Lemma 12.14.12 (011L)—The Stacks project

Lemma 12.14.12. Notation and assumptions as in Lemma 12.14.10. Let $\alpha : A^\bullet \to B^\bullet$ , $\beta : B^\bullet \to C^\bullet$ be the given morphisms of complexes. Suppose $(s')^ n : C^ n \to B^ n$ and $(\pi ')^ n : B^ n \to A^ n$ is a second choice of splittings. Write $(s')^ n = s^ n + \alpha ^ n \circ h^ n$ and $(\pi ')^ n = \pi ^ n + g^ n \circ \beta ^ n$ for some unique morphisms $h^ n : C^ n \to A^ n$ and $g^ n : C^ n \to A^ n$ . Then

$g^ n = - h^ n$ , and
the family of maps $\{ g^ n : C^ n \to A[1]^{n - 1}\}$ is a homotopy between $\delta , \delta ' : C^\bullet \to A[1]^\bullet$ , more precisely $(\delta ')^ n = \delta ^ n + g^{n + 1} \circ d_ C^ n + d_{A[1]}^{n - 1} \circ g^ n$ .

Proof. As $(s')^ n$ and $(\pi ')^ n$ are splittings we have $(\pi ')^ n \circ (s')^ n = 0$ . Hence

$0 = ( \pi ^ n + g^ n \circ \beta ^ n ) \circ ( s^ n + \alpha ^ n \circ h^ n ) = g^ n \circ \beta ^ n \circ s^ n + \pi ^ n \circ \alpha ^ n \circ h^ n = g^ n + h^ n$

which proves (1). We compute $(\delta ')^ n$ as follows

$( \pi ^{n + 1} + g^{n + 1} \circ \beta ^{n + 1} ) \circ d_ B^ n \circ ( s^ n + \alpha ^ n \circ h^ n ) = \delta ^ n + g^{n + 1} \circ d_ C^ n + d_ A^ n \circ h^ n$

Since $h^ n = -g^ n$ and since $d_{A[1]}^{n - 1} = -d_ A^ n$ we conclude that (2) holds. $\square$

Comments (3)

Comment #295 by arp on September 06, 2013 at 08:49

Typos: In the statement of the lemma, the target of the map $\alpha$ is missing, should read $\alpha: A^{\bullet} \rightarrow B^{\bullet}$ . Also $\beta$ should map to $C^{\bullet}$ not $C^n$ .

Comment #5086 by R on May 05, 2020 at 21:49

A direct definition of $g^n$ is $(\pi')^ns^n$ (apply $s^n$ on the right in the definition), and similarly for $h^n$ .

This lemma generalises without too much difficulty to the case of a morphism of termwise split short exact sequences, which gives a more direct way than the somewhat muddled Tags 13.12.1 and 13.12.3 to show that the canonical $\delta$ -functor does on termwise split short exact sequences (and morphisms thereof) what you think it does.

Comment #5297 by Johan on June 16, 2020 at 19:15

@#5086: Are you saying this would improve on Lemma 13.12.1 (which is not about termwise split short exact sequences of complexes) or are you saying this would improve 13.12.3?

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2 comment(s) on Section 12.14: Homotopy and the shift functor

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