perfect infinity-stack


(,1)(\infty,1)-Topos Theory

(∞,1)-topos theory





Extra stuff, structure and property



structures in a cohesive (∞,1)-topos

Higher geometry



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


Let kk be a field of characteristic 0. Let TT be the Lawvere theory of commutative associative algebras over kk. When this is regarded as an (∞,1)-algebraic theory, the TT-\infty-algebras are modeled (by the monoidal Dold-Kan correspondence equivalently) by the

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

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

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


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

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

given by

SpecAAMod, Spec A \mapsto A Mod \,,

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


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

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

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


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

Xlim iU i=lim iSpecA i. X \simeq {\lim_\to}_i U_i = {\lim_\to}_i Spec A_i \,.

We have then

QC(X)lim iQC(U i)lim iA iMod. QC(X) \simeq {\lim_\leftarrow}_i QC(U_i) \simeq {\lim_\leftarrow}_i A_i Mod \,.

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


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

(Ben-ZviFrancisNadler, section 3.1).


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

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

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


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

  • it has affine diagonal XX×XX \to X \times X;

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

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

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

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


Equivalent reformulations


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


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

  • XX is perfect

  • QC(X)QC(X) is

Geometric \infty-function theory

The assigmnent

QC:XQC(X) QC : X \mapsto QC(X)

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



Perfect stacks cover a broad array of spaces of interest, with notable exceptions being the (constant ∞-stack on a) classifying space G\mathcal{B}G of a topological group GG such as the circle S 1S^1 \simeq \mathcal{B} \mathbb{Z} or the classifying spaces of most algebraic groups in non-zero characteristic. This is because if XX is perfect, then the global sections functor Γ\Gamma must preserve colimits, which fails when the global sections Γ(X,𝒪 X)\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/GX/G of a quasi-projective derived scheme XX by a linear action of an affine group (for kk of characteristic 0).


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

Revised on December 14, 2010 00:48:27 by Urs Schreiber (