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Tannaka duality for geometric stacks

Context

Topos Theory

topos theory

Background

Toposes

Internal Logic

Topos morphisms

Extra stuff, structure, properties

Cohomology and homotopy

In higher category theory

Theorems

Higher geometry

Contents

Idea

Under mild conditions, a given site CTAlg opC \subset T Alg^{op} of formal duals of algebras over an algebraic theory admits Isbell duality exhibited by an adjunction

(𝒪Spec):(TAlg Δ) opSpec𝒪Sh (,1)(C) (\mathcal{O} \dashv Spec) : (T Alg^{\Delta})^{op} \stackrel{\overset{\mathcal{O}}{\leftarrow}}{\underset{Spec}{\to}} Sh_{(\infty,1)}(C)

as described at function algebras on ∞-stacks Here 𝒪(X)\mathcal{O}(X) is an (,1)(\infty,1)-algebra of functions on XX.

This entry describes for certain algebraic stacks an analog of this situation where the 1-algebras are replaced by 2-algebras in the form of commutative algebra objects in the 2-category of abelian categories: abelian symmetric monoidal categories, and where the function algebras 𝒪(X)\mathcal{O}(X) are replaced with category QC(X)QC(X) of quasicoherent sheaves.

The replacement of the 1-algebra 𝒪(X)\mathcal{O}(X) by the 2-algebra QC(X)QC(X) is the starting point for what is called derived noncommutative geometry.

Setup

Ringed toposes

All toposes that we consider are Grothendieck toposes. A ringed topos (S,𝒪 S)(S, \mathcal{O}_S) is a topos SS equipped with a ring object 𝒪 S\mathcal{O}_S – a sheaf of rings – called the structure sheaf – on whatever site SS is the category of sheaves on. We write 𝒪 SMod\mathcal{O}_S Mod for the category of modules in SS (sheaves of modules) over 𝒪 S\mathcal{O}_S.

We write RngdToposRngdTopos for the category of ringed toposes.

For XX a scheme or more generally an algebraic stack, write Sh(X et)Sh(X_{et}) for its little etale topos.

Definition

A ringed topos (S,𝒪 S)(S,\mathcal{O}_S) is a locally ringed topos with respect to the étale topology if for every object USU \in S and every family {SpecR iSpec𝒪 S(U)}\{Spec R_i \to Spec \mathcal{O}_S(U)\} of étale morphisms such that

𝒪 S(U) iR i \mathcal{O}_S(U) \to \prod_i R_i

is faithfully flat, there exists morphisms E iEE_i \to E in SS and factorizations 𝒪 S(U)R i𝒪 S(E i)\mathcal{O}_S(U) \to R_i \to \mathcal{O}_S(E_i) such that

iE iE \coprod_i E_i \to E

is an epimorphism.

Proposition

If SS has enough points then (S,𝒪 S)(S, \mathcal{O}_S) is local for the étale topology precisely if the stalk 𝒪 S(x)\mathcal{O}_S(x) at every point x:SetSx : Set \to S is a strictly Henselian local ring.

This is (Lurie, remark 4.4).

Example

Abelian tensor categories

Definition

An abelian tensor category (for the purposes of the present discusission) is a symmetric monoidal category (C,)(C, \otimes) such that

  • CC is an abelian category;

  • for every xCx \in C the functor ()x:CC(-) \otimes x : C\to C is additive and right-exact: it commutes with finite colimits.

A complete abelian tensor category is an abelian tensor category such that

An abelian tensor category is called tame if for any short exact sequence

0MMM0 0 \to M'\to M \to M''\to 0

with MM'' a flat object (such that xxMx \mapsto x \otimes M'' is an exact functor) and any NCN \in C also the induced sequence

0MNMNMN0 0 \to M'\otimes N \to M\otimes N \to M''\otimes N \to 0

is exact.

This appears as (Lurie, def. 5.2) together with the paragraph below remark 5.3.

Definition

For C,DC,D two complete abelian tensor categories write

Func (C,D)Func(C,D) Func_\otimes(C,D) \subset Func(C,D)

for the core of the subcategory of the functor category on those functors that

Write

TCAbTens TCAbTens

for the (strict) (2,1)-category of tame complete abelian tensor categories with hom-groupoids given by this Func Func_\otimes.

This appears as (Lurie, def 5.9) together with the following remarks.

Example

For kk a ring, write kModk Mod for its abelian symmetric monoidal category of modules

Let (S,𝒪 S)(S,\mathcal{O}_S) be a ringed topos. Then

𝒪 SMod \mathcal{O}_S Mod

(the category of sheaves of 𝒪 S\mathcal{O}_S-modules) is a tame complete abelian tensor category.

This is (Lurie, example 5.7).

Example

For XX an algebraic stack, write

QC(X) QC(X)

for its category quasicoherent sheaves.

This is a complete abelian tensor category

Lemma

If XX is a Noetherian geometric stack, then QC(X)QC(X) is the category of ind-objects of its full subcategory Coh(X)QC(X)Coh(X) \subset QC(X) of coherent sheaves

QC(X)Ind(Coh(X)). QC(X) \simeq Ind(Coh(X)) \,.

This appears as (Lurie, lemma 3.9).

Geometric stacks

Definition

A geometric stack is

  • an algebraic stack XX over SpecSpec \mathbb{Z}

  • that is quasi-compact, in particular there is an epimorphism SpecAXSpec A \to X;

  • with affine and representable diagonal XX×XX \to X \times X.

Example

The geometricity condition on an algebraic stack implies that there are “enough” quasicoherent sheaves on it, as formalized by the following statement.

Theorem

If XX is a geometric stack then the bounded-below derived category of quasicoherent sheaves on XX is naturally equivalent to the full subcategory of the left-bounded derived category of smooth-etale 𝒪 X\mathcal{O}_X-modules whose chain cohomology sheaves are quasicoherent.

This is (Lurie, theorem 3.8).

Tannaka duality for geometric stacks

Theorem

Let XX be a geometric stack.

Then for every ring AA there is an equivalence of categories

RngdTopos(Sh((SpecA) et),Sh(X et))Hom (QC(X),AMod) RngdTopos(Sh((Spec A)_{et}),Sh(X_{et})) \simeq Hom_\otimes(QC(X), A Mod)

hence (by the 2-Yoneda lemma)

X(SpecA)Hom (QC(X),QC(SpecA)). X(Spec A) \simeq Hom_\otimes(QC(X), QC(Spec A)) \,.

More generally, for (S,𝒪 S)(S, \mathcal{O}_S) any etale-locally ringed topos, we have

RngdTopos(S,Sh(X et))Hom (QC(X),𝒪 SMod). RngdTopos(S,Sh(X_{et})) \simeq Hom_\otimes(QC(X), \mathcal{O}_S Mod) \,.

This is (Lurie, theorem 5.11) in view of (Lurie, remark 4.5).

Remark

It follows that forming quasicoherent sheaves constitutes a full and faithful (2,1)-functor

QC:GeomStacksTCAbTens op QC : GeomStacks \to TCAbTens^{op}

from geometric stacks to tame complete abelian tensor categories.

This statement justifies thinking of QC(X)QC(X) as being the “2-algebra” of functions on XX. This perspective is the basis for derived noncommutative geometry.

References

The above material is taken from

The generalization to geometric stacks in the context of Spectral Schemes is in

Related discussion is in

Revised on February 4, 2013 18:21:26 by Urs Schreiber (82.113.99.102)