We say a complex manifold $X$ has the *resolution property* if every coherent sheaf $\mathcal{M}$ on $X$ admits a surjection $\mathcal{E}\twoheadrightarrow \mathcal{M}$ by some finite rank locally free sheaf $\mathcal{E}$.


It is well-known that any projective manifold has the resolution property ([SGA 6, Expose II, 2.2]). In addition, any smooth compact complex surface has the resolution property too. (Schuster, *Locally free resolutions of coherent sheaves on surfaces*, 1982)



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> > My question is: if $X$ and $Y$ have the resolution property, is it true that $X\times Y$ must have the resolution property? In particular does the product of two smooth compact complex surfaces always have the resolution property?
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Does $X\times Y$ have the resolution property if both $X$ and $Y$ have?
Please find below a short argument in the case of schemes. The answer is positive for algebraic spaces too; in that case it can be proven using the characterization: $X$ has the resolution property $\Leftrightarrow$ $X = X'/GL\_n$ with $X'$ quasi-affine $+$ products of quasi-affines are quasi-affine $+$ a bit more work.


Lemma. Let $S$ be a separated scheme. Let $X$ and $Y$ be quasi-compact and quasi-separated schemes over $S$. Then if $X$ and $Y$ have the resolution property, so does $X \times\_S Y$.


Proof. Let $X = U\_1 \cup \ldots \cup U\_n$ and $Y = V\_1 \cup \ldots \cup V\_m$ be affine open coverings. Choose finite type quasi-coherent ideals $\mathcal{I}\_i \subset \mathcal{O}\_X$ such that $U\_i = X \setminus V(\mathcal{I}\_i)$. Choose finite type quasi-coherent ideals $\mathcal{J}\_j \subset \mathcal{O}\_Y$ such that $V\_j = Y \setminus V(\mathcal{J}\_j)$. Denote $\mathcal{K}\_{ij} = (\text{pr}\_1^{-1}\mathcal{I}\_i)(\text{pr}\_2^{-1}\mathcal{J}\_j) \subset \mathcal{O}\_{X \times\_S Y}$ the finite type quasi-coherent ideal generated by the product of the pullback of $\mathcal{I}\_i$ and the pullback of $\mathcal{J}\_j$. Then the affine open $U\_i \times\_S V\_j$ is equal to $X \times\_S Y \setminus V(\mathcal{K}\_{ij})$. As $X$ and $Y$ have the resloution property there exist surjections $\mathcal{E}\_i \to \mathcal{I}\_i$ and $\mathcal{F}\_j \to \mathcal{J}\_j$ with $\mathcal{E}\_i$ finite locally free on $X$ and $\mathcal{F}\_j$ finite locally free on $Y$. Then we see that there is a surjection $\text{pr}\_1^\*\mathcal{E}\_i \otimes \text{pr}\_2^\*\mathcal{F}\_j \to \mathcal{K}\_{ij}$. Since $X \times\_S Y = \bigcup\_{i = 1, \ldots, n, j = 1, \ldots, m} U\_i \times\_S V\_j$ we conclude $X \times\_S Y$ has the resolution property by [Lemma Tag 0F89](https://stacks.math.columbia.edu/tag/0F89).


