A note on group actions on algebraic stacks

代数堆的讲座

Lectures on algebraic stacks

本文为2022年秋季在台北NCTS的11讲课程准备的代数堆笔记，涵盖核心概念与方法，为初学者提供系统性理解。

Moduli Spaces of Coherent Sheaves on Projective Deligne-Mumford Stacks over Algebraic Spaces

In this paper, we study the geometric invariant theory on algebraic spaces, and construct te moduli spaces of $\mathcal{H}$-semistable sheaves on projective Deligne-Mumford stacks over algebraic spaces $S$. We prove that this moduli space is projective over $S$ as an algebraic space.

A note on finite abelian gerbes over toric Deligne-Mumford stacks

Any toric Deligne-Mumford stack is a $μ$-gerbe over the underlying toric orbifold for a finite abelian group $μ$. In this paper we give a sufficient condition so that certain kinds of gerbes over a toric Deligne-Mumford stack are again toric Deligne-Mumford stacks.

Logarithmic geometry, minimal free resolutions and toric algebraic stacks

In this paper we will introduce a certain type of morphisms of log schemes (in the sense of Fontaine, Illusie, and Kato) and investigate their moduli. Then by applying this we define a notion of toric algebraic stacks over arbitrary schemes, which may be regarded as torus embeddings within the framework of algebraic stacks, and study some basic properties. Furthermore, we study the stack-theoretic analogue of toroidal embeddings.

Cohomological invariants of algebraic stacks

The purpose of this paper is to lay the foundations of a theory of invariants in étale cohomology for smooth Artin stacks. We compute the invariants in the case of the stack of elliptic curves, and we use the theory we developed to get some results regarding Brauer groups of algebraic spaces.

Grothendieck's Existence Theorem for Relatively Perfect Complexes on Algebraic Stacks

We prove Grothendieck's existence theorem for relatively perfect complexes on an algebraic stack that is proper and flat over an $I$-adically complete Noetherian ring $A$. This generalizes an earlier result of Lieblich in the setting of algebraic spaces.

Cohomology and base change for algebraic stacks

We prove that cohomology and base change holds for algebraic stacks, generalizing work of Brochard in the tame case. We also show that Hom-spaces on algebraic stacks are represented by abelian cones, generalizing results of Grothendieck, Brochard, Olsson, Lieblich, and Roth--Starr. To accomplish all of this, we prove that a wide class of Ext-functors in algebraic geometry are coherent (in the sense of M. Auslander).

代数栈的动机同伦理论

Motivic Homotopy Theory of Algebraic Stacks

本文利用∞-范畴语言和增强运算映射，将动机同伦理论及六函子形式体系从概形推广到一大类代数栈，并证明其满足同伦不变性、局部化和纯性。

Stable points on algebraic stacks

This paper is largely concerned with constructing coarse moduli spaces for Artin stacks. The main purpose of this paper is to introduce the notion of stability on an arbitrary Artin stack and construct a coarse moduli space for the open substack of stable points. Also, we present an application to coherent cohomology of Artin stacks.

The notion of category over an algebraic stack

The goal of this note is to spell out the (apparently well-known and intuitively clear) notion of abelian category over an algebraic stack. In the future we will discuss the (much less evident) notion, when instead of an abelian category one considers a triangulated one.

On some finiteness questions for algebraic stacks

We prove that under a certain mild hypothesis, the DG category of D-modules on a quasi-compact algebraic stack is compactly generated. We also show that under the same hypothesis, the functor of global sections on the DG category of quasi-coherent sheaves is continuous.

The Fundamental Group of an Algebraic Stack

We define the notion of fundamental group of an algebraic stack, prove a comparison theorem between the fundamental group of a stack over the complex numbers and that of the associated analytic orbifold, show that this notion coincides with that defined with simplicial schemes by Oda and finally prove some short exact sequences involving these groups.

Relative dimension of morphisms and dimension for algebraic stacks

Motivated by applications in moduli theory, we introduce a flexible and powerful language for expressing lower bounds on relative dimension of morphisms of schemes, and more generally of algebraic stacks. We show that the theory is robust and applies to a wide range of situations. Consequently, we obtain simple tools for making dimension-based deformation arguments on moduli spaces. Additionally, in a complementary direction we develop the basic properties of codimension for algebraic stacks. One of our goals is to provide a comprehensive toolkit for working transparently with dimension statem…

结点代数曲线上主$\Bbb C^*$-丛的模栈的上同调

Cohomology of Moduli Stacks of Principal $\Bbb C^*$-Bundles Over Nodal Algebraic Curves

研究结点曲线上主$\Bbb C^*$-丛模栈，用陈类确定其有理上同调代数。

Coherent Tannaka duality and algebraicity of Hom-stacks

We establish Tannaka duality for noetherian algebraic stacks with affine stabilizer groups. Our main application is the existence of Hom-stacks in great generality.

