nLab Tannaka duality for geometric stacks

Idea

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

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

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

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 $\mathcal{O}(X)$ are replaced with category $QC(X)$ of quasicoherent sheaves.

The replacement of the 1-algebra $\mathcal{O}(X)$ by the 2-algebra $QC(X)$ 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 $(S, \mathcal{O}_S)$ is a topos $S$ equipped with a ring object $\mathcal{O}_S$ – a sheaf of rings – called the structure sheaf – on whatever site $S$ is the category of sheaves on. We write $\mathcal{O}_S Mod$ for the category of modules in $S$ (sheaves of modules) over $\mathcal{O}_S$.

We write $RngdTopos$ for the category of ringed toposes.

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

Definition

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

$$\mathcal{O}_S(U) \to \prod_i R_i$$

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

$$\coprod_i E_i \to E$$

is an epimorphism.

Proposition

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

This is (Lurie, remark 4.4).

Example

• The little étale topos $Sh(X_{et})$ of a Deligne-Mumford stack $X$ is locally ringed with respect to the étale topology.

Abelian tensor categories

Definition

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

• $C$ is an abelian category;

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

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

• it satisfies the axiom AB5 at additive and abelian categories;

• $(-) \otimes x$ commutes with all small colimits.

  (equivalently, we have a closed monoidal category).

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

$$0 \to M'\to M \to M''\to 0$$

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

$$0 \to M'\otimes N \to M\otimes N \to M''\otimes N \to 0$$

is exact.

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

Definition

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

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

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

• are symmetric monoidal functors;

• commute with all small colimits (which implies they are additive and right exact)

• preserve flat objects and short exact sequences whose last object is flat.

Write

$$TCAbTens$$

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

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

Example

For $k$ a ring, write $k Mod$ for its abelian symmetric monoidal category of modules

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

$$\mathcal{O}_S Mod$$

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

This is (Lurie, example 5.7).

Example

For $X$ an algebraic stack, write

$$QC(X)$$

for its category quasicoherent sheaves.

This is a complete abelian tensor category

Lemma

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

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

This appears as (Lurie, lemma 3.9).

Geometric stacks

Definition

A geometric stack is

• an algebraic stack $X$ over $Spec \mathbb{Z}$

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

• with affine and representable diagonal $X \to X \times X$.

Example

• A quasicompact separated scheme is a geometric stack.

• The classifying stack of a smooth affine group scheme is a geometric stack.

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

Theorem

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

This is (Lurie, theorem 3.8).

Tannaka duality for geometric stacks

Theorem

Let $X$ be a geometric stack.

Then for every ring $A$ there is an equivalence of categories

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

hence (by the 2-Yoneda lemma)

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

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

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

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

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

from geometric stacks to tame complete abelian tensor categories.

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

Related concepts

• analytification

• 2-algebraic geometry

• spectrum of a tensor triangulated category

• prime spectrum of a symmetric monoidal stable (∞,1)-category

• Tannaka duality for Lie groupoids

• Bondal-Orlov reconstruction theorem

• 2-ring

References

The above material is taken from

• Jacob Lurie, Tannaka duality for geometric stacks, (arXiv:math.AG/0412266)

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

• Jacob Lurie, Quasi-Coherent Sheaves and Tannaka Duality Theorems

Related discussion is in

• Martin Brandenburg, Alexandru Chirvasitu, Tensor functors between categories of quasi-coherent sheaves (arXiv:abs/1202.5147)