nLab
perfect infinity-stack

Idea

A stack or infinity-stack $X$, which might be a derived stack, is called perfect if the canonical (infinity,1)-category of “modules over its function algebra” is particularly well behaved.

In the context of geometric infinity-function theory these modules play the role of generalized functions, perfect $\mathrm{\infty }$-stacks are those for which the fundamental theorem of $\mathrm{\infty }$-function theory (see therenfinity-function theory)) holds.

A perfect $\mathrm{\infty }$-stack is an $\mathrm{\infty }$-stack whose “space of $\mathrm{\infty }$-functions” — the (infinity,1)-category $\mathrm{QC}(X)$ of quasicoherent sheaves on $X$ — is locally the inductive limit of a category of ‘finite’ objects.

Definition

To define a perfect stack $X$, we must first define which quasicoherent sheaves on $X$ we shall consider to be ‘finite’. These are called the perfect complexes.

Definition

Let $A$ be a derived commutative ring. An $A$-module is perfect if it lies in the smallest $\mathrm{\infty }$-subcategory of ${\mathrm{Mod}}_A$ containing $A$ and closed under finite colimits and retracts. For a derived stack $X$, the $\mathrm{\infty }$-category $\mathrm{Perf}(X)$ is the full $\mathrm{\infty }$-subcategory of $\mathrm{QC}(X)$ consisting of those sheaves $M$ whose restriction $f^{*}M$ to any affine $f:U\to X$ over $X$ is a perfect module.

Now we can define what it means for a stack to be perfect, as well as for morphisms between them.

Definition

A derived stack $X$ is said to be perfect if it has affine diagonal and the $(\mathrm{\infty }\mathrm{,1})$-category $\mathrm{QC}(X)$ is the (infinity,1)-category of ind-objects

(1)$$\mathrm{QC}(X)\simeq IndPerf(X)$$QC(X) \simeq \Ind \Perf(X)

of the full $(\mathrm{\infty }\mathrm{,1})$-subcategory $\mathrm{Perf}(X)$ of perfect complexes on $X$.

A morphism $X\to Y$ is said to be perfect if its fibers $X{\times }_{Y}U$ over affines $U\to Y$ are perfect.

Bruce: I’d like to get a better grip on this definition at the ordinary categorical level, as opposed to the $\mathrm{\infty }$-categorical level.

I’d like to see some statement to the effect “A space $X$ is compact if and only if its space of functions $C(X)$ has a certain property”, where that “certain property” was clearly seen to be some kind of decategorified analogue of (a) ind-limit of perfect objects (b) generated by compact objects, etc.

The best I’ve seen so far is: A space $X$ is compact if and only if $\mathrm{Hom}(X,\cdot )$ preserves all filtered colimits. I’d like to see a similar statement at the level of functions though to make the analogy clearer.

Toby: Do we need a page perfect stack?? And we could certainly use some exposition at compact object!

Bruce: I’m not sure if ‘perfect stack’ makes sense at the non-derived level.

Remarks

• Perfect stacks cover a broad array of spaces of interest, with notable exceptions being the classifying space of a topological group such as $S^1$ or the classifying spaces of most algebraic groups in non-zero characteristic. This is because if $X$ is perfect, then the global sections functor $\Gamma$ must preserve colimits, which fails when the global sections $\Gamma (X,{𝒪}_X)$ of the structure sheaf is ‘too large’, as in the previous cases.

Properties

The following proposition establishes that the concept of ‘perfect stack’ is robust in that various other roads would have led to the same destination.

Proposition

For a derived stack $X$ with affine diagonal, the following are equivalent:

• $X$ is perfect
• $\mathrm{QC}(X)$ is compactly generated?, and its compact and dualizable objects coincide.

References

The concept of a perfect stack in the context of (infinity,1)-categories was introduced in

• Ben-Zvi, Francis and Nadler, Integral transforms and Drinfeld centers in derived algebraic geometry (arXiv)