Algebraic Stacks Research Articles

λ-ring structures on the K-theory of algebraic stacks

λ-ring structures on the K-theory of algebraic stacks

Motivic homotopy theory of algebraic stacks

Motivic homotopy theory of algebraic stacks

Effective Morphisms and Quotient Stacks

Abstract We give a valuative criterion for when a smooth algebraic stack with a separated good moduli space is the quotient of a separated Deligne–Mumford stack by a torus. For doing so, we introduce a new class of morphisms, the effective morphisms, which are a generalization of separated morphisms.

The Atiyah class on algebraic stacks

Abstract We generalize Illusie’s definition of the Atiyah class to complexes with quasi-coherent cohomology on arbitrary algebraic stacks. We show that this gives a global obstruction theory for moduli stacks of complexes in algebraic geometry without derived methods. We give a similar generalization of the reduced Atiyah class, and we show various useful properties for working with Atiyah classes, such as compatibilities between the reduced and ordinary Atiyah class, and compatibility with tensor products and determinants.

Cox ring of an algebraic stack

We give a proper definition of the multiplicative structure of the following rings: Cox ring of invertible sheaves on a general algebraic stack; Cox ring of rank one reflexive sheaves on a normal and excellent algebraic stack. We show that such Cox rings always exist and establish its (non-)uniqueness in terms of an Ext-group. Moreover, we compare this definition with the classical construction of a Cox ring on a variety. Finally, we give an application to the theory of Mori dream stacks.

The Br = Br' question for some classifying stacks

In this paper we consider the Br = Br ′ question for classifying stacks by various group schemes. These are algebraic stacks that do not necessarily admit a finite flat cover by a scheme for which Br = Br ′ holds, hence are not amenable to the usual argument of pushing forward a twisted vector bundle. We provide two classes of examples satisfying Br ≠ Br ′ that do not arise from the scheme case. Communicated by Tony Varilly-Alvarado

Quantum parameters of the geometric Langlands theory

Fix a smooth, complete algebraic curve X over an algebraically closed field k of characteristic zero. To a reductive group G over k, we associate an algebraic stack {text {Par}}_G of quantum parameters for the geometric Langlands theory. Then we construct a family of (quasi-)twistings parametrized by {text {Par}}_G, whose module categories give rise to twisted {mathcal {D}}-modules on {text {Bun}}_G as well as quasi-coherent sheaves on the DG stack {text {LocSys}}_G.

An arithmetic valuative criterion for proper maps of tame algebraic stacks

The valuative criterion for proper maps of schemes has many applications in arithmetic, e.g. specializing Qp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {Q}_{p}$$\\end{document}-points to Fp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {F}_{p}$$\\end{document}-points. For algebraic stacks, the usual valuative criterion for proper maps is ill-suited for these kind of arguments, since it only gives a specialization point defined over an extension of the residue field, e.g. a Qp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {Q}_{p}$$\\end{document}-point will specialize to an Fpn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {F}_{p^{n}}$$\\end{document}-point for some n. We give a new valuative criterion for proper maps of tame stacks which solves this problem and is well-suited for arithmetic applications. As a consequence, we prove that the Lang–Nishimura theorem holds for tame stacks.

The Nisnevich motive of an algebraic stack

The Nisnevich motive of an algebraic stack

Brauer spaces of spectral algebraic stacks

AbstractWe study the question of whether the Brauer group is isomorphic to the cohomological one in spectral algebraic geometry. For this, we prove the compact generation of the derived category of twisted sheaves in the setting of spectral algebraic stacks. In particular, we obtain the compact generation of the ‐category of quasi‐coherent sheaves and the existence of compact perfect complexes with prescribed support for such stacks. We extend these results to derived algebraic geometry by studying the relationship between derived and spectral algebraic stacks.

An Improvement on the Base-Change Theorem and the Functor f^!

