The Stacks project

Lemma 10.75.3. Let $(A_{\bullet , \bullet }, d, \delta )$ be a double complex such that

  1. Each row $A_{\bullet , j}$ is a resolution of $R(A)_ j$.

  2. Each column $A_{i, \bullet }$ is a resolution of $U(A)_ i$.

Then there are canonical isomorphisms

\[ H_ i(R(A)_\bullet ) \cong H_ i(U(A)_\bullet ). \]

The isomorphisms are functorial with respect to morphisms of double complexes with the properties above.

Proof. We will show that $H_ i(R(A)_\bullet ))$ and $H_ i(U(A)_\bullet )$ are canonically isomorphic to a third group. Namely

\[ \mathbf{H}_ i(A) := \frac{ \{ (a_{i, 0}, a_{i-1, 1}, \ldots , a_{0, i}) \mid d(a_{i, 0}) = \delta (a_{i-1, 1}), \ldots , d(a_{1, i-1}) = \delta (a_{0, i}) \} }{ \{ d(a_{i + 1, 0}) + \delta (a_{i, 1}), d(a_{i, 1}) + \delta (a_{i-1, 2}), \ldots , d(a_{1, i}) + \delta (a_{0, i + 1}) \} } \]

Here we use the notational convention that $a_{i, j}$ denotes an element of $A_{i, j}$. In other words, an element of $\mathbf{H}_ i$ is represented by a zig-zag, represented as follows for $i = 2$

\[ \xymatrix{ a_{2, 0} \ar@{|->}[r] & d(a_{2, 0}) = \delta (a_{1, 1}) & \\ & a_{1, 1} \ar@{|->}[u] \ar@{|->}[r] & d(a_{1, 1}) = \delta (a_{0, 2}) \\ & & a_{0, 2} \ar@{|->}[u] \\ } \]

Naturally, we divide out by “trivial” zig-zags, namely the submodule generated by elements of the form $(0, \ldots , 0, -\delta (a_{t + 1, t-i}), d(a_{t + 1, t-i}), 0, \ldots , 0)$. Note that there are canonical homomorphisms

\[ \mathbf{H}_ i(A) \to H_ i(R(A)_\bullet ), \quad (a_{i, 0}, a_{i-1, 1}, \ldots , a_{0, i}) \mapsto \text{class of image of }a_{0, i} \]


\[ \mathbf{H}_ i(A) \to H_ i(U(A)_\bullet ), \quad (a_{i, 0}, a_{i-1, 1}, \ldots , a_{0, i}) \mapsto \text{class of image of }a_{i, 0} \]

First we show that these maps are surjective. Suppose that $\overline{r} \in H_ i(R(A)_\bullet )$. Let $r \in R(A)_ i$ be a cocycle representing the class of $\overline{r}$. Let $a_{0, i} \in A_{0, i}$ be an element which maps to $r$. Because $\delta (r) = 0$, we see that $\delta (a_{0, i})$ is in the image of $d$. Hence there exists an element $a_{1, i-1} \in A_{1, i-1}$ such that $d(a_{1, i-1}) = \delta (a_{0, i})$. This in turn implies that $\delta (a_{1, i-1})$ is in the kernel of $d$ (because $d(\delta (a_{1, i-1})) = \delta (d(a_{1, i-1})) = \delta (\delta (a_{0, i})) = 0$. By exactness of the rows we find an element $a_{2, i-2}$ such that $d(a_{2, i-2}) = \delta (a_{1, i-1})$. And so on until a full zig-zag is found. Of course surjectivity of $\mathbf{H}_ i \to H_ i(U(A))$ is shown similarly.

To prove injectivity we argue in exactly the same way. Namely, suppose we are given a zig-zag $(a_{i, 0}, a_{i-1, 1}, \ldots , a_{0, i})$ which maps to zero in $H_ i(R(A)_\bullet )$. This means that $a_{0, i}$ maps to an element of $\mathop{\mathrm{Coker}}(A_{i, 1} \to A_{i, 0})$ which is in the image of $\delta : \mathop{\mathrm{Coker}}(A_{i + 1, 1} \to A_{i + 1, 0}) \to \mathop{\mathrm{Coker}}(A_{i, 1} \to A_{i, 0})$. In other words, $a_{0, i}$ is in the image of $\delta \oplus d : A_{0, i + 1} \oplus A_{1, i} \to A_{0, i}$. From the definition of trivial zig-zags we see that we may modify our zig-zag by a trivial one and assume that $a_{0, i} = 0$. This immediately implies that $d(a_{1, i-1}) = 0$. As the rows are exact this implies that $a_{1, i-1}$ is in the image of $d : A_{2, i-1} \to A_{1, i-1}$. Thus we may modify our zig-zag once again by a trivial zig-zag and assume that our zig-zag looks like $(a_{i, 0}, a_{i-1, 1}, \ldots , a_{2, i-2}, 0, 0)$. Continuing like this we obtain the desired injectivity.

If $\Phi : (A_{\bullet , \bullet }, d, \delta ) \to (B_{\bullet , \bullet }, d, \delta )$ is a morphism of double complexes both of which satisfy the conditions of the lemma, then we clearly obtain a commutative diagram

\[ \xymatrix{ H_ i(U(A)_\bullet ) \ar[d] & \mathbf{H}_ i(A) \ar[r] \ar[l] \ar[d] & H_ i(R(A)_\bullet ) \ar[d] \\ H_ i(U(B)_\bullet ) & \mathbf{H}_ i(B) \ar[r] \ar[l] & H_ i(R(B)_\bullet ) \\ } \]

This proves the functoriality. $\square$

Comments (0)

There are also:

  • 2 comment(s) on Section 10.75: Tor groups and flatness

Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 00M1. Beware of the difference between the letter 'O' and the digit '0'.