Characterizing Artin stacks

We study properties of morphisms of stacks in the context of the homotopy theory of presheaves of groupoids on a small site . There is a natural method for extending a property P of morphisms of sheaves on to a property ${\mathcal{P}}$ of morphisms of presheaves of groupoids. We prove that the property ${\mathcal{P}}$ is homotopy invariant in the local model structure on when P is stable under pullback and local on the target. Using the homotopy invariance of the properties of being a representable morphism, representable in algebraic spaces, and of being a cover, we obtain homotopy theoretic characterizations of algebraic and Artin stacks as those which are equivalent to simplicial objects in satisfying certain analogues of the Kan conditions. The definition of Artin stack can naturally be placed within a hierarchy which roughly measures how far a stack is from being representable. We call the higher analogues of Artin stacks n-algebraic stacks, and provide a characterization of these in terms of simplicial objects. A consequence of this characterization is that, for presheaves of groupoids, n-algebraic is the same as 3-algebraic for all n ≥ 3. As an application of these results we show that a stack is n-algebraic if and only if the homotopy orbits of a group action on it is.     

Functors of log Artin rings

This paper gives a generalization of the theory of functors of Artin rings in the framework of log geometry. In the final section we apply it to the log smooth deformation theory. Received: 4 August 1997 / Revised version: 13 January 1998     

On proper coverings of Artin stacks

We prove that every separated Artin stack of finite type over a noetherian base scheme admits a proper covering by a quasi-projective scheme. An application of this result is a version of the Grothendieck existence theorem for Artin stacks.     

Note on local structure of Artin stacks

In this Note we show that an Artin stack with finite inertia stack is étale locally isormorphic to the quotient of an affine scheme by an action of a general linear group.     

On the local quotient structure of Artin stacks

We show that near closed points with linearly reductive stabilizer, Artin stacks are formally locally quotient stacks by the stabilizer. We conjecture that the statement holds étale locally and we provide some evidence for this conjecture. In particular, we prove that if the stabilizer of a point is linearly reductive, the stabilizer acts algebraically on a miniversal deformation space, generalizing the results of Pinkham and Rim. We provide a generalization and stack-theoretic proof of Luna’s étale slice theorem which shows that GIT quotient stacks are étale locally quotients stacks by the stabilizer.     

Presenting higher stacks as simplicial schemes

We show that an n$n$-geometric stack may be regarded as a special kind of simplicial scheme, namely a Duskin n$n$-hypergroupoid in affine schemes, where surjectivity is defined in terms of covering maps, yielding Artin n$n$-stacks, Deligne–Mumford n$n$-stacks and n$n$-schemes as the notion of covering varies. This formulation adapts to all HAG contexts, so in particular works for derived n$n$-stacks (replacing rings with simplicial rings). We exploit this to describe quasi-coherent sheaves and complexes on these stacks, and to draw comparisons with Kontsevich’s dg-schemes. As an application, we show how the cotangent complex controls infinitesimal deformations of higher and derived stacks.     

There is no degree map for 0-cycles on Artin stacks

We show that there is no way to define degrees of 0-cycles on Artin stacks with proper good moduli spaces so that (1) the degree of an ordinary point is non-zero, and (1) degrees are compatible with closed immersions.     

Product systems over right-angled Artin semigroups

We build upon Mac Lane's definition of a tensor category to introduce the concept of a product system that takes values in a tensor groupoid . We show that the existing notions of product systems fit into our categorical framework, as do the -graphs of Kumjian and Pask. We then specialize to product systems over right-angled Artin semigroups; these are semigroups that interpolate between free semigroups and free abelian semigroups. For such a semigroup we characterize all product systems which take values in a given tensor groupoid . In particular, we obtain necessary and sufficient conditions under which a collection of -graphs form the coordinate graphs of a -graph.     

The six operations for sheaves on Artin stacks I: Finite coefficients

In this paper we develop a theory of Grothendieck’s six operations of lisse-étale constructible sheaves on Artin stacks locally of finite type over certain excellent schemes of finite Krull dimension. We also give generalizations of the classical base change theorems and Kunneth formula to stacks, and prove new results about cohomological descent for unbounded complexes.     

Motivic integration over Deligne-Mumford stacks

The aim of this article is to develop the theory of motivic integration over Deligne-Mumford stacks and to apply it to the birational geometry of Deligne-Mumford stacks.     

Categorical representation of locally noetherian log schemes

In this paper, we show that for a certain fairly general class of log schemes, the structure of the log scheme may be recovered entirely from the purely categorical structure of a certain associated category of log schemes of finite type over the given log scheme. This result is motivated partly by Grothendieck's anabelian philosophy and partly by general philosophical considerations concerning the importance of categories as a foundation for mathematics.     

Nearby cycles for log smooth families

We calculate l-adic nearby cycles in the étale cohomology for families with log smooth reduction using log étale cohomology. In particular, nearby cycles for log smooth families coincide with tame nearby cycles, as L. Illusie expected, and nearby cycles for semistable families depend only on the first infinitesimal neighborhood of the special fiber.     

Canonical key formula for projective abelian schemes

In this paper we prove a refined version of the canonical key formula for projective abelian schemes in the sense of Moret-Bailly (cf. Astérisque 129, 1985), we also extend this discussion to the context of Arakelov geometry. Precisely, let \({\pi: A \to S}\) be a projective abelian scheme over a locally noetherian scheme S with unit section \({e: S \to A}\) and let L be a symmetric, rigidified, relatively ample line bundle on A. Denote by ω _A the determinant of the sheaf of differentials of π and by d the rank of the locally free sheaf π_* L. In this paper, we shall prove the following results: (i). there is an isomorphism $${\rm det}(\pi_*L)^{\otimes 24} \cong (e^*\omega_A^\vee)^{\otimes 12d}$$ which is canonical in the sense that it can be chosen to be functorial, namely it is compatible with arbitrary base-change; (ii). if the generic fibre of S is separated and smooth, then there exist a positive integer m and canonical metrics on L and on ω _A such that there exists an isometry $${\rm det}(\pi_*\overline{L})^{\otimes 2m} \cong (e^*\overline{\omega}_A^\vee)^{\otimes md}$$ which is canonical in the sense of (i). Here the constant m only depends on g, d and is independent of L.     

The Relative Transpose over Cohen-Macaulay Finite Artin Algebras

The relative transpose via Gorenstein projective modules is introduced, and some corresponding results on the Auslander-Reiten sequences and the Auslander-Reiten formula to this relative version are generalized.     

Finite Flat Group Schemes over Local Artin Rings

Let R be a local Artin ring with maximal ideal m and residue class field of characteristic p > 0. We show that every finite flat group scheme over R is annihilated by its rank, whenever m ^p = pm = 0. This implies that any finite flat group scheme over an Artin ring the square of whose maximal ideal is zero, is annihilated by its rank.