nLab
localization of abelian groups

Contents

Context

Group Theory

Homological algebra

homological algebra

(also nonabelian homological algebra)

Introduction

Context

Basic definitions

Stable homotopy theory notions

Constructions

Lemmas

diagram chasing

Homology theories

Theorems

Contents

Idea

The general concept of localization applied to the (derived) category of abelian groups yield the concept of localization of abelian groups.

The two main examples are

  1. classical localization at/aways from primes;

  2. completion at a prime

at prime numbers pp. Here “classical pp-localization” is localization at the morphism 0/p0 \to \mathbb{Z}/p\mathbb{Z}, while pp-completion is localization at the morphism 0[p 1]0 \to \mathbb{Z}[p^{-1}].

Definition

Remark

Recall that Ext-groups Ext (A,B)\Ext^\bullet(A,B) between abelian groups A,BA, B \in Ab are concentrated in degrees 0 and 1 (prop.). Since

Ext 0(A,B)Hom(A,B) Ext^0(A,B) \simeq Hom(A,B)

is the plain hom-functor, this means that there is only one possibly non-vanishing Ext-group Ext 1Ext^1, therefore often abbreviated to just “ExtExt”:

Ext(A,B)Ext 1(A,B). Ext(A,B) \coloneqq Ext^1(A,B) \,.
Definition

Let KK be an abelian group.

Then an abelian group AA is called KK-local if all the Ext-groups from KK to AA vanish:

Ext (K,A)0 Ext^\bullet(K,A) \simeq 0

hence equivalently (remark ) if

Hom(K,A)0andExt(K,A)0. Hom(K,A) \simeq 0 \;\;\;\;\; and \;\;\;\;\; Ext(K,A) \simeq 0 \,.

A homomorphism of abelian groups f:BCf \colon B \longrightarrow C is called KK-local if for all KK-local groups AA the function

Hom(f,A):Hom(B,A)Hom(A,A) Hom(f,A) \;\colon\; Hom(B,A) \longrightarrow Hom(A,A)

is a bijection.

(Beware that here it would seem more natural to use Ext Ext^\bullet instead of HomHom, but we do use HomHom. See (Neisendorfer 08, remark 3.2).

A homomorphism of abelian groups

η:AL KA \eta \;\colon\; A \longrightarrow L_K A

is called a KK-localization if

  1. L KAL_K A is KK-local;

  2. η\eta is a KK-local morphism.

We now discuss two classes of examples of localization of abelian groups

  1. Classical localization at/away from primes;

  2. Formal completion at primes.

Classical localization at/away from primes

For nn \in \mathbb{N}, write /n\mathbb{Z}/n\mathbb{Z} for the cyclic group of order nn.

Lemma

For nn \in \mathbb{N} and AAbA \in Ab any abelian group, then

  1. the hom-group out of /n\mathbb{Z}/n\mathbb{Z} into AA is the nn-torsion subgroup of AA

    Hom(/n,A){aA|pa=0} Hom(\mathbb{Z}/n\mathbb{Z}, A) \simeq \{ a \in A \;\vert\; p \cdot a = 0 \}
  2. the first Ext-group out of /n\mathbb{Z}/n\mathbb{Z} into AA is

    Ext 1(/n,A)A/nA. Ext^1(\mathbb{Z}/n\mathbb{Z},A) \simeq A/n A \,.
Proof

Regarding the first item: Since /p\mathbb{Z}/p\mathbb{Z} is generated by its element 1 a morphism /pA\mathbb{Z}/p\mathbb{Z} \to A is fixed by the image aa of this element, and the only relation on 1 in /p\mathbb{Z}/p\mathbb{Z} is that p1=0p \cdot 1 = 0.

Regarding the second item:

Consider the canonical free resolution

0p()/p0. 0 \to \mathbb{Z} \overset{p \cdot (-)}{\longrightarrow} \mathbb{Z} \longrightarrow \mathbb{Z}/p\mathbb{Z} \to 0 \,.

By the general discusson of derived functors in homological algebra this exhibits the Ext-group in degree 1 as part of the following short exact sequence

0Hom(,A)Hom(n(),A)Hom(,A)Ext 1(/n,A)0, 0 \to Hom(\mathbb{Z},A) \overset{Hom(n\cdot(-),A)}{\longrightarrow} Hom(\mathbb{Z}, A) \longrightarrow Ext^1(\mathbb{Z}/n\mathbb{Z},A) \to 0 \,,

where the morphism on the left is equivalently An()AA \overset{n \cdot (-)}{\to} A.

