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Baer's criterion

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Homological algebra

homological algebra

and

nonabelian homological algebra

Context

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Contents

Idea

Baer’s criterion is a criterion for detecting injective objects in a category of modules: injective modules.

Statemen

Let RR be a commutative ring and C=RC = R Mod the category of RR-modules.

Proposition

(Baer's criterion)

An object QRModQ \in R Mod is injective precisely if for II any left RR-ideal regarded as an RR-module, any morphism g:IQg : I \to Q in CC can be extended to all of RR along the inclusion IRI \hookrightarrow R.

Sketch of proof

Let i:MNi \colon M \hookrightarrow N be a mono in RModR Mod, and let f:MQf \colon M \to Q be a map. We must extend ff to a map h:NQh \colon N \to Q. Consider the poset whose elements are pairs (M,f)(M', f') where MM' is an intermediate submodule between MM and NN and f:MQf' \colon M' \to Q is an extension of ff, ordered by (M,f)(M,f)(M', f') \leq (M'', f'') if MM'' contains MM' and ff'' extends ff'. By an application of Zorn's lemma, this poset has a maximal element, say (M,f)(M', f'). Suppose MM' is not all of NN, and let xNx \in N be an element not in MM'; we show that ff' extends to a map M=x+MQM'' = \langle x \rangle + M' \to Q, contradiction.

The set {rR:rxM}\{r \in R: r x \in M'\} is an ideal II of RR, and we have a module map g:IQg \colon I \to Q defined by g(r)=f(rx)g(r) = f'(r x). By hypothesis, we may extend gg to a module map k:RQk \colon R \to Q. Writing a general element of MM'' as rx+yr x + y where yMy \in M', it may be shown that

f(rx+y)=k(r)+g(y)f''(r x + y) = k(r) + g(y)

is well-defined and extends ff', as desired.

Consequences

Corollary

Let RR be a Noetherian ring, and let {Q j} jJ\{Q_j\}_{j \in J} be a collection of injective modules over RR. Then the direct sum Q= jJQ jQ = \bigoplus_{j \in J} Q_j is also injective.

Proof

By Baer’s criterion, it suffices to show that for any ideal II of RR, a module map f:IQf \colon I \to Q extends to a map RQR \to Q. Since RR is Noetherian, II is finitely generated as an RR-module, say by elements x 1,,x nx_1, \ldots, x_n. Let p j:QQ jp_j \colon Q \to Q_j be the projection, and put f j=p jff_j = p_j \circ f. Then for each x ix_i, f j(x i)f_j(x_i) is nonzero for only finitely many summands. Taking all of these summands together over all ii, we see that ff factors through

jJQ j= jJQ jQ\prod_{j \in J'} Q_j = \bigoplus_{j \in J'} Q_j \hookrightarrow Q

for some finite JJJ' \subset J. But a product of injectives is injective, hence ff extends to a map R jJQ jR \to \prod_{j \in J'} Q_j, which completes the proof.

Conversely, a result of Bass and Papp is that RR is Noetherian if direct sums of injective RR-modules are injective. See Lam, Theorem 3.46.

Revised on September 26, 2012 14:29:08 by Urs Schreiber (131.174.191.22)