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perfect infinity-stack

Idea

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

Definition

Let $k$ be a field of characteristic 0. Let $T$ be the Lawvere theory of commutative associative algebras over $k$. When this is regarded as an (∞,1)-algebraic theory, the $T$-$\mathrm{\infty }$-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

$$C\subset T{\mathrm{Alg}}_{\mathrm{\infty }}^{\mathrm{op}}$$C \subset T Alg_\infty^{op}

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

Definition

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

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

given by

$$\mathrm{Spec}A\mapsto A\mathrm{Mod},$$Spec A \mapsto A Mod \,,

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

Definition

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

$$\mathrm{QC}(X):={\mathrm{PSh}}_{(\mathrm{\infty },2)}(C)(X,\mathrm{Mod})$$QC(X) := PSh_{(\infty,2)}(C)\left( X, Mod \right)

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

Remark

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

$$X\simeq {\underrightarrow{\lim }}_i{U}_i={\underrightarrow{\lim }}_i\mathrm{Spec}{A}_i.$$X \simeq {\lim_\to}_i U_i = {\lim_\to}_i Spec A_i \,.

We have then

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

This appears as (Ben-ZviFrancisNadler, section 3.1).

Proposition

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

• is a stable (∞,1)-category

• that has all (∞,1)-colimits.

(Ben-ZviFrancisNadler, section 3.1).

Definition

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

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

This appears as (Ben-ZviFrancisNadler, definition 3.1).

Definition

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

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

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

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

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

A morphism $X\to Y$ is said to be perfect morphism if its fibers $X{\times }_{Y}U$ over affines $U\to Y$ 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 object 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 a ∞-stack $X\in {\mathrm{Sh}}_{(\mathrm{\infty },1)}(C)$ with affine diagonal, the following are equivalent:

• $X$ is perfect

• $\mathrm{QC}(X)$ is

  • compactly generated,

  • and its compact and dualizable objects coincide.

Geometric $\mathrm{\infty }$-function theory

The assigmnent

$$\mathrm{QC}:X\mapsto \mathrm{QC}(X)$$QC : X \mapsto QC(X)

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

(Ben-ZviFrancisNadler

Examples

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

But the following are examples of perfect $\mathrm{\infty }$-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)