Example

An abelian group AA is /p\mathbb{Z}/p\mathbb{Z}-local precisely if the operation

p():AA p \cdot (-) \;\colon\; A \longrightarrow A

of multiplication by pp is an isomorphism, hence if “pp is invertible in AA”.

Proof

By the first item of lemma we have

Hom(/p,A){aA|pa=0} Hom(\mathbb{Z}/p\mathbb{Z}, A) \simeq \{ a \in A \;\vert\; p \cdot a = 0 \}

By the second item of lemma we have

Ext 1(/p,A)A/pA. Ext^1(\mathbb{Z}/p\mathbb{Z},A) \simeq A/p A \,.

Hence by def. AA is /p\mathbb{Z}/p\mathbb{Z}-local precisely if

{aA|pa=0}0 \{ a \in A \;\vert\; p \cdot a = 0 \} \simeq 0

and if

A/pA0. A / p A \simeq 0 \,.

Both these conditions are equivalent to multiplication by pp being invertible.

Definition

For JJ \subset \mathbb{N} a set of prime numbers, consider the direct sum pJ/p\underset{p \in J}{\oplus} \mathbb{Z}/p\mathbb{Z} of cyclic groups of order pp.

The operation of pJ/p\underset{p \in J}{\oplus} \mathbb{Z}/p\mathbb{Z}-localization of abelian groups according to def. is called inverting the primes in JJ.

Specifically

  1. for J={p}J = \{p\} a single prime then /p\mathbb{Z}/p\mathbb{Z}-localization is called localization away from pp;

  2. for JJ the set of all primes except pp then pJ/p\underset{p \in J}{\oplus} \mathbb{Z}/p\mathbb{Z}-localization is called localization at pp;

  3. for JJ the set of all primes, then pJ/p\underset{p \in J}{\oplus} \mathbb{Z}/p\mathbb{Z}-localizaton is called rationalization..

Definition

For JPrimesJ \subset Primes \subset \mathbb{N} a set of prime numbers, then

[J 1] \mathbb{Z}[J^{-1}] \hookrightarrow \mathbb{Q}

denotes the subgroup of the rational numbers on those elements which have an expression as a fraction of natural numbers with denominator a product of elements in JJ.

This is the abelian group underlying the localization of a commutative ring of the ring of integers at the set JJ of primes.

If J=Primes{p}J = Primes - \{p\} is the set of all primes except pp one also writes

(p)[Primes{p}]. \mathbb{Z}_{(p)} \coloneqq \mathbb{Z}[Primes - \{p\}] \,.

Notice the parenthesis, to distinguish from the notation p\mathbb{Z}_{p} for the p-adic integers (def. below).

Remark

The terminology in def. is motivated by the following perspective of arithmetic geometry:

Generally for RR a commutative ring, then an RR-module is equivalently a quasicoherent sheaf on the spectrum of the ring Spec(R)Spec(R).

In the present case R=R = \mathbb{Z} is the integers and abelian groups are identified with \mathbb{Z}-modules. Hence we may think of an abelian group AA equivalently as a quasicoherent sheaf on Spec(Z).

The “ring of functions” on Spec(Z) is the integers, and a point in Spec()Spec(\mathbb{Z}) is labeled by a prime number pp, thought of as generating the ideal of functions on Spec(Z) which vanish at that point. The residue field at that point is 𝔽 p=/p\mathbb{F}_p = \mathbb{Z}/p\mathbb{Z}.

Inverting a prime means forcing pp to become invertible, which, by this characterization, it is precisely away from that point which it labels. The localization of an abelian group at /p\mathbb{Z}/p\mathbb{Z} hence corresponds to the restriction of the corresponding quasicoherent sheaf over Spec()Spec(\mathbb{Z}) to the complement of the point labeled by pp.

Similarly localization at pp is localization away from all points except pp.

See also at function field analogy for more on this.

Proposition

For JJ \subset \mathbb{N} a set of prime numbers, a homomorphism of abelian groups f:AlookrightarrowBf \;\colon\; A \lookrightarrow B is local (def. ) with respect to pJ/p\underset{p \in J}{\oplus} \mathbb{Z}/p\mathbb{Z} (def. ) if under tensor product of abelian groups with [J 1]\mathbb{Z}[J^{-1}] (def. ) it becomes an isomorphism

f[J 1]:X[J 1]Y[J 1]. f \otimes \mathbb{Z}[J^{-1}] \;\colon\; X \otimes \mathbb{Z}[J^{-1}] \overset{\simeq}{\longrightarrow} Y \otimes \mathbb{Z}[J^{-1}] \,.

