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Algebraic stack

An algebraic stack is a mathematical structure in algebraic geometry that generalizes schemes and algebraic spaces to parametrize families of geometric objects with symmetries, defined as a stack in groupoids over the big fppf site of schemes such that the diagonal morphism is representable by algebraic spaces and the stack is representable by a smooth surjective morphism from a scheme.^[1] This framework addresses moduli problems where classical schemes fail due to non-trivial automorphisms, allowing the incorporation of group actions and quotient constructions in a geometrically meaningful way.^[2] Algebraic stacks were first introduced by Pierre Deligne and David Mumford in 1969 to resolve the moduli problem for stable curves of genus greater than or equal to 2, proving the irreducibility of the corresponding moduli space by using stacks to handle the automorphisms of curves.^[3] Their work laid the foundation for Deligne-Mumford stacks, a subclass where the stack admits an étale surjective presentation by a scheme, ensuring a well-behaved geometry suitable for compactifications and deformation theory.^[1] Subsequent developments by Michael Artin in the 1970s extended the theory to Artin stacks, which allow smooth presentations and apply to broader classes of problems, such as principal bundles and gerbes.^[2] A key result in the theory is the Keel-Mori theorem (1999), which guarantees the existence of a coarse moduli space—a scheme approximating the stack by forgetting automorphisms—for any algebraic stack with finite (quasi-finite and separated) diagonal, providing a bridge back to classical algebraic geometry.^[4] Notable examples include the moduli stack of elliptic curves, which is a Deligne-Mumford stack over the complex numbers and admits a coarse moduli space given by the j-invariant line, and quotient stacks [X/G] for a scheme X acted upon by an algebraic group G, capturing geometric quotients.^[5] Algebraic stacks have since become indispensable in modern algebraic geometry, enabling advances in enumerative geometry, mirror symmetry, and the study of derived categories on singular spaces.^[2]

Introduction

Motivation from moduli spaces

Moduli spaces arise in algebraic geometry as a way to classify families of geometric objects, such as curves or sheaves, up to isomorphism. Formally, a moduli problem can be viewed as a functor from the category of schemes to sets, where for a scheme S, the value of the functor assigns the set of isomorphism classes of objects (e.g., elliptic curves) over S. A classic example is the moduli space of elliptic curves, denoted \mathcal{M}_{1,1}, which parametrizes elliptic curves up to isomorphism. However, points in this space often correspond to objects with non-trivial automorphisms; for instance, most elliptic curves have automorphism group \{ \pm 1 \}, while special ones like y^2 = x^3 + x (j-invariant 1728) have \mu_4, and y^2 + y = x^3 (j-invariant 0) have \mu_6. These automorphisms prevent the existence of a fine moduli space representable by a scheme with a universal family, as isomorphisms would not be uniquely determined.^[6]^[7] Naive schemes fail to adequately represent such moduli problems due to their rigidity, which cannot account for the "stacky" nature introduced by automorphisms. Coarse moduli spaces, like the j-line \mathbb{A}^1 for elliptic curves, provide a separated scheme but lose information about automorphisms and fail to support a universal family; for example, maps to the coarse space do not lift uniquely to families of elliptic curves because of varying automorphism groups at special points. This leads to issues like non-separatedness in the quotient or infinite discrete components from infinite automorphisms in higher genus cases, necessitating a "good" moduli interpretation that preserves more structure while remaining scheme-like. Algebraic stacks resolve this by allowing points to have associated automorphism groups, effectively turning the moduli functor into a stack over the site of schemes, where objects over S form a groupoid rather than a set.^[6]^[5] A specific illustration of this rigidity problem is the contrast between the Hilbert scheme and the stack of coherent sheaves. The Hilbert scheme \mathrm{Hilb}_{X/B} parametrizes flat families of subschemes of a scheme X over a base B, representable as an algebraic space under suitable conditions like finite presentation and properness, but it rigidly fixes the embedding and ignores sheaf isomorphisms beyond the scheme structure. In contrast, the stack of coherent sheaves \mathrm{Coh}_{X/B} classifies families of coherent sheaves on X, incorporating descent data and isomorphisms, making it an algebraic stack that handles automorphisms and non-flat cases more flexibly; for instance, it generalizes the Hilbert functor via the Quot scheme but extends to all quasi-coherent modules with proper support. This stack structure is essential for moduli problems where sheaves have non-trivial endomorphisms, avoiding the limitations of scheme representability.^[8] The historical motivation for algebraic stacks traces back to David Mumford's work on Picard groups of moduli problems, where he implicitly treated the moduli of elliptic curves as a stack to compute its Picard group, revealing the need for a framework beyond schemes to handle line bundles on families with symmetries. Mumford's analysis extended ideas from Teichmüller spaces in complex geometry, which parametrize marked Riemann surfaces and exhibit orbifold-like structures due to mapping class group actions, motivating an algebraic analogue to study deformations and automorphisms uniformly across characteristics.^[9]^[10]

