In this paper, we investigate the theory of heights in a family of stacky curves following recent work of Ellenberg, Satriano, and Zureick-Brown. We first give an elementary construction of a height which is seen to be dual to theirs. We count rational points having bounded ESZ-B height on a particular stacky curve, answering a question of Ellenberg, Satriano, and Zureick-Brown. We also show that when the Euler characteristic of stacky curves is non-positive, the ESZ-B height coming from the anti-canonical divisor class fails to have the Northcott property. We prove that a stacky version of a conjecture of Vojta is equivalent to the $abc$-conjecture.

We consider a Deligne–Mumford stack $X$ which is the quotient of an affine scheme $\operatorname {Spec}A$ by the action of a finite group $G$ and show that the Balmer spectrum of the tensor triangulated category of perfect complexes on $X$ is homeomorphic to the space of homogeneous prime ideals in the group cohomology ring $H^*(G,A)$.

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.

We study the Chow ring of the moduli stack $\mathfrak {M}_{g,n}$ of prestable curves and define the notion of tautological classes on this stack. We extend formulas for intersection products and functoriality of tautological classes under natural morphisms from the case of the tautological ring of the moduli space $\overline {\mathcal {M}}_{g,n}$ of stable curves. This paper provides foundations for the paper [BS21].

In the appendix (jointly with J. Skowera), we develop the theory of a proper, but not necessary projective, pushforward of algebraic cycles. The proper pushforward is necessary for the construction of the tautological rings of $\mathfrak {M}_{g,n}$ and is important in its own right. We also develop operational Chow groups for algebraic stacks.

We study the action of the inertia operator on the motivic Hall algebra and prove that it is diagonalizable. This leads to a filtration of the Hall algebra, whose associated graded algebra is commutative. In particular, the degree 1 subspace forms a Lie algebra, which we call the Lie algebra of virtually indecomposable elements, following Joyce. We prove that the integral of virtually indecomposable elements admits an Euler characteristic specialization. In order to take advantage of the fact that our inertia groups are unit groups in algebras, we introduce the notion of algebroid.

Let $X$ be a quasi-compact and quasi-separated scheme. There are two fundamental and pervasive facts about the unbounded derived category of $X$: (1) $\mathsf{D}_{\text{qc}}(X)$ is compactly generated by perfect complexes and (2) if $X$ is noetherian or has affine diagonal, then the functor $\unicode[STIX]{x1D6F9}_{X}:\mathsf{D}(\mathsf{QCoh}(X))\rightarrow \mathsf{D}_{\text{qc}}(X)$ is an equivalence. Our main results are that for algebraic stacks in positive characteristic, the assertions (1) and (2) are typically false.

We develop a theory of unbounded derived categories of quasi-coherent sheaves on algebraic stacks. In particular, we show that these categories are compactly generated by perfect complexes for stacks that either have finite stabilizers or are local quotient stacks. We also extend Toën and Antieau–Gepner’s results on derived Azumaya algebras and compact generation of sheaves on linear categories from derived schemes to derived Deligne–Mumford stacks. These are all consequences of our main theorem: compact generation of a presheaf of triangulated categories on an algebraic stack is local for the quasi-finite flat topology.

We prove a general criterion for an algebraic stack to admit a good moduli space. This result may be considered as a generalization of the Keel–Mori theorem, which guarantees the existence of a coarse moduli space for a separated Deligne–Mumford stack. We apply this result to prove that the moduli stacks $\overline{{\mathcal{M}}}_{g,n}(\unicode[STIX]{x1D6FC})$ parameterizing $\unicode[STIX]{x1D6FC}$-stable curves introduced in [J. Alper et al., Second flip in the Hassett–Keel program: a local description, Compositio Math. 153 (2017), 1547–1583] admit good moduli spaces.