Moreover, for AA any abelian group then its pJ/p\underset{p \in J}{\oplus} \mathbb{Z}/p\mathbb{Z}-localization exists and is given by the canonical projection morphism

AA[J 1]. A \longrightarrow A \otimes \mathbb{Z}[J^{-1}] \,.

(e.g. Neisendorfer 08, theorem 4.2)

Formal completion at primes

We have seen above in remark that classical localization of abelian groups at a prime number is geometrically interpreted as restricting a quasicoherent sheaf over Spec(Z) to a single point, the point that is labeled by that prime number.

Alternatively one may restrict to the “infinitesimal neighbourhood” of such a point. Technically this is called the formal neighbourhood, because its ring of functions is, by definition, the ring of formal power series (regarded as an adic ring or pro-ring). The corresponding operation on abelian groups is accordingly called formal completion or adic completion or just completion, for short, of abelian groups.

It turns out that if the abelian group is finitely generated then the operation of p-completion coincides with an operation of localization in the sense of def. , namely localization at the p-primary component (p )\mathbb{Z}(p^\infty) of the group /\mathbb{Q}/\mathbb{Z} (def. below). On the one hand this equivalence is useful for deducing some key properties of p-completion, this we discuss below. On the other hand this situation is a shadow of the relation between localization of spectra and nilpotent completion of spectra, which is important for characterizing the convergence properties of Adams spectral sequences.

Definition

For pp a prime number, then the p-adic completion of an abelian group AA is the abelian group given by the limit

A p lim(A/p 3AA/p 2AA/pA), A^\wedge_p \coloneqq \underset{\longleftarrow}{\lim} \left( \cdots \longrightarrow A / p^3 A \longrightarrow A / p^2 A \longrightarrow A/p A \right) \,,

where the morphisms are the evident quotient morphisms. With these understood we often write

A p lim nA/p nA A^\wedge_p \coloneqq \underset{\longleftarrow}{\lim}_n A/p^n A

for short. Notice that here the indexing starts at n=1n = 1.

Example

The p-adic completion (def. ) of the integers \mathbb{Z} is called the p-adic integers, often written

p p lim n/p n, \mathbb{Z}_p \coloneqq \mathbb{Z}^\wedge_p \coloneqq \underset{\longleftarrow}{\lim}_n \mathbb{Z}/p^n \mathbb{Z} \,,

which is the original example that gives the general concept its name.

With respect to the canonical ring-structure on the integers, of course pp \mathbb{Z} is a prime ideal.

Compare this to the ring 𝒪 \mathcal{O}_{\mathbb{C}} of holomorphic functions on the complex plane. For xx \in \mathbb{C} any point, it contains the prime ideal generated by (zx)(z-x) (for zz the canonical coordinate function on 𝕫\mathbb{z}). The formal power series ring [[(z.x)]]\mathbb{C}[ [(z.x)] ] is the adic completion of 𝒪 \mathcal{O}_{\mathbb{C}} at this ideal. It has the interpretation of functions defined on a formal neighbourhood of XX in \mathbb{C}.

Analogously, the p-adic integers p\mathbb{Z}_p may be thought of as the functions defined on a formal neighbourhood of the point labeled by pp in Spec(Z).

Lemma

There is a short exact sequence

0 pp() p/p0. 0 \to \mathbb{Z}_p \overset{p \cdot (-)}{\longrightarrow} \mathbb{Z}_p \longrightarrow \mathbb{Z}/p\mathbb{Z} \to 0 \,.
Proof

Consider the following commuting diagram

/p 3 p() /p 4 /p /p 2 p() /p 3 /p /p p() /p 2 /p 0 /p /p. \array{ \vdots && \vdots && \vdots \\ \downarrow && \downarrow && \downarrow \\ \mathbb{Z}/p^3\mathbb{Z} &\overset{p\cdot (-)}{\longrightarrow}& \mathbb{Z}/p^4 \mathbb{Z} &\longrightarrow& \mathbb{Z}/p\mathbb{Z} \\ \downarrow && \downarrow && \downarrow \\ \mathbb{Z}/p^2\mathbb{Z} &\overset{p\cdot (-)}{\longrightarrow}& \mathbb{Z}/p^3 \mathbb{Z} &\longrightarrow& \mathbb{Z}/p\mathbb{Z} \\ \downarrow && \downarrow && \downarrow \\ \mathbb{Z}/p\mathbb{Z} &\overset{p\cdot (-)}{\longrightarrow}& \mathbb{Z}/p^2 \mathbb{Z} &\longrightarrow& \mathbb{Z}/p\mathbb{Z} \\ \downarrow && \downarrow && \downarrow \\ 0 &\longrightarrow& \mathbb{Z}/p\mathbb{Z} &\longrightarrow& \mathbb{Z}/p\mathbb{Z} } \,.

