The Stacks project

113.3 Homological algebra

Remark 113.3.1. The following remarks are obsolete as they are subsumed in Homology, Lemmas 12.24.11 and 12.25.3. Let $\mathcal{A}$ be an abelian category. Let $\mathcal{C} \subset \mathcal{A}$ be a weak Serre subcategory (see Homology, Definition 12.10.1). Suppose that $K^{\bullet , \bullet }$ is a double complex to which Homology, Lemma 12.25.3 applies such that for some $r \geq 0$ all the objects ${}'E_ r^{p, q}$ belong to $\mathcal{C}$. Then all the cohomology groups $H^ n(sK^\bullet )$ belong to $\mathcal{C}$. Namely, the assumptions imply that the kernels and images of ${}'d_ r^{p, q}$ are in $\mathcal{C}$. Whereupon we see that each ${}'E_{r + 1}^{p, q}$ is in $\mathcal{C}$. By induction we see that each ${}'E_\infty ^{p, q}$ is in $\mathcal{C}$. Hence each $H^ n(sK^\bullet )$ has a finite filtration whose subquotients are in $\mathcal{C}$. Using that $\mathcal{C}$ is closed under extensions we conclude that $H^ n(sK^\bullet )$ is in $\mathcal{C}$ as claimed. The same result holds for the second spectral sequence associated to $K^{\bullet , \bullet }$. Similarly, if $(K^\bullet , F)$ is a filtered complex to which Homology, Lemma 12.24.11 applies and for some $r \geq 0$ all the objects $E_ r^{p, q}$ belong to $\mathcal{C}$, then each $H^ n(K^\bullet )$ is an object of $\mathcal{C}$.


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