In 2009, Lipman published his polished book (Lipman in Foundations of Grothendieck duality for diagrams of schemes, Lecture Notes in Mathematics, vol. 1960, Springer, Berlin, pp 1–259, 2009), the product of a decade’s work, giving the definitive, state-of-the-art treatment of Grothendieck duality. In this article, we achieve a sharp improvement: we begin by giving a new proof of the base-change theorem, which can handle unbounded complexes and work in the generality of algebraic stacks (subject to mild technical restrictions). This means that our base-change theorem must be subtle and delicate, the unbounded version is right at the boundary of known counterexamples—counterexamples (in the world of schemes) that had led the experts to believe that major parts of the theory could only be developed in the bounded-below derived category. Having proved our new base-change theorem, we then use it to define the functor f^! on the unbounded derived category and establish its functoriality properties. In Sect. 1, we will use this to clarify the relation among all the various constructions of Grothendieck duality. One illustration of the power of the new methods is that we can improve Lipman (Lecture Notes in Mathematics, vol. 1960, Springer, Berlin, pp 1–259, 2009, Theorem 4.9.4) to handle complexes that are not necessarily bounded. There are also applications to the theory developed by Avramov, Iyengar, Lipman and Nayak on the connection between Grothendieck duality and Hochschild homology and cohomology but, to keep this paper from becoming even longer, these are being relegated to separate articles. See for example (Neeman in K-Theory—Proceedings of the International Colloquium, Mumbai, 2016, Hindustan Book Agency, New Delhi, pp 91–126, 2018).

Enumerating odd-degree hyperelliptic curves and abelian surfaces over {mathbb {P}}^1

Given asymptotic counts in number theory, a question of Venkatesh asks what is the topological nature of lower order terms. We consider the arithmetic aspect of the inertia stack of an algebraic stack over finite fields to partially answer this question. Subsequently, we acquire new sharp enumerations of quasi-admissible odd-degree hyperelliptic curves over {mathbb {F}}_q(t) ordered by discriminant height.

Mapping stacks and categorical notions of properness

One fundamental consequence of a scheme $X$ being proper is that the functor classifying maps from $X$ to any other suitably nice scheme or algebraic stack is representable by an algebraic stack. This result has been generalized by replacing $X$ with a proper algebraic stack. We show, however, that it also holds when $X$ is replaced by many examples of algebraic stacks which are not proper, including many global quotient stacks. This leads us to revisit the definition of properness for stacks. We introduce the notion of a formally proper morphism of stacks and study its properties. We develop methods for establishing formal properness in a large class of examples. Along the way, we prove strong $h$-descent results which hold in the setting of derived algebraic geometry but not in classical algebraic geometry. Our main applications are algebraicity results for mapping stacks and the stack of coherent sheaves on a flat and formally proper stack.

Heights on stacks and a generalized Batyrev–Manin–Malle conjecture

Abstract We define a notion of height for rational points with respect to a vector bundle on a proper algebraic stack with finite diagonal over a global field, which generalizes the usual notion for rational points on projective varieties. We explain how to compute this height for various stacks of interest (for instance: classifying stacks of finite groups, symmetric products of varieties, moduli stacks of abelian varieties, weighted projective spaces). In many cases, our uniform definition reproduces ways already in use for measuring the complexity of rational points, while in others it is something new. Finally, we formulate a conjecture about the number of rational points of bounded height (in our sense) on a stack $\mathcal {X}$ , which specializes to the Batyrev–Manin conjecture when $\mathcal {X}$ is a scheme and to Malle’s conjecture when $\mathcal {X}$ is the classifying stack of a finite group.

