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The Stacks project

Remark 12.22.6 (Variant). Let \mathcal{A} be an abelian category and let S, T : \mathcal{A} \to \mathcal{A} be shift functors, i.e., isomorphisms of categories. Assume that TS = ST as functors. Consider pairs (A, d) consisting of an object A of \mathcal{A} and a morphism d : A \to SA such that d \circ S^{-1}d = 0. The category of these objects is abelian. We define H(A, d) = \mathop{\mathrm{Ker}}(d)/\mathop{\mathrm{Im}}(S^{-1}d) and we observe that H(SA, Sd) = SH(A, d) (canonical isomorphism). Given a short exact sequence

0 \to (A, d) \to (B, d) \to (C, d) \to 0

we obtain a long exact homology sequence

\ldots \to S^{-1}H(C, d) \to H(A, d) \to H(B, d) \to H(C, d) \to SH(A, d) \to \ldots

(note the shifts in the boundary maps). Since ST = TS the functor T defines a shift functor on pairs by setting T(A, d) = (TA, Td). Next, let \alpha : (A, d) \to T^{-1}(A, d) be injective with cokernel (Q, d). Then we get an exact couple as in Remark 12.21.5 with shift functors TS and T given by

(H(A, d), S^{-1}H(Q, d), \overline{\alpha }, f, g)

where \overline{\alpha } : H(A, d) \to T^{-1}H(A, d) is induced by \alpha , the map f : S^{-1}H(Q, d) \to H(A, d) is the boundary map and g : H(A, d) \to TH(Q, d) = TS(S^{-1}H(Q, d)) is induced by the quotient map A \to TQ. Thus we get a spectral sequence as above with E_1 = S^{-1}H(Q, d) and differentials d_ r : E_ r \to T^ rSE_ r. As above we set E_0 = S^{-1}Q and d_0 : E_0 \to SE_0 given by S^{-1}d : S^{-1}Q \to Q. If according to our conventions we define B_ r \subset Z_ r \subset E_0, then we have for r \geq 1 that

  1. SB_ r is the image of

    (T^{-r + 1}\alpha \circ \ldots \circ T^{-1}\alpha )^{-1} \mathop{\mathrm{Im}}(T^{-r}S^{-1}d)

    under the natural map T^{-1}A \to Q,

  2. Z_ r is the image of

    (S^{-1}T^{-1}d)^{-1} \mathop{\mathrm{Im}}(\alpha \circ \ldots \circ T^{r - 1}\alpha )

    under the natural map S^{-1}T^{-1}A \to S^{-1}Q.

The differentials can be described as follows: if x \in Z_ r, then pick x' \in S^{-1}T^{-1}A mapping to x. Then S^{-1}T^{-1}d(x') is (\alpha \circ \ldots \circ T^{r - 1}\alpha )(y) for some y \in T^{r - 1}A. Then d_ r(x) \in T^ rSE_ r is represented by the class of the image of y in T^ rSE_0 = T^ rQ modulo T^ rSB_ r.


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