Subsection 1.1. (0NZ0) · ScalingStacks

1.1. Perfect stacks

For an arbitrary derived stack $X$ , the $\infty$ -category $\qc(X)$ is difficult to control algebraically. For example, it may contain large objects that are impossible to construct in terms of concrete, locally-finite objects.

To get a handle on $\qc(X)$ , we need to know that it has a small $\infty$ -subcategory $\qc(X)^{\circ}$ of “generators” which are “finite” in an appropriate sense. There are two common notions of when a small subcategory $\mathcal{C}^{\circ}$ generates a category $\mathcal{C}$ , and they have natural $\infty$ -analogues. On the one hand, we could ask that $\mathcal{C}$ be the inductive limit or ind-category $\operatorname{Ind}\mathcal{C}^{\circ}$ (i.e., that $\mathcal{C}$ is freely generated from $\mathcal{C}^{\circ}$ by taking inductive limits), and on the other hand, we could ask that in $\mathcal{C}$ the right orthogonal of $\mathcal{C}^{\circ}$ vanishes. When $\mathcal{C}$ is $\qc(X)$ , there are three common notions of which objects may be considered finite: perfect objects, dualizable objects, and compact objects, which refer respectively to the geometry, monoidal structure, and categorical structure of $\qc(X)$ . We review all of these notions and the relations between them in Section 3.1 (in particular, perfect and dualizable objects always coincide).

To have a tractable and broadly applicable class of derived stacks, we introduce the notion of a perfect stack. By definition, an object $M$ of $\qc(X)$ is a perfect complex if locally for any affine $U\to X$ its restriction to $U$ is a perfect module (finite complex of projective modules). Equivalently, $M$ is dualizable with respect to the monoidal structure on $\qc(X)$ . A derived stack $X$ is said to be perfect if it has affine diagonal and the $\infty$ -category $\qc(X)$ is the inductive limit

\qc(X)\simeq\operatorname{Ind}\operatorname{Perf}(X)

of the full $\infty$ -subcategory $\operatorname{Perf}(X)$ of perfect complexes. A morphism $X\to Y$ is said to be perfect if its fibers $X\times_{Y}U$ over affines $U\to Y$ are perfect.

On a perfect stack $X$ , compact objects of $\qc(X)$ are the same thing as perfect complexes (which in turn are always the same thing as dualizable objects). In fact, we have the following alternative formulation: a derived stack $X$ is perfect if it has affine diagonal, $\qc(X)$ is compactly generated (that is, there is no right orthogonal to the compact objects), and compact objects and dualizable objects coincide.

0NVX

Remark 1.1. The notion of compactly generated categories is a standard one in homotopy theory, especially in conjunction with themes such as Brown representability and Bousfield localization. Schwede and Shipley [SSh] prove in great generality that compactly generated categories can be expressed as categories of modules.

In algebraic geometry, the importance of the interplay between compact and perfect objects was originally recognized and put to great use by Thomason [TT]. These ideas were combined with homotopical techniques by Bökstedt and Neeman [BoN], and further developed and enhanced by Neeman [N1, N2] and many others [Ke, BV, To1]. The key property of derived categories of quasi-coherent sheaves on quasi-compact, separated schemes identified in these papers is that on the one hand, they are compactly generated, and on the other hand, their compact and perfect objects coincide. Such categories appear as unital algebraic stable homotopy categories in the general axiomatic framework developed by Hovey, Palmieri and Strickland [HPS]. This combination of properties underlies the definition of a perfect stack.

In Section 3.2, we establish base change and the projection formula for perfect morphisms following arguments in Lurie’s thesis [L1] (in fact, the arguments here only use that the structure sheaf is relatively compact).

In Section 3.3, we show that the class of perfect stacks is very broad. We show that the following are all perfect stacks:

(1)

Quasi-compact derived schemes with affine diagonal (following arguments of [N2]).
(2)

The total space of a quasi-projective morphism over a perfect base.
(3)

In characteristic zero, the quotient $X/G$ of a quasi-projective derived scheme $X$ by a linear action of an affine group $G$ .
(4)

The quotient $X/G$ of a perfect stack $X$ by a finite affine group scheme in “good” characteristics for $G$ .
(5)

The mapping stack $X^{\Sigma}=\Map(\Sigma,X)$ , for a perfect stack $X$ and a finite simplicial set $\Sigma$ .
(6)

Fiber products of perfect stacks.

We also show that any morphism $X\to Y$ between perfect stacks is itself perfect.

Though the above examples show that perfect stacks cover a broad array of spaces of interest, it is worth pointing out that there are many commonly arising derived stacks that are imperfect. Since categories of quasi-coherent sheaves are usually compactly generated, the typical reason for $X$ to be imperfect is that the structure sheaf $\mathcal{O}_{X}$ (which is always dualizable) fails to be compact (or equivalently, the global sections functor fails to preserve colimits). This can occur if the cohomology $\Gamma(X,\mathcal{O}_{X})$ is too big such as for (1) the classifying space of a finite group in modular characteristic (for the simplest example, one can take $B\mathbb{Z}/2\mathbb{Z}$ when $2$ is not invertible), (2) the classifying space of a topological group such as $S^{1}$ , or (2) an ind-scheme such as the formal disc $\operatorname{Spf}k[[t]]$ . In the opposite direction, categories of ${\mathcal{D}}$ -modules on smooth schemes (which can be considered as quasi-coherent sheaves on the corresponding de Rham stacks) have compact unit, but also many compact objects (such as ${\mathcal{D}}$ itself) which are not dualizable.

David Ben-Zvi, John Francis, David Nadler

Original source: arXiv:0805.0157v5