# usual tensor product of chain complexes of sheaves and flatness

Let $Sh(\mathcal{O}$ be the category of sheaves of ${\mathcal{O}}_X$-modules over a scheme $X$. Also let $Ch(R)$ be the category of chain complexes of (left) $R$-modules. We know that in both of $Ch(R)$ and $Ch(O)$(the category of chain complexes of sheaves of $\mathcal{O}_X$-modules), one can consider the usual tensor product of chain complexes and ${\rm Hom}$ functor. But it is known that the usual tensor product dose not characterize flatness as it dose in the category of modules. In paricula we have a complex$X$ with $X\bigotimes -$ exact and yet $X$ is not flat. Actually if we know only that $X^n$ is flat then $X\bigotimes -$ exact.

This motivated Enochs and Rozas to define a new tensor product of chain complexes which characterize flatness in $Ch(R)$ (Enochs and Rozas, Tensor Product of Chain Complexes, Math Jor of Okayam Univ, 1997).

Question: Can the usual tensor product of chain complexes of sheves characterize flatness in $Ch(\mathcal{O})$? If the answer is No, how can we define a new tensor product to deduce that $F$ is flat if and only if $F\bigotimes -$ is exact?

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