flatness and Completion 16 / 27

SLIDE 77

3. The Module of Decaying Functions

The canonical homomorphism A → A induces a homomorphism Ffin(Z, A) → Fdec(Z, A). Theorem 3.2. Let Z be any set.

1. The A-module Fdec(Z,

A) is isomorphic to the a-adic completion of the A-module Ffin(Z, A).

2. The A-module Fdec(Z,

A) is a-adically complete.

3. The A-module Fdec(Z,

A) is flat.

Amnon Yekutieli (BGU) Flatness and Completion 16 / 27

SLIDE 78

3. The Module of Decaying Functions

The canonical homomorphism A → A induces a homomorphism Ffin(Z, A) → Fdec(Z, A). Theorem 3.2. Let Z be any set.

1. The A-module Fdec(Z,

A) is isomorphic to the a-adic completion of the A-module Ffin(Z, A).

2. The A-module Fdec(Z,

A) is a-adically complete.

3. The A-module Fdec(Z,

A) is flat.

Amnon Yekutieli (BGU) Flatness and Completion 16 / 27

SLIDE 79

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 Fdec(Z, M). By direct calculation I prove the following facts:

◮ The functor M → Fdec(Z,

M) is exact on Modf A.

◮ For any M ∈ Modf A the canonical homomorphism

M ⊗A Fdec(Z, A) → Fdec(Z, M) is bijective.

Amnon Yekutieli (BGU) Flatness and Completion 17 / 27

SLIDE 80

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 Fdec(Z, M). By direct calculation I prove the following facts:

◮ The functor M → Fdec(Z,

M) is exact on Modf A.

◮ For any M ∈ Modf A the canonical homomorphism

M ⊗A Fdec(Z, A) → Fdec(Z, M) is bijective.

Amnon Yekutieli (BGU) Flatness and Completion 17 / 27

SLIDE 81

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 Fdec(Z, M). By direct calculation I prove the following facts:

◮ The functor M → Fdec(Z,

M) is exact on Modf A.

◮ For any M ∈ Modf A the canonical homomorphism

M ⊗A Fdec(Z, A) → Fdec(Z, M) is bijective.

Amnon Yekutieli (BGU) Flatness and Completion 17 / 27

SLIDE 82

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 Fdec(Z, M). By direct calculation I prove the following facts:

◮ The functor M → Fdec(Z,

M) is exact on Modf A.

◮ For any M ∈ Modf A the canonical homomorphism

M ⊗A Fdec(Z, A) → Fdec(Z, M) is bijective.

Amnon Yekutieli (BGU) Flatness and Completion 17 / 27

SLIDE 83

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 Fdec(Z, M). By direct calculation I prove the following facts:

◮ The functor M → Fdec(Z,

M) is exact on Modf A.

◮ For any M ∈ Modf A the canonical homomorphism

M ⊗A Fdec(Z, A) → Fdec(Z, M) is bijective.

Amnon Yekutieli (BGU) Flatness and Completion 17 / 27

SLIDE 84

3. The Module of Decaying Functions

We see that the functor M → M ⊗A Fdec(Z, A) is exact on Modf A. Hence the A-module Fdec(Z, A) is flat. Another consequence is that for every i ∈ N the canonical homomorphism Ai ⊗A Fdec(Z, A) → Ffin(Z, Ai) is bijective. By Theorem 2.2 it follows that Fdec(Z, A) is a-adically complete.

Amnon Yekutieli (BGU) Flatness and Completion 18 / 27

SLIDE 85

3. The Module of Decaying Functions

We see that the functor M → M ⊗A Fdec(Z, A) is exact on Modf A. Hence the A-module Fdec(Z, A) is flat. Another consequence is that for every i ∈ N the canonical homomorphism Ai ⊗A Fdec(Z, A) → Ffin(Z, Ai) is bijective. By Theorem 2.2 it follows that Fdec(Z, A) is a-adically complete.

Amnon Yekutieli (BGU) Flatness and Completion 18 / 27

SLIDE 86

3. The Module of Decaying Functions

We see that the functor M → M ⊗A Fdec(Z, A) is exact on Modf A. Hence the A-module Fdec(Z, A) is flat. Another consequence is that for every i ∈ N the canonical homomorphism Ai ⊗A Fdec(Z, A) → Ffin(Z, Ai) is bijective. By Theorem 2.2 it follows that Fdec(Z, A) is a-adically complete.

Amnon Yekutieli (BGU) Flatness and Completion 18 / 27

SLIDE 87

3. The Module of Decaying Functions

We see that the functor M → M ⊗A Fdec(Z, A) is exact on Modf A. Hence the A-module Fdec(Z, A) is flat. Another consequence is that for every i ∈ N the canonical homomorphism Ai ⊗A Fdec(Z, A) → Ffin(Z, Ai) is bijective. By Theorem 2.2 it follows that Fdec(Z, A) is a-adically complete.

Amnon Yekutieli (BGU) Flatness and Completion 18 / 27

SLIDE 88

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

SLIDE 89

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

SLIDE 90

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

SLIDE 91

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 : Fdec(Z, A) → M such that φ(δz) = f(z) for every z ∈ Z. The formula is φ(g) = ∑

z∈Z

g(z)f(z) ∈ M for g ∈ Fdec(Z, A).

Amnon Yekutieli (BGU) Flatness and Completion 20 / 27

SLIDE 92

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 : Fdec(Z, A) → M such that φ(δz) = f(z) for every z ∈ Z. The formula is φ(g) = ∑

z∈Z

g(z)f(z) ∈ M for g ∈ Fdec(Z, A).

Amnon Yekutieli (BGU) Flatness and Completion 20 / 27

SLIDE 93

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 : Fdec(Z, A) → M such that φ(δz) = f(z) for every z ∈ Z. The formula is φ(g) = ∑

z∈Z

g(z)f(z) ∈ M for g ∈ Fdec(Z, A).

Amnon Yekutieli (BGU) Flatness and Completion 20 / 27

SLIDE 94

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 : Fdec(Z, A) → M such that φ(δz) = f(z) for every z ∈ Z. The formula is φ(g) = ∑

z∈Z

g(z)f(z) ∈ M for g ∈ Fdec(Z, A).

Amnon Yekutieli (BGU) Flatness and Completion 20 / 27

SLIDE 95

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 Fdec(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 A0-module A0 ⊗A P. This cardinality is called the a-adic rank

f P.

Amnon Yekutieli (BGU) Flatness and Completion 21 / 27

SLIDE 96

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 Fdec(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 A0-module A0 ⊗A P. This cardinality is called the a-adic rank

f P.

Amnon Yekutieli (BGU) Flatness and Completion 21 / 27

SLIDE 97

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 Fdec(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 A0-module A0 ⊗A P. This cardinality is called the a-adic rank

f P.

Amnon Yekutieli (BGU) Flatness and Completion 21 / 27

SLIDE 98

5. Sheaves of Complete Flat Modules
5. Sheaves of Complete Flat Modules