nLab perfect infinity-stack

Idea

In the context of dg-geometry an ∞-stack $X$ is called perfect if its (∞,1)-category $QC(X)$ of quasicoherent ∞-stacks (of modules over the structure sheaf $\mathcal{O}(X)$) is generated from compact objects/dualizable objects: modules that are locally perfect chain complexes.

Definition

Let $k$ be a field of characteristic zero. Let $T$ be the Lawvere theory of commutative associative algebras over $k$. When this is regarded as an (∞,1)-algebraic theory, the $T$-$\infty$-algebras are modeled (by the monoidal Dold-Kan correspondence equivalently) by the

• model structure on simplicial T-algebras;

• model structure on dg-algebras (over $k$, in non-positive degree, with positively graded differential).

The higher geometry/derived geometry over formal duals of these algebras is sometimes called dg-geometry: a general space in this context is given by an ∞-stack over a full sub-(∞,1)-site

$$C \subset T Alg_\infty^{op}$$

of the opposite (∞,1)-category of these $\infty$-algebras.

Definition

The (∞,2)-presheaf of quasicoherent ∞-stacks is

$$Mod \colon C^{op} \to (\infty,1)Cat$$

given by

$$Spec A \mapsto A Mod \,,$$

where on the right we take the $(\infty,1)$-category of $\infty$-modules over the $\infty$-algebra $A$, regarded as an unbounded dg-algebra.

Definition

For $X \in Sh_{(\infty,1)}(C)$ an ∞-stack in dg-geometry, write

$$QC(X) \coloneqq PSh_{(\infty,2)}(C)\left( X, Mod \right)$$

for the $(\infty,1)$-category of quasicoherent $\infty$-stacks on $X$.

Remark

By the co-Yoneda lemma we may express every $X \in Sh_{(\infty,1)}(C)$ as an (∞,1)-colimit of representables

$$X \simeq {\lim_\to}_i U_i = {\lim_\to}_i Spec A_i \,.$$

We have then

$$QC(X) \simeq {\lim_\leftarrow}_i QC(U_i) \simeq {\lim_\leftarrow}_i A_i Mod \,.$$

This appears in Ben-Zvi, Francis & Nadler, section 3.1.

Proposition

For all $X \in \mathbf{H}$, we have that $QC(X)$

• is a stable (∞,1)-category

• that has all (∞,1)-colimits.

(Ben-Zvi, Francis & Nadler, section 3.1).

Definition

Let $A \in T Alg_\infty$ . An $A$-module is a perfect module if it lies in the smallest sub-(∞,1)-category of $A Mod$ containing $A$ and closed under finite (∞,1)-colimits and retracts.

For a ∞-stack $X \in Sh_{(\infty,1)}(C)$, the $\infty$-category $Perf(X)$ is the full sub-$(\infty,1)$-category of $QC(X)$ consisting of those modules that are prefect over every affine $U\to X$.

This appears in Ben-Zvi, Francis & Nadler, definition 3.1.

Definition

A ∞-stack $X \in Sh_{(\infty,1)}(C)$ is called a perfect stack if

• it has affine diagonal $X \to X \times X$;

• and $QC(X)$ is the (∞,1)-category of ind-objects

  $$QC(X) \simeq \Ind \Perf(X)$$

  of the full sub-(∞,1)-category $Perf(X) \subset QC(X)$ of perfect complexes of modules on $X$.

A morphism $X \rightarrow Y$ is said to be perfect morphism if its fibers $X \times_Y U$ over affines $U \rightarrow Y$ are perfect.

Properties

Equivalent reformulations

Definition

A stable (∞,1)-category $C$ is compactly generated if it has a small set $\{c_i\}_{i \in I}$ of compact objects that are generators in the sense that if for $N \in C$ we have that $C(c_i, N)$ is equivalent to the zero morphism, then $N$ is the zero object.

Proposition

For an ∞-stack $X \in Sh_{(\infty,1)}(C)$ with affine diagonal, the following are equivalent:

• $X$ is perfect

• $QC(X)$ is

  • compactly generated,

  • and its compact and dualizable objects coincide.

Geometric $\infty$-function theory

The assigmnent

$$QC \colon X \mapsto QC(X)$$

of the $(\infty,2)$-algebras $QC(X)$ of quasicoherent ∞-stacks to perfect $\infty$-stacks $X$ constitutes a geometric ∞-function theory: this assignment commutes with (∞,1)-pullbacks and admits a good pull-push theory of integral transforms on sheaves.

(Ben-Zvi, Francis& Nadler)

Examples

Perfect stacks cover a broad array of spaces of interest, with notable exceptions being the (constant ∞-stack on a) classifying space $\mathcal{B}G$ of a topological group $G$ such as the circle $S^1 \simeq \mathcal{B} \mathbb{Z}$ 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 $\Gamma(X, \mathcal{O}_X)$ of the structure sheaf is ‘too large’, as in the previous cases.

But the following are examples of perfect $\infty$-stacks

• quasi-compact derived schemes with affine diagonal;

• the total space of a quasi-projective morphism over a perfect base;

• a quasi-projective derived scheme;

• the quotient $X/G$ of a quasi-projective derived scheme $X$ by a linear action of an affine group (for $k$ of characteristic 0).

References

The concept of a perfect stack in the context of dg-geometry is considered in

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