The Stacks project

Lemma 15.31.3. Let $A \to B$ be a ring map. Let $f_1, \ldots , f_ r$ be a sequence in $B$ such that $B/(f_1, \ldots , f_ r)$ is $A$-flat. Let $A \to A'$ be a ring map. Then the canonical map

\[ H_1(K_\bullet (B, f_1, \ldots , f_ r)) \otimes _ A A' \longrightarrow H_1(K_\bullet (B', f'_1, \ldots , f'_ r)) \]

is surjective. Here $B' = B \otimes _ A A'$ and $f_ i' \in B'$ is the image of $f_ i$.

Proof. The sequence

\[ \wedge ^2(B^{\oplus r}) \to B^{\oplus r} \to B \to B/J \to 0 \]

is a complex of $A$-modules with $B/J$ flat over $A$ and cohomology group $H_1 = H_1(K_\bullet (B, f_1, \ldots , f_ r))$ in the spot $B^{\oplus r}$. If we tensor this with $A'$ we obtain a complex

\[ \wedge ^2((B')^{\oplus r}) \to (B')^{\oplus r} \to B' \to B'/J' \to 0 \]

which is exact at $B'$ and $B'/J'$. In order to compute its cohomology group $H'_1 = H_1(K_\bullet (B', f'_1, \ldots , f'_ r))$ at $(B')^{\oplus r}$ we split the first sequence above into the exact sequences $0 \to J \to B \to B/J \to 0$, $0 \to K \to B^{\oplus r} \to J \to 0$, and $\wedge ^2(B^{\oplus r}) \to K \to H_1 \to 0$. Tensoring over $A$ with $A'$ we obtain the exact sequences

\[ \begin{matrix} 0 \to J \otimes _ A A' \to B \otimes _ A A' \to (B/J) \otimes _ A A' \to 0 \\ K \otimes _ A A' \to B^{\oplus r} \otimes _ A A' \to J \otimes _ A A' \to 0 \\ \wedge ^2(B^{\oplus r}) \otimes _ A A' \to K \otimes _ A A' \to H_1 \otimes _ A A' \to 0 \end{matrix} \]

where the first one is exact as $B/J$ is flat over $A$, see Algebra, Lemma 10.39.12. We conclude that $J' = J \otimes _ A A' \subset B'$ and that $K \otimes _ A A' \to \mathop{\mathrm{Ker}}((B')^{\oplus r} \to B')$ is surjective. Thus

\begin{align*} H_1 \otimes _ A A' & = \mathop{\mathrm{Coker}}\left(\wedge ^2(B^{\oplus r}) \otimes _ A A' \to K \otimes _ A A'\right) \\ & \to \mathop{\mathrm{Coker}}\left( \wedge ^2((B')^{\oplus r}) \to \mathop{\mathrm{Ker}}((B')^{\oplus r} \to B') \right) = H'_1 \end{align*}

is surjective too. $\square$

Comments (3)

Comment #7125 by Xuande Liu on

I cannot figure out why . Could someone give me a hint?

Comment #7126 by on

Oops! The lemma should say that the map is just surjective and this follows because surjects onto the kernel of . I just checked and the lemma originally used to just say this but in an edit in 2016 changed the statement to "isomorphism" without justifying why! Luckily, the only lemmas using this lemma rely on the original statement of surjectivity! I'm going to fix this right now. Thanks very much!

Comment #7127 by on

Thanks again for pointing this out. Fixed here. The webpage should be updated in a few minutes. Furthermore, in case you are interested, I observe that if is flat over , then is a kernel of a surjection of flat -modules, hence is itself flat over and we do get injective and we do get an isomorphism in the statement of the lemma.

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