Each horizontal sequence is exact. Taking the limit over the vertical sequences yields the sequence in question. Since limits commute over limits, the result follows.

We now consider a concept of pp-completion that is in general different from def. , but turns out to coincide with it in finitely generated abelian groups.

Definition

For pp a prime number, write

[1/p]lim(p()p()) \mathbb{Z}[1/p] \coloneqq \underset{\longrightarrow}{\lim} \left( \mathbb{Z} \overset{p \cdot (-)}{\longrightarrow} \mathbb{Z} \overset{p \cdot (-)}{\longrightarrow} \mathbb{Z} \overset{}{\longrightarrow} \cdots \right)

for the colimit (in Ab) over iterative applications of multiplication by pp on the integers.

This is the abelian group generated by formal expressions 1p k\frac{1}{p^k} for kk \in \mathbb{N}, subject to the relations

(pn)1p k+1=n1p k. (p \cdot n) \frac{1}{p^{k+1}} = n \frac{1}{p^k} \,.

Equivalently it is the abelian group underlying the ring localization [1/p]\mathbb{Z}[1/p].

Definition

For pp a prime number, then localization of abelian groups (def. ) at [1/p]\mathbb{Z}[1/p] (def. ) is called pp-completion of abelian groups.

Lemma

An abelian group AA is pp-complete according to def. precisely if both the limit as well as the lim^1 of the sequence

ApApApA \cdots \overset{}{\longrightarrow} A \overset{p}{\longrightarrow} A \overset{p}{\longrightarrow} A \overset{p}{\longrightarrow} A

vanishes:

lim(ApApApA)0 \underset{\longleftarrow}{\lim} \left( \cdots \overset{}{\longrightarrow} A \overset{p}{\longrightarrow} A \overset{p}{\longrightarrow} A \overset{p}{\longrightarrow} A \right) \simeq 0

and

lim 1(ApApApA)0. \underset{\longleftarrow}{\lim}^1 \left( \cdots \overset{}{\longrightarrow} A \overset{p}{\longrightarrow} A \overset{p}{\longrightarrow} A \overset{p}{\longrightarrow} A \right) \simeq 0 \,.
Proof

By def. the group AA is [1/p]\mathbb{Z}[1/p]-local precisely if

Hom([1/p],A)0andExt 1([1/p],A)0. Hom(\mathbb{Z}[1/p], A) \simeq 0 \;\;\;\;\;\;\; and \;\;\;\;\;\;\; Ext^1(\mathbb{Z}[1/p], A) \simeq 0 \,.

Now use the colimit definition [1/p]=lim n\mathbb{Z}[1/p] = \underset{\longrightarrow}{\lim}_n \mathbb{Z} (def. ) and the fact that the hom-functor sends colimits in the first argument to limits to deduce that

Hom([1/p],A) =Hom(lim n,A) lim nHom(,A) lim nA. \begin{aligned} Hom(\mathbb{Z}[1/p], A) & = Hom( \underset{\longrightarrow}{\lim}_n \mathbb{Z}, A ) \\ & \simeq \underset{\longleftarrow}{\lim}_n Hom(\mathbb{Z},A) \\ & \simeq \underset{\longleftarrow}{\lim}_n A \end{aligned} \,.

This yields the first statement. For the second, use that for every cotower over abelian groups B B_\bullet there is a short exact sequence of the form

0lim n 1Hom(B n,A)Ext 1(lim nB n,A)lim nExt 1(B n,A)0 0 \to \underset{\longleftarrow}{\lim}^1_n Hom(B_n, A) \longrightarrow Ext^1( \underset{\longrightarrow}{\lim}_n B_n, A ) \longrightarrow \underset{\longleftarrow}{\lim}_n Ext^1( B_n, A) \to 0

(by this lemma).

In the present case all B nB_n \simeq \mathbb{Z}, which is a free abelian group, hence a projective object, so that all the Ext-groups out of it vannish. Hence by exactness there is an isomorphism

Ext 1(lim n,A)lim n 1Hom(,A)lim n 1A. Ext^1( \underset{\longrightarrow}{\lim}_n \mathbb{Z}, A ) \simeq \underset{\longleftarrow}{\lim}^1_n Hom(\mathbb{Z}, A) \simeq \underset{\longleftarrow}{\lim}^1_n A \,.