Hom stacks

We study Hom 2-functors parameterizing 1-morphisms of algebraic stacks, and prove that it is representable by an algebraic stack under certain conditions, using Artin's criterion. As an application we study Picard 2-functors which parameterizes line bundles on algebraic stacks.

Algebraicity and smoothness of fixed point stacks

We study algebraicity and smoothness of fixed point stacks for flat group schemes which have a finite composition series whose factors are either reductive or proper, flat, finitely presented, acting on algebraic stacks with affine, finitely presented diagonal. For this, we extend some theorems of [SGA3.2] on functors of homomorphisms Hom(G, H) and functors of reductive subgroups Sub(H) for an affine, possibly non-flat group scheme H.

代数栈的紧逼近与下降

Compact approximation and descent for algebraic stacks

本文研究代数栈上复形导出范畴的逼近与生成，证明紧对象逼近可沿拟有限平坦覆盖下降，推广了概形与代数空间的相关结果，并探讨了导出前推与拉回下生成的行为。

代数堆的弗罗贝尼乌斯生成

Frobenius generation for algebraic stacks

本文探讨了正特征下代数堆导出范畴中弗罗贝尼乌斯映射的生成性质，并在德林费尔德-穆尔堆中界定了生成所需的迭代次数。

Dimension theory and components of algebraic stacks

We prove some basic results on the dimension theory of algebraic stacks, and on the multiplicities of their irreducible components, for which we do not know a reference.

关于代数曲线上主丛模栈的 Frobenius 态射作用

On actions of Frobenius morphisms for moduli stacks of principal bundles over algebraic curves

本文研究代数曲线上主丛模栈的算术与几何 Frobenius 态射，并利用 Chern 类明确其在 ℓ-adic 上同调上的作用。

代数栈上的Morel-Voevodsky构造

The Morel-Voevodsky Construction over Algebraic Stacks

本文以Morel-Voevodsky精神构造了代数栈的（非）稳定motivic同伦范畴，证明其与Chowdhury-D'Angelo的定义一致，并将Bachmann谱刚性定理、带框motivic同伦范畴及Hoyois重建定理推广至代数栈。

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 …

Steenrod operations on the de Rham cohomology of algebraic stacks

Building up on work of Epstein, May and Drury, we define and investigate the mod $p$ Steenrod operations on the de Rham cohomology of smooth algebraic stacks over a field of characteristic $p>0$. We then compute the action of the operations on the de Rham cohomology of classifying stacks for finite groups, connected reductive groups for which $p$ is not a torsion prime, and (special) orthogonal groups when $p=2$.

Note on orbifold Chow ring of semi-projective toric Deligne-Mumford stacks

We prove a formula for the orbifold Chow ring of semi-projective toric DM stacks, generalizing the orbifold Chow ring formula of projective toric DM stacks by Borisov-Chen-Smith. We also consider a special kind of semi-projective toric DM stacks, the Lawrence toric DM stacks. We prove that the orbifold Chow ring of a Lawrence toric DM stack is isomorphic to the orbifold Chow ring of its associated hypertoric DM stack studied in \cite{JT}.

Brauer Groups of Algebraic Stacks and GIT-Quotients

In this paper we consider the Brauer groups of algebraic stacks and GIT quotients: the only algebraic stacks we consider in this paper are quotient stacks [X/G], where X is a smooth scheme of finite type over a field k, and G is a linear algebraic group over k and acting on X, as well as various moduli stacks of principal G-bundles on a smooth projective curve X, associated to a reductive group G. We also consider the Brauer groups of the corresponding coarse moduli spaces, which most often identify with the corresponding GIT-quotients. One conclusion that we seem to draw then is that the Brau…

Finiteness theorems for the Picard objects of an algebraic stack

We prove some finiteness theorems for the Picard functor of an algebraic stack, in the spirit of SGA 6, exp. XII and XIII. In particular, we give a stacky version of Raynaud's relative representability theorem, we give sufficient conditions for the existence of the torsion component of the Picard functor, and for the finite generation of the Neron-Severi groups or of the Picard group itself. We give some examples and applications. In an appendix, we prove the semicontinuity theorem for a (non necessarily tame) algebraic stack.

Noetherian approximation of algebraic spaces and stacks

We show that every scheme/algebraic space/stack that is quasi-compact with quasi-finite diagonal can be approximated by a noetherian scheme/algebraic space/stack. More generally, we show that any stack which is etale-locally a global quotient stack can be approximated. Examples of applications are generalizations of Chevalley's, Serre's and Zariski's theorems and Chow's lemma to the non-noetherian setting. We also show that every quasi-compact algebraic stack with quasi-finite diagonal has a finite generically flat cover by a scheme.