Assuming Vojta’s conjecture, and building on recent work of the authors, we prove that, for a fixed number field $K$ and a positive integer $g$, there is an integer $m_{0}$ such that for any $m>m_{0}$ there is no principally polarized abelian variety $A/K$ of dimension $g$ with full level-$m$ structure. To this end, we develop a version of Vojta’s conjecture for Deligne–Mumford stacks, which we deduce from Vojta’s conjecture for schemes.

In this paper, we show that there is an equivalence between the 2-category of smooth Deligne–Mumford stacks with torus embeddings and actions and the 1-category of stacky fans. To this end, we prove two main results. The first is related to a combinatorial aspect of the 2-category of toric algebraic stacks defined by I. Iwanari [Logarithmic geometry, minimal free resolutions and toric algebraic stacks, Preprint (2007)]; we establish an equivalence between the 2-category of toric algebraic stacks and the 1-category of stacky fans. The second result provides a geometric characterization of toric algebraic stacks. Logarithmic geometry in the sense of Fontaine–Illusie plays a central role in obtaining our results.

For a tame Deligne-Mumford stack X with the resolution property, we show that the Cartan-Eilenberg resolutions of unbounded complexes of quasicoherent sheaves are K-injective resolutions. This allows us to realize the derived category of quasi-coherent sheaves on X as a reflexive full subcategory of the derived category of X-modules.

We then use the results of Neeman and recent results of Kresch to establish the localization theorem of Thomason-Trobaugh for the K-theory of perfect complexes on stacks of above type which have coarse moduli schemes. As a byproduct, we get a generalization of Krause's result about the stable derived categories of schemes to such stacks.

We prove Thomason's classification of thick triangulated tensor subcategories of D(perf / X). As the final application of our localization theorem, we show that the spectrum of D(perf / X) as defined by Balmer, is naturally isomorphic to the coarse moduli scheme of X, answering a question of Balmer for the tensor triangulated categories arising from Deligne-Mumford stacks.

We study the deformation theory of algebraic stacks. The 2-category of deformations using Ext groups of the cotangent complex is described. Then we show that an algebraic stack has a versal deformation under certain conditions.

We define stacks of uniform cyclic covers of Brauer–Severi schemes, proving that they can be realized as quotient stacks of open subsets of representations, and compute the Picard group for the open substacks parametrizing smooth uniform cyclic covers. Moreover, we give an analogous description for stacks parametrizing triple cyclic covers of Brauer–Severi schemes of rank 1 that are not necessarily uniform, and give a presentation of the Picard group of the substacks corresponding to smooth triple cyclic covers.

Let $Y \rightrightarrows X$ be a finite flat groupoid scheme with $X$ a quasi-projective variety and let $S$ be its coarse moduli scheme. We associate to the groupoid scheme a coherent sheaf of algebras $\mathcal{O}_{X / Y}$ on $S$ which we call the non-commutative coordinate ring of the groupoid scheme. We show that when $X$ is a smooth curve and the groupoid action is generically free, the non-commutative coordinate rings which can occur are, up to Morita equivalence, the hereditary orders on smooth curves. This gives a bijective correspondence between smooth one-dimensional Deligne–Mumford stacks of finite type and Morita equivalence classes of hereditary orders on smooth curves.

In this paper we establish Riemann–Roch and Lefschtez–Riemann–Roch theorems for arbitrary proper maps of finite cohomological dimension between algebraic stacks in the sense of Artin. The Riemann–Roch theorem is established as a natural transformation between the G-theory of algebraic stacks and topological G-theory for stacks: we define the latter as the localization of G-theory by topological K-homology. The Lefschtez–Riemann–Roch is an extension of this including the action of a torus for Deligne–Mumford stacks. This generalizes the corresponding Riemann–Roch theorem (Lefschetz–Riemann–Roch theorem) for proper maps between schemes (that are also equivariant for the action of a torus, respectively) making use of some fundamental results due to Vistoli and Toen. A key result established here is that topological G-theory (as well as rational G-theory) has cohomological descent on the isovariant étale site of an algebraic stack. This extends cohomological descent for topological G-theory on schemes as proved by Thomason.