This gives the second statement.

Example

For pp a prime number, the p-primary cyclic groups of the form /p n\mathbb{Z}/p^n \mathbb{Z} are pp-complete (def. ).

Proof

By lemma we need to check that

lim(p/p np/p np/p n)0 \underset{\longleftarrow}{\lim} \left( \cdots \overset{p}{\longrightarrow} \mathbb{Z}/p^n \mathbb{Z} \overset{p}{\longrightarrow} \mathbb{Z}/p^n \mathbb{Z} \overset{p}{\longrightarrow} \mathbb{Z}/p^n \mathbb{Z} \right) \simeq 0

and

lim 1(p/p np/p np/p n)0. \underset{\longleftarrow}{\lim}^1 \left( \cdots \overset{p}{\longrightarrow} \mathbb{Z}/p^n \mathbb{Z} \overset{p}{\longrightarrow} \mathbb{Z}/p^n \mathbb{Z} \overset{p}{\longrightarrow} \mathbb{Z}/p^n \mathbb{Z} \right) \simeq 0 \,.

For the first statement observe that nn consecutive stages of the tower compose to the zero morphism. First of all this directly implies that the limit vanishes, secondly it means that the tower satisfies the Mittag-Leffler condition (def.) and this implies that the lim 1\lim^1 also vanishes (prop.).

Definition

For pp a prime number, write

(p )[1/p]/ \mathbb{Z}(p^\infty) \coloneqq \mathbb{Z}[1/p]/\mathbb{Z}

(the p-primary part of /\mathbb{Q}/\mathbb{Z}), where [1/p]=lim(pp)\mathbb{Z}[1/p] = \underset{\longrightarrow}{\lim}(\mathbb{Z}\overset{p}{\to} \mathbb{Z} \overset{p}{\to} \mathbb{Z} \to \cdots ) from def. .

Since colimits commute over each other, this is equivalently

(p )lim(0/p/p 2). \mathbb{Z}(p^\infty) \simeq \underset{\longrightarrow}{\lim} ( 0 \hookrightarrow \mathbb{Z}/p\mathbb{Z} \hookrightarrow \mathbb{Z}/p^2 \mathbb{Z} \hookrightarrow \cdots ) \,.
Theorem

For pp a prime number, the [1/p]\mathbb{Z}[1/p]-localization

AL [1/p]A A \longrightarrow L_{\mathbb{Z}[1/p]} A

of an abelian group AA (def. , def. ), hence the pp-completion of AA according to def. , is given by the morphism

AExt 1((p ),A) A \longrightarrow Ext^1( \mathbb{Z}(p^\infty), A )

into the first Ext-group into AA out of (p )\mathbb{Z}(p^\infty) (def. ), which appears as the first connecting homomorphism δ\delta in the long exact sequence of Ext-groups

0Hom((p ),A)Hom([1/p],A)Hom(,A)δ)Ext 1((p ),A). 0 \to Hom(\mathbb{Z}(p^\infty),A) \longrightarrow Hom(\mathbb{Z}[1/p],A) \longrightarrow Hom(\mathbb{Z},A) \overset{\delta)}{\longrightarrow} Ext^1(\mathbb{Z}(p^\infty), A) \to \cdots \,.

that is induced (via this prop.) from the defining short exact sequence

0[1/p](p )0 0 \to \mathbb{Z} \longrightarrow \mathbb{Z}[1/p] \longrightarrow \mathbb{Z}(p^\infty) \to 0

(def. ).

e.g. (Neisendorfer 08, p. 16)

Proposition

If AA is a finitely generated abelian group, then its pp-completion (def. , for any prime number pp) is equivalently its p-adic completion (def. )

[1/p]AA p . \mathbb{Z}[1/p] A \simeq A^\wedge_p \,.
Proof

By theorem the pp-completion is Ext 1((p ),A)Ext^1(\mathbb{Z}(p^\infty),A). By def. there is a colimit

(p )=lim(/p/p 2/p 3). \mathbb{Z}(p^\infty) = \underset{\longrightarrow}{\lim} \left( \mathbb{Z}/p\mathbb{Z} \to \mathbb{Z}/p^2 \mathbb{Z} \to \mathbb{Z}/p^3 \mathbb{Z} \to \cdots \right) \,.

