Ext and localisation

This post and the next are related, but I found this result interesting enough for a post of its own.

Lemma. Let M be a finitely presented R-module, and let S \sbq R be a multiplicative subset. Then

    \[ \Hom_R(M,N)[S^{-1}] = \Hom_{R[S^{-1}]}(M[S^{-1}],N[S^{-1}]). \]

Proof. The result is true when M is finite free, since

    \[ \Hom_R(R^n, N)[S^{-1}] = (N^n)[S^{-1}] = N[S^{-1}]^n, \]

whereas

    \[ \Hom_{R[S^{-1}]}(R[S^{-1}]^n, N[S^{-1}]) = N[S^{-1}]^n. \]

Now consider a finite presentation F_1 \ra F_0 \ra M \ra 0 of M. Since \Hom is left exact and localisation is exact, we get a commutative diagram

Rendered by QuickLaTeX.com

with exact rows (where the \Hom in the bottom row is over R[S^{-1}]). The right two vertical maps are isomorphisms, hence so is the one on the left. \qedsymbol

Definition. Let M be an R-module. Then M is k-finitely presented if there exists finite free modules F_0, \ldots, F_k and an exact sequence

    \[ F_k \ra F_{k-1} \ra \ldots \ra F_0 \ra M \ra 0. \]

For example, M is finitely generated if and only if it is 0-finitely presented, and finitely presented if and only if it is 1-finitely presented. Over a Noetherian ring, any finitely generated module is k-finitely presented for any k \in \Z_{\geq 0}.

[I do not know if this is standard terminology, but it should be.]

Corollary. Let k \geq 1, let M be a k-finitely presented module, and let S \sbq R be a multiplicative subset. Then

    \[ \Ext{R}{k-1}(M,N)[S^{-1}] = \Ext{R[S^{-1}]}{k-1}(M[S^{-1}],N[S^{-1}]). \]

Proof. Given an exact sequence F_k \ra \ldots \ra F_0 \ra M \ra 0 with F_i finite free, let M' be the kernel of F_0 \ra M. Then M' is (k-1)-finitely presented, and we have a short exact sequence

    \[ 0 \ra M' \ra F_0 \ra M \ra 0. \]

Now the result follows by induction, using the long exact sequence for \Ext{}{}. \qedsymbol

Remark. As Sebastian pointed out to me, we never used any specific properties of localisation, and the same result (with the same proof) works for any flat R-algebra.

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