The Stacks project

Remark 57.3.4. Let $k$ be a field and let $K^{\bullet , \bullet }$ be a double complex of finite dimensional $k$-vector spaces which is also a matress as in Remark 57.3.3. We claim that the dual double complex $L^{p, q} = \mathop{\mathrm{Hom}}\nolimits _ k(K^{-q, -p}, k)$ is also a matress. The exactness of rows and columns and the 3-periodicity are immediate. To see the additional condition holds, consider the linear map

\[ \partial : K^{p, q} \oplus K^{p - 1, q + 1} \oplus K^{p - 2, q + 2} \longrightarrow K^{p, q + 1} \oplus K^{p - 1, q + 2} \oplus K^{p - 2, q + 3} \]

sending $(\alpha , \beta , \gamma )$ to $(d_2 \alpha - d_1 \beta , d_2 \gamma - d_1 \alpha , d_1 \gamma - d_2 \beta )$ with identifications as in Remark 57.3.3. The condition of being a matress is that

\[ \mathop{\mathrm{Im}}(\partial ) \cap \left( 0 \oplus K^{p - 1, q + 2} \oplus K^{p - 2, q + 3} \right) = \mathop{\mathrm{Im}}(\partial |_{0 \oplus 0 \oplus K^{p - 2, q + 2}}) \]

Denoting ${}^\wedge $ the dual of a vector space or map, taking duals we get

\[ \mathop{\mathrm{Ker}}(\partial ^\wedge ) + \left(L^{-p, -q - 1} \oplus 0 \oplus 0 \right) = \mathop{\mathrm{Ker}}(\text{pr}_{L^{-p + 2, -q - 2}} \circ \partial ^\wedge ) \]

where the $\text{pr}$ term indicates projection. This implies that $\mathop{\mathrm{Im}}(\partial ^\wedge ) \cap (L^{-p, -q} \oplus L^{-p + 1, -q - 1} \oplus 0)$ is equal to $\mathop{\mathrm{Im}}(\partial ^\wedge |_{L^{-p, -q - 1} \oplus 0 \oplus 0})$ and this translates into the matress condition for $L^{\bullet , \bullet }$. Some details omitted.

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