# Tag Info

### Does module Hom commute with tensor product in the second variable?

You can think about tensor products as a kind of colimit; you're asking the hom functor $\text{Hom}_A(L, -)$ to commute with this colimit in the second variable, but usually the hom functor only ...
• 107k
Accepted

• 2,699

• 799

### When are two resolutions of a coherent sheaf homotopic

No. The simplest example is given by the following two resolutions of the structure sheaf of a point $P \in \mathbb{P}^1$:  0 \to \mathcal{O}_{\mathbb{P}^1}(-1) \to \mathcal{O}_{\mathbb{P}^1} \to \...
• 31.8k
Accepted

### When are two resolutions of a coherent sheaf homotopic

If this were true, then any short exact sequence of vector bundles would split. Indeed, if $0 \to \mathscr E_1 \to \mathscr E_2 \to \mathscr E_3 \to 0$ is a short exact sequence of vector bundles, ...