First question here. Apologies for any rookie mistakes, gladly take comments on how to improve phrasing etc.
Conventions: I define $[k]$ to be the one-dimensional vector space concentrated in degree $-k$. Define $A[k]:= [k]\otimes A$. I also work with the Koszul sign convention.
Let $f: A\to B$ be a map of cochain complexes of degree $n$. Let's say I want to define a map $f': A[k]\to B[l]$ using $f$. There is at least two ways of doing so.
- $f'= ([l]\otimes -)\circ f \circ ([-k]\otimes- )$,
- $f'= ([l-k]\otimes -)\circ 1(\otimes f)$,
where $([x]\otimes -)$ means "tensoring from the left". Option 2 will then come with an extra sign $(-1)^{nk}$.
My question is: What are the pros/cons of choosing one over the other?
Naively I would like to use 1. since it produces less signs. But in the literature, especially when defining shifted cochain complexes I have seen option 2 more frequently used (e.g. let $(A,d)$ be a cochain complex, then $(A[k],d_{[k]})$, where $d_{[k]}(a[k])=(-)^k(da)[k]$ is the shifted cochain complex).