Let $A,B,C$ be three locally free sheaves on some complex manifold or variety $(X,\mathscr O_X)$.
Q: Can we say $Ext^1(A,B)=Ext^1(A\otimes C,B\otimes C)$?
Ext preserves the tensor
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sheaf-theory
sheaf-cohomology
derived-functors
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0Do these sheaves live on an affine scheme $(X,\mathcal{O}_X)$? If so, what category are you taking $Ext$ in? If its the category of quasi-coherent modules, then $Ext$ will vanish since locally free sheaves are in particular projective there. – 2017-02-20
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0Is $C$ of rank 1 ? Otherwise it is clearly false. – 2017-02-20
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0@Roland Yes, how to show this case? Could you explain more details. – 2017-02-20
1 Answers
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As said in the comments, we need $C$ of rank 1. Otherwise $C=\mathcal{O}_X^{\oplus 2}$ is a counter-example whenever $\operatorname{Ext}^1(A,B)\neq 0$.
But if $C$ is of rank 1, then it is invertible. In particular, the functors $B\mapsto\operatorname{Hom}(A,B)$ and $B\mapsto\operatorname{Hom}(A\otimes C,B\otimes C)$ are naturally isomorphic. Thus, so are their derived functors.