Alternative definition of sheaf associated to a module?

Question

Given an affine scheme $X = \text{Spec}(A)$, then from an $A$-module $M$ we can form the associated sheaf $\tilde{M}$, where $$ \tilde{M}(D_f) = M_f = M \otimes_A A_f $$

This agrees with the presheaf $$ M'(U) := M \otimes_A \mathcal{O}_X(U), $$ on the basis of principal open sets.

Is $M'$ always a sheaf? If not, what is an example where $M'$ and $\tilde{M}$ do not agree?

The only thing I know from here is that $\tilde{M}$ is the sheafification of $M'$.

The reason I am interested is that for affinoid varieties, $M'$ defined in this way does form a sheaf.

Answer 1

As an example where $M'$ is not a sheaf take $A = k[x,y]$ and $M = A/(x,y)$. Then ${\cal O}_X(D_x \cup D_y) = A$ hence $M'(D_x \cup D_y) = M$. But $M'(D_x) = M'(D_y) = 0$.