Unramified F–divided objects and the étale fundamental pro-groupoid in positive characteristic

Fix a scheme $S$ of characteristic $p$. Let $\mathscr{M}$ be an $S$-algebraic stack and let $\mbox{Fdiv}(\mathscr{M})$ be the stack of $\mbox{F}$-divided objects, that is sequences of objects $x_i\in\mathscr{M}$ with isomorphisms $\sigma_i:x_i\to \mbox{F}^*x_{i+1}$. Let $\mathscr{X}$ be a flat, finitely presented $S$-algebraic stack and $\mathscr{X}\to \Pi_1(\mathscr{X}/S)$ the \'etale fundamental pro-groupoid, constructed in the present text. We prove that if $\mathscr{M}$ is a quasi-separated Deligne-Mumford stack and $\mathscr{X}\to S$ has geometrically reduced fibres, there is a bifunctorial isomorphism of stacks \[\mathscr{H}\!om(\Pi_1(\mathscr{X}/S),\mathscr{M}) \simeq \mathscr{H}\!om(\mathscr{X},\mbox{Fdiv}(\mathscr{M})).\] In particular, the system of relative Frobenius morphisms $\mathscr{X}\to \mathscr{X}^{p/S}\to \mathscr{X}^{p^2/S}\to\dots$ allows to recover the space of connected components $\pi_0(\mathscr{X}/S)$ and the relative \'etale fundamental gerbe. In order to obtain these results, we study the existence and properties of relative perfection for algebras in characteristic $p$.

Synthetic spectra and the cellular motivic category

To an Adams-type homology theory we associate the notion of a synthetic spectrum; this is a product-preserving sheaf on the site of finite spectra with projective E-homology. We show that the $$\infty $$ -category of synthetic spectra based on E is in a precise sense a deformation of the $$\infty $$ -category of spectra into quasi-coherent sheaves over a certain algebraic stack, and show that this deformation encodes the $$E_{*}$$ -based Adams spectral sequence. We describe a symmetric monoidal functor from the $$\infty $$ -category of cellular motivic spectra over $$\text {Spec}({\mathbb {C}})$$ into an even variant of synthetic spectra based on $$\textrm{MU}$$ and show that it induces an equivalence between the $$\infty $$ -categories of p-complete objects for all primes p. In particular, it follows that the p-complete cellular motivic category can be described purely in terms of chromatic homotopy theory.

Categorical Milnor squares and K-theory of algebraic stacks

We introduce a notion of Milnor square of stable $$\infty $$ -categories and prove a criterion under which algebraic K-theory sends such a square to a cartesian square of spectra. We apply this to prove Milnor excision and proper excision theorems in the K-theory of algebraic stacks with affine diagonal and nice stabilizers. This yields a generalization of Weibel’s conjecture on the vanishing of negative K-groups for this class of stacks.

Logarithmic intersections of double ramification cycles

We describe a theory of logarithmic Chow rings and tautological subrings for logarithmically smooth algebraic stacks, via a generalisation of the notion of piecewise-polynomial functions. Using this machinery we prove that the double-double ramification cycle lies in the tautological subring of the (classical) Chow ring of the moduli space of curves, and that the logarithmic double ramification cycle is divisorial (as conjectured by Molcho, Pandharipande, and Schmitt).

The logarithmic Picard group and its tropicalization

We construct the logarithmic and tropical Picard groups of a family of logarithmic curves and realize the latter as the quotient of the former by the algebraic Jacobian. We show that the logarithmic Jacobian is a proper family of logarithmic abelian varieties over the moduli space of Deligne–Mumford stable curves, but does not possess an underlying algebraic stack. However, the logarithmic Picard group does have logarithmic modifications that are representable by logarithmic schemes, all of which are obtained by pullback from subdivisions of the tropical Picard group.

Costello’s pushforward formula: errata and generalization

Costello’s pushforward formula relates virtual fundamental classes of virtually birational algebraic stacks. Its original formulation omits a necessary hypothesis, whose addition is not sufficient to correct the proof. We supply a substitute for Costello’s notion of pure degree and prove the pushforward formula with this definition. We also show the hypotheses of the corrected pushforward formula are satisfied in a variety of its applications. Some adjustments to the original proofs are required in several cases, including the original one.