ag.algebraic geometry - When can we cancel vector bundles from tensor products?

The answer to your question is yes, if $F$ is a direct sum of line bundles. So, let us assume that and its rank is $m$. And as you observed, we will assume that we are working over $mathbb{P}^2$. Let $0to F_0to F_1to Eto 0$ be the minimal resolution of $E$, where $F_i$ are direct sum of line bundles with rank of $F_0=n$ so that rank of $F_1=n+m$. Tensoring with $G$, we get $0to Gotimes F_0to Gotimes F_1to Gotimes Fto 0$, the last by the assumption. Taking cohomologies, letting $H_*^0(G)=M,H_*^1(G)=N$, we get an exact sequence $N^nto N^{n+m}to N^m$, which for length considerations (length of $N$ is finite) can easily seen to be exact on the left (and right). In particular, we have surjectivity of global sections and thus an exact sequence $0to M^nto M^{n+m}to M^mto 0$. This splits (I have forgotten whose theorem it is, but I think Ihave read it recently in some comment by Graham Leuschke) which is impossible since the first map has all entries in the maximal ideal unless $n=0$.

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