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10.73 Functorialities for Ext

In this section we briefly discuss the functoriality of $\mathop{\mathrm{Ext}}\nolimits $ with respect to change of ring, etc. Here is a list of items to work out.

  1. Given $R \to R'$, an $R$-module $M$ and an $R'$-module $N'$ the $R$-module $\mathop{\mathrm{Ext}}\nolimits ^ i_ R(M, N')$ has a natural $R'$-module structure. Moreover, there is a canonical $R'$-linear map $\mathop{\mathrm{Ext}}\nolimits ^ i_{R'}(M \otimes _ R R', N') \to \mathop{\mathrm{Ext}}\nolimits ^ i_ R(M, N')$.

  2. Given $R \to R'$ and $R$-modules $M$, $N$ there is a natural $R$-module map $\mathop{\mathrm{Ext}}\nolimits ^ i_ R(M, N) \to \text{Ext}^ i_ R(M, N \otimes _ R R')$.

Lemma 10.73.1. Given a flat ring map $R \to R'$, an $R$-module $M$, and an $R'$-module $N'$ the natural map

\[ \mathop{\mathrm{Ext}}\nolimits ^ i_{R'}(M \otimes _ R R', N') \to \text{Ext}^ i_ R(M, N') \]

is an isomorphism for $i \geq 0$.

Proof. Choose a free resolution $F_\bullet $ of $M$. Since $R \to R'$ is flat we see that $F_\bullet \otimes _ R R'$ is a free resolution of $M \otimes _ R R'$ over $R'$. The statement is that the map

\[ \mathop{\mathrm{Hom}}\nolimits _{R'}(F_\bullet \otimes _ R R', N') \to \mathop{\mathrm{Hom}}\nolimits _ R(F_\bullet , N') \]

induces an isomorphism on homology groups, which is true because it is an isomorphism of complexes by Lemma 10.14.3. $\square$

Comments (1)

Comment #683 by Keenan Kidwell on

In 0879, it says "the natural maps," but there is only one map, right? Also, in the proof, "homology group" should be "homology groups."

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