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

  1. $g^ n = - h^ n$, and

  2. 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

Typos: In the statement of the lemma, the target of the map is missing, should read . Also should map to not .

Comment #5086 by R on

A direct definition of is (apply on the right in the definition), and similarly for .

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 -functor does on termwise split short exact sequences (and morphisms thereof) what you think it does.

Comment #5297 by on

@#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?

There are also:

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

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