TITLE: What is the tensor product of a sheaf and a module?
QUESTION [8 upvotes]: The following object was studied in section III.12 of Hartshorne's Algebraic Geometry book. 
Let $A$ be a noetherian ring, $Y=\mathrm{Spec}A$, and $M$ be an $A$-module.  Let $f:X\to Y$ be a morphism, and $\mathcal{F}$ be a quasicoherent sheaf on $X$. Then Hartshorne writes $\mathcal{F}\otimes_A M$ without defining what it is (technically, he already used it in Coroally III.9.4). 
Here is what I have guessed.  For each affine $U\subset X$, $A_U:=\mathcal{O}_X(U)$ is an $A$-algebra. So we have a presheaf by assigning $U$ with the module $\mathcal{F}(U)\otimes_A M$.  Then we will call the associated sheaf of this presheaf $\mathcal{F}\otimes_A M$. 
But what puzzles me is that in Proposition 12.2, it was claimed that if $C^*(\mathfrak{U}, \mathcal{F})$ is the Čech complex of $\mathcal{F}$, then for any $i_0, \ldots, i_p$, we have 
$$\Gamma(U_{i_0,\ldots, i_p}, \mathcal{F}\otimes_A M)=\Gamma(U_{i_0,\ldots, i_p}, \mathcal{F})\otimes_A M.$$
In other words, it seems to me that the book claims that the presheaf that I have defined is in fact a sheaf, which is not obvious to me at all.  Could anyone explain what I am missing here?
Edit: In proposition 12.2, it was assumed that $\mathcal{F}$ is flat over $Y$.

REPLY [7 votes]: I cannot write comments yet so sorry for only slightly elaborating on points that have already been made. It is certain that Hartshorne forgot to assume that the cover $\mathfrak{U}$ in the proof of Proposition 12.2 consists of affine(!) open subschemes. Without this assumption there will not even be any connection between Cech and sheaf cohomology. He also needs to assume that the cover is a finite(!) cover for the tensor product to commute with the products occuring in the Cech complex.
The most natural description of $\mathcal{F}\otimes_A M$ is indeed $\mathcal{F}\otimes_{\mathcal{O}_X} f^* \tilde{M}$. If $U\subset X$ is an affine open subscheme, we have $(\mathcal{F}\otimes_A M)(U)=\mathcal{F}(U)\otimes_A M$. To see this note that 
$(\mathcal{F}\otimes_{\mathcal{O}_X} f^* \tilde{M})_{|U}=\mathcal{F}_{|U}\otimes_{\mathcal{O}_U} (f_{|U})^*\tilde{M}$. But if $U=\mathrm{Spec}\, B$ then by the description of the pullback of quasi-coherent sheaves between affine schemes we simply have $(f_{|U})^*\tilde{M}=\widetilde{B\otimes_A M}$ and hence $(\mathcal{F}_{|U}\otimes_{\mathcal{O}_U} (f_{|U})^*\tilde{M})(U)=\mathcal{F}(U)\otimes_B (B\otimes_A M)=\mathcal{F}(U)\otimes_A M$ as claimed.
Hence if the cover is a finite cover consisting of open affine subschemes and if the scheme is separated (so that all the intersections occuring in the Cech complex are affine) his claim about the Cech complex of $\mathcal{F}\otimes_A M$ is true.