前代数栈上的有理动机

Rational motives on pro-algebraic stacks

该研究定义了前代数栈的有理动机∞-范畴，证明其对特定态射类有6函子形式体系且在前概形上符合Liu-Zheng定义，为不可表示代数栈和非有限型态射的动机理论提供新途径。

An introduction to moduli stacks, with a view towards Higgs bundles on algebraic curves

This article is based in part on lecture notes prepared for the summer school "The Geometry, Topology and Physics of Moduli Spaces of Higgs Bundles" at the Institute for Mathematical Sciences at the National University of Singapore in July of 2014. The aim is to provide a brief introduction to algebraic stacks, and then to give several constructions of the moduli stack of Higgs bundles on algebraic curves. The first construction is via a "bootstrap" method from the algebraic stack of vector bundles on an algebraic curve. This construction is motivated in part by Nitsure's GIT construction of a…

Perfect complexes on algebraic stacks

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 t…

真代数栈的基本群是有限表示的

Fundamental groups of proper algebraic stacks are finitely presented

本文证明了代数闭域上具有有限惯性的真代数栈的平展基本群是拓扑有限表示的，推广了 Lara、Srinivas 和 Stix 关于真概形的结果。

A Luna étale slice theorem for algebraic stacks

We prove that every algebraic stack, locally of finite type over an algebraically closed field with affine stabilizers, is étale-locally a quotient stack in a neighborhood of a point with a linearly reductive stabilizer group. The proof uses an equivariant version of Artin's algebraization theorem proved in the appendix. We provide numerous applications of the main theorems.

Tame stacks in positive characteristic

We introduce and study a class of algebraic stacks with finite inertia in positive and mixed characteristic, which we call tame algebraic stacks. They include tame Deligne-Mumford stacks, and are arguably better behaved than general Deligne-Mumford stacks. We also give a complete characterization of finite flat linearly reductive schemes over an arbitrary base. Our main result is that tame algebraic stacks are étale locally quotient by actions of linearly reductive finite group schemes. In a subsequent paper we will show that tame algebraic stacks admit a good theory of stable maps.

Equivalence of two notions of log moduli stacks

We show the equivalence between two notions of log moduli stacks which appear in literatures. In particular, we generalize M.Olsson's theorem of representation of log algebraic stacks and answer a question posted by him (\cite{Ol4} 3.5.3). As an application, we obtain several fundamental results of algebraic log stacks which resemble to those in algebraic stacks.

Essentially Finite Vector Bundles on Normal Pseudo-proper Algebraic Stacks

Let $X$ be a normal, connected and projective variety over an algebraically closed field $k$. It is known that a vector bundle $V$ on $X$ is essentially finite if and only if it is trivialized by a proper surjective morphism $f:Y\to X$. In this paper we introduce a different approach to this problem which allows to extend the results to normal, connected and strongly pseudo-proper algebraic stack of finite type over an arbitrary field $k$.

Note on the Bondal-Orlov Functors for Toric DM Stacks

We calculate explicit formulas for the general equivariant Bondal-Orlov functors on the localized K-theory groups for a crepant birational transformation of toric DM stacks. We recall some facts that the Bondal-Orlov functors give equivalences on the bounded derived categories. Applying twice of these functors we get the Seidel-Thomas spherical twists for the derived category.

Stacky Abelianization of an Algebraic Group

Let G be a connected algebraic group and let [G,G] be its commutator subgroup. We prove a conjecture of Drinfeld about the existence of a connected etale group cover H of [G,G], characterized by the following properties: every central extension of G, by a finite etale group scheme, splits over H, and the commutator map of G lifts to H. We prove, moreover, that the quotient stack of G by the natural action of H is the universal Deligne-Mumford Picard stack to which G maps.

Essential dimension and algebraic stacks

We define and study the essential dimension of an algebraic stack. We compute the essential dimension of the stacks Mgn and MgnBar of smooth, or stable, n-pointed curves of genus g. We also prove a general lower bound for the essential dimension of algebraic groups with a non-trivial center. Using this, we find new exponential lower bounds for the essential dimension of spin groups and new formulas for the essential dimension of some finite p-groups. Finally, we apply the lower bound for spin groups to the theory of the Witt ring of quadratic forms over a field k.

Tangencial base points on algebraic stacks

The notion of tangencial base point is well-known for schemes in characteristic zero. We show that the definition in terms of Puiseux series generalizes to the case of Deligne--Mumford stacks, over a field of arbitrary characteristic. We then apply this construction to moduli spaces of smooth curves, generalizing a result of Ihara-Nakamura.