Together this implies, by this lemma, that there is a short exact sequence of the form

0lim 1Hom(/p n,A)X p lim nExt 1(/p n,A)0. 0 \to \underset{\longleftarrow}{\lim}^1 Hom(\mathbb{Z}/p^n \mathbb{Z},A) \longrightarrow X^\wedge_p \longrightarrow \underset{\longleftarrow}{\lim}_n Ext^1(\mathbb{Z}/p^n \mathbb{Z}, A) \to 0 \,.

By lemma the lim^1 on the left is over the p np^n-torsion subgroups of AA, as nn ranges. By the assumption that AA is finitely generated, there is a maximum nn such that all torsion elements in AA are annihilated by p np^n. This means that the Mittag-Leffler condition (def.) is satisfied by the tower of pp-torsion subgroups, and hence the lim^1-term vanishes (prop.).

Therefore by exactness of the above sequence there is an isomorphism

L [1/p]X lim nExt 1(/p n,A) lim nA/p nA, \begin{aligned} L_{\mathbb{Z}[1/p]}X & \simeq \underset{\longleftarrow}{\lim}_n Ext^1(\mathbb{Z}/p^n \mathbb{Z}, A) \\ & \simeq \underset{\longleftarrow}{\lim}_n A/p^n A \end{aligned} \,,

where the second isomorphism is by lemma .

Proposition

If AA is a pp-divisible group in that Ap()AA \overset{p \cdot (-)}{\longrightarrow} A is an isomorphism, then its pp-completion (def. ) vanishes.

Proof

By theorem the localization morphism δ\delta sits in a long exact sequence of the form

0Hom((p ),A)Hom([1/p],A)ϕHom(,A)δExt 1((p ),A). 0 \to Hom(\mathbb{Z}(p^\infty),A) \longrightarrow Hom(\mathbb{Z}[1/p],A) \overset{\phi}{\longrightarrow} Hom(\mathbb{Z},A) \overset{\delta}{\longrightarrow} Ext^1(\mathbb{Z}(p^\infty), A) \to \cdots \,.

Here by def. and since the hom-functor takes colimits in the first argument to limits, the second term is equivalently the limit

Hom([1/p],A)lim(Ap()Ap()A). Hom(\mathbb{Z}[1/p],A) \simeq \underset{\longleftarrow}{\lim} \left( \cdots \to A \overset{p \cdot (-)}{\longrightarrow} A \overset{p \cdot (-)}{\longrightarrow} A \right) \,.

But by assumption all these morphisms p()p \cdot (-) that the limit is over are isomorphisms, so that the limit collapses to its first term, which means that the morphism ϕ\phi in the above sequence is an isomorphism. But by exactness of the sequence this means that δ=0\delta = 0.

Corollary

Let pp be a prime number. If AA is a finite abelian group, then its pp-completion (def. ) is equivalently its p-primary part.

Proof

By the fundamental theorem of finite abelian groups, AA is a finite direct sum

Ai/p i k i A \simeq \underset{i}{\oplus} \mathbb{Z}/p_i^{k_i}\mathbb{Z}

of cyclic groups of ordr p i k 1p_i^{k_1} for p ip_i prime numbers and k ik_i \in \mathbb{N} (thm.).

Since finite direct sums are equivalently products (biproducts: Ab is an additive category) this means that

Ext 1((p ),A)iExt 1((p ),/p i k 1). Ext^1( \mathbb{Z}(p^\infty), A ) \simeq \underset{i}{\prod} Ext^1( \mathbb{Z}(p^\infty), \mathbb{Z}/p_i^{k_1}\mathbb{Z} ) \,.

By theorem the iith factor here is the pp-completion of /p i k i\mathbb{Z}/p_i^{k_i}\mathbb{Z}, and since p()p \cdot(-) is an isomorphism on /p i k i\mathbb{Z}/p_i^{k_i}\mathbb{Z} if p ipp_i \neq p (because its kernel evidently vanishes), prop. says that in this case the factor vanishes, so that only the factors with p i=pp_i = p remain. On these however Ext 1((p ),)Ext^1(\mathbb{Z}(p^\infty),-) is the identity by example .

References

  • Joseph NeisendorferA Quick Trip through Localization, in Alpine perspectives on algebraic topology, Third Ariolla Conference 2008 (pdf)

  • Peter May, Kate Ponto, section 10.1 of More concise algebraic topology: Localization, completion, and model categories (pdf)

Last revised on March 28, 2017 at 04:40:18. See the history of this page for a list of all contributions to it.