Existence of functorial (K-)flat resolutions?

Question

I am wondering about the following: Suppose $X$ is a reasonable scheme or stack with the resolution property. (So, all quasi-coherent sheaves admit a surjection from a flat sheaf.) Then I believe that all sheaves in the unbounded category of cochains $\operatorname{Ch}(X)=\operatorname{Ch}(\operatorname{QCoh}(X))$ admit a $K$-flat resolution, in the sense of Spaltenstein.

Is it the case, however, that $\operatorname{Ch}(X)$ admits functorial $K$-flat resolutions?

Though I'm open to any information in this regard, here I might specifically seek an endofunctor \begin{equation} T:\operatorname{Ch}(X)\to \operatorname{Ch}(X) \end{equation} which takes $K$-flat values and comes equipped with a natural transformation $\epsilon_M:T(M)\to M$ which is a quasi-isomorphism at all $M$. Or, maybe more reasonably, I might seek such an endofunctor $T:\mathscr{K}(X)\to \mathscr{K}(X)$ and transformation at the level of the homotopy $\infty$-category.

In the presence of a suitable model structure I believe that such things would exist, but I cant find any specific information in this regard in the literature.

Answer 1

Another and more "modern" approach would be using the construction of the complete cotorsion pair ("a half of an abelian model structure") generated by a generating set of objects in a Grothendieck abelian category.

Let $F$ be a flat quasi-coherent sheaf that is a generator of $\operatorname{QCoh}(X)$, as in my previous answer. Consider the Grothendieck abelian category $\operatorname{Ch}(X)=\operatorname{Ch}(\operatorname{QCoh}(X))$ of complexes in $\operatorname{QCoh}(X)$. Denote by $S^\bullet$ the direct sum of all shifts of the one-term (stalk) complex $\dotsb\to0\to F\to0\to\dotsb$ and all shifts of the contractible two-term complex $\dotsb\to0\to F\overset{\mathrm{id}}\to F\to0\to\dotsb$. Then $S^\bullet$ is a $K$-flat complex and a generator of $\operatorname{Ch}(X)$.

Using the small object argument (see, e.g., the paper Eklof, Trlifaj "How to make Ext vanish"), one can construct for any complex $A^\bullet\in\operatorname{Ch}(X)$ a surjective morphism of complexes $G^\bullet\to A^\bullet$ with the following properties. The complex $G^\bullet$ is a transfinitely iterated extension of copies of the complex $S^\bullet$. The kernel $K^\bullet$ of the morphism of complexes $G^\bullet\to A^\bullet$ has the property that $$ \operatorname{Ext}^1_{\operatorname{Ch}(X)}(S^\bullet,K^\bullet)=0. $$

For any two complexes $C^\bullet$ and $D^\bullet$ in an abelian category $\mathsf{A}$, the group of all cochain homotopy classes of morphisms $C^\bullet\to D^\bullet[1]$ is naturally a subgroup of the group $\operatorname{Ext}^1_{\operatorname{Ch}(\mathsf A)}(C^\bullet,D^\bullet)$. Specifically, this is the subgroup formed by all the termwise (degreewise) split extensions.

In the situation at hand, it follows that all the maps of complexes $S^\bullet\to K^\bullet[-1]$ are homotopic to zero. Since $F$ is a generator of $\operatorname{QCoh}(X)$, one easily concludes that the complex $K^\bullet$ is acyclic. Thus $G^\bullet\to A^\bullet$ is a quasi-isomorphism.

On the other hand, the class of all $K$-flat complexes of flat quasi-coherent sheaves is closed under extensions and direct limits in $\operatorname{Ch}(X)$. Hence it is also closed under transfinitely iterated extensions. Therefore, the complex $G^\bullet$ is $K$-flat.

The small object argument is functorial. So one can produce the termwise surjective morphism of complexes $G^\bullet\to A^\bullet$ as above in such a way that it depends functorially on a complex $A^\bullet$.

Answer 2

I'd guess that if $X$ is a reasonable scheme or stack, then $\operatorname{QCoh}(X)$ is, at least, a Grothendieck abelian category. In particular, it has a generator $G$.

If $X$ has resolution property, then $G$ is a quotient sheaf of a flat quasi-coherent sheaf $F$. So $F$ is a flat quasi-coherent sheaf on $X$ that is a generator of $\operatorname{QCoh}(X)$.

Now, for every quasi-coherent sheaf $M$ on $X$, there is a natural, functorial surjective morphism $F_M=F^{(\operatorname{Hom}(F,M))}\to M$ onto $M$ from the direct sum of copies of $F$ indexed over the set $\operatorname{Hom}(F,M)$ of all morphisms $F\to M$ in $\operatorname{QCoh}(X)$. Notice that any direct sum of copies of $F$ is a flat quasi-coherent sheaf on $X$. So we have a functorial surjection $F_M\to M$ onto every quasi-coherent sheaf $M$ from a flat quasi-coherent sheaf $F_M$ on $X$.

Using this construction of a functorial surjection onto an arbitrary quasi-coherent sheaf from a flat one, one can construct a functorial resolution of any quasi-coherent sheaf on $M$ by flat quasi-coherent sheaves. Moreover, one can even construct a functorial resolution of any complex of quasi-coherent sheaves $M^\bullet$ concentrated in the cohomological degrees $\le0$ by a complex of flat quasi-coherent sheaves.

Now, when you are given an unbounded complex $A^\bullet$ of quasi-coherent sheaves on $X$, you should be able to apply the original Spaltenstein's construction in order to produce a $K$-flat complex of quasi-coherent sheaves mapping quasi-isomorphically to $A^\bullet$. See Spaltenstein's paper "Resolutions of unbounded complexes", Proposition 5.6. The context in Spaltenstein is a bit different (sheaves of $\mathcal O_X$-modules instead of quasi-coherent sheaves), but I think that, given a functorial surjection $F_M\to M$ as above, the same construction as in Spaltenstein should be applicable, functorially.