代数叠的S-等价

S-equivalence for algebraic stacks

本文将S-等价概念从半稳定向量丛推广至任意代数叠，用于描述模空间中的点等同，并恢复经典结果及探讨与分离性的关联。

Approximation of sheaves on algebraic stacks

Raynaud--Gruson characterized flat and pure morphisms between affine schemes in terms of projective modules. We give a similar characterization for non-affine morphisms. As an application, we show that every quasi-coherent sheaf is the union of its finitely generated quasi-coherent subsheaves on any quasi-compact and quasi-separated algebraic stack.

Geometry of Moduli Stacks of $(k,l)$-stable vector bundles over an algebraic curve

We study the geometry of the moduli stack of vector bundles of fixed rank and degree over an algebraic curve by introducing a filtration made of open substacks build from $(k, l)$-stable vector bundles. $(k, l)$-stability was introduced by Narasimhan and Ramanan to study the geometry of the coarse moduli space of stable bundles. We will exhibit the stacky picture and analyse the geometric and cohomological properties of the moduli stacks of $(k, l)$-stable vector bundles. For particular pairs $(k, l) $ of integers we also show that these moduli stacks admit coarse moduli spaces and we discuss …

Further remarks on derived categories of algebraic stacks

Let $X$ be an algebraic stack with quasi-affine diagonal of finite type over a field $k$ of characteristic $0$. We extend the well-known equivalence $\mathsf{D}^+(\mathsf{QCoh}(X)) \simeq \mathsf{D}_{\mathrm{qc}}^+(X)$ to unbounded derived categories. We also prove that if $X$ is smooth over $k$, then $\mathsf{D}_{\mathrm{qc}}(X)$ is compactly generated. We accomplish the former using the descendable algebras of Mathew. We also establish related results in positive and mixed characteristics.

关于代数栈的动机同伦类型

On the Motivic Homotopy Type of Algebraic Stacks

本文构造了在Morel-Voevodsky A1-同伦范畴中为局部满射的代数栈的光滑展示，并证明光滑栈的动机具有与光滑概形动机相似的许多性质。

代数栈的Cox环

Cox rings of algebraic stacks

本文严格定义了一般代数栈及正规优秀代数栈上的Cox环乘法结构，证明了其存在性及基于Ext群的（非）唯一性，并与簇上经典构造对比，应用于Mori dream stack理论。

代数栈上伪凝聚复形的有限性定理

Finiteness theorems for pseudo-coherent complexes on algebraic stacks

本文证明代数栈的固有驯服态射下，相对伪凝聚复形的直接像保持伪凝聚性，解决 Illusie 猜想，并推广至非驯服情形，方法基于导出代数几何。

正特征下代数栈的上同调不变量与Brauer群

Cohomological invariants and Brauer groups of algebraic stacks in positive characteristic

引入特征p下代数栈的带模p^r系数上同调不变量理论，并据此完成了任意域上椭圆曲线栈的Brauer群与上同调不变量的计算。

A note on toric Deligne-Mumford stacks

We give a new description of the data needed to specify a morphism from a scheme to a toric Deligne-Mumford stack. The description is given in terms of a collection of line bundles and sections which satisfy certain conditions. As applications, we characterize any toric Deligne-Mumford stack as a product of roots of line bundles over the rigidified stack, describe the torus action, describe morphisms between toric Deligne-Mumford stacks with complete coarse moduli spaces in terms of homogeneous polynomials, and compare two different definitions of toric stacks.

代数栈上的 Atiyah 类

The Atiyah class on algebraic stacks

本文将 Illusie 的 Atiyah 类定义推广至任意代数栈上具拟凝聚上同调的复形，为复形模栈提供了无导出方法的全局阻碍理论，并证明了约化与普通 Atiyah 类及其与张量积和行列式的相容性。

代数栈导出范畴的对偶性

Dualizability of derived categories of algebraic stacks

本文证明了具有拟仿形对角线的 Noetherian 代数栈上拟凝聚层稳定 $\infty$-范畴的对偶性，等价于其几何点处稳定子约化恒等分量为环面，并指出该条件等价于其他范畴条件。

代数栈的orbifold公式

An orbifold formula for algebraic stacks

本文推广orbifold公式至与光滑代数栈双有理的簇，提供计算弦论E-函数的新方法，并应用于klt奇点研究，涉及动机积分及Bruhat-Tits理论。

The Hilbert Stack

Let π: X -> S be a morphism of algebraic stacks that is locally of finite presentation with affine stabilizers. We prove that there is an algebraic S-stack, the Hilbert stack, parameterizing proper algebraic stacks mapping quasi-finitely to X. This was previously unknown, even for a morphism of schemes.

代数栈的平展局部结构

The étale local structure of algebraic stacks

本文证明了具有仿射稳定化子的代数栈在具有线性约化稳定化子的点附近在平展局部上是商栈，并给出了 adic 序列的凝聚完备性等基础结果及相关应用。