Historical context

The foundational ideas underlying algebraic stacks trace back to Alexander Grothendieck's work in the 1960s on descent theory and fibered categories, developed as part of the Séminaire de Géométrie Algébrique (SGA). In SGA 1 (1960–1961), Grothendieck introduced fibered categories in the context of étale coverings and the fundamental group of schemes, providing tools to handle descent data for morphisms. This framework was expanded in SGA 4 (1963–1964), where effective descent for quasi-coherent sheaves under fpqc morphisms was established, laying groundwork for gluing constructions essential to stack theory. A pivotal milestone came in 1969 with Pierre Deligne and David Mumford's paper "The irreducibility of the space of curves of given genus," which introduced the concept of algebraic stacks to resolve issues in compactifying the moduli space of smooth curves by incorporating stable curves with automorphisms.^[11] Their Deligne–Mumford stacks, defined via étale presentations by schemes with finite stabilizers, proved the irreducibility of the moduli stack \overline{\mathcal{M}}_g for genus g \geq 2. This addressed the need for a proper moduli space while accounting for nontrivial automorphisms, a problem arising in classical moduli theory.^[11] In the 1970s, Michael Artin generalized these ideas by extending his axioms for algebraic spaces—developed in the late 1960s—to algebraic stacks in his 1974 paper "Versal deformations and algebraic stacks."^[12] Artin defined algebraic stacks as categories fibered in groupoids over the site of schemes that are locally of finite presentation and admit smooth atlases, allowing for stacks with infinite stabilizers and broader representability criteria. This framework unified Deligne–Mumford stacks as a special case and facilitated applications to deformation theory and quotient constructions.^[12] From the 1990s onward, modern refinements included Kai Behrend and Behrang Noohi's 2005 work on uniformization of Deligne–Mumford curves, comparing algebraic, analytic, and topological stacks,^[13] and Behrang Noohi's 2002 study of fundamental groups for algebraic stacks incorporating inertia structures.^[14] Martin Olsson provided comprehensive treatments in his 2007 lecture notes^[15] and 2016 book "Algebraic Spaces and Stacks,"^[16] emphasizing sheaves and deformation theory on stacks. Angelo Vistoli's 2004 expository notes standardized terminology for fibered categories, Grothendieck topologies, and descent in the stack context, becoming a key reference for the field's foundational machinery.^[17]

Foundational Concepts

Stacks as fibered categories

A fibered category, also known as a category fibred over another category, provides a framework for organizing objects and morphisms relative to a base category, allowing for base change operations. Formally, given categories \mathcal{C} and \mathcal{S}, a functor p: \mathcal{S} \to \mathcal{C} defines a fibration if, for every object x \in \mathcal{S} lying over U \in \mathrm{Ob}(\mathcal{C}) and every morphism f: V \to U in \mathcal{C}, there exists a morphism \varphi: y \to x in \mathcal{S} (called a Cartesian lift) such that p(\varphi) = f, and this lift satisfies a universal property: for any z \to x over a morphism g: W \to U that factors as g = f \circ h with h: W \to V, there is a unique morphism z \to y over h making the triangle commute.^[18] This structure ensures that objects can be "pulled back" along base morphisms in a coherent way.^[19] A cleavage of the fibration further refines this by selecting, for each f: V \to U and x \in \mathcal{S}_U (the fiber category over U), a specific Cartesian lift f^*x \to x, thereby equipping the fibration with pullback functors f^*: \mathcal{S}_U \to \mathcal{S}_V that are functorial in f. These functors satisfy compatibility conditions with composition in \mathcal{C}, such as (g \circ f)^* \cong f^* \circ g^*, up to canonical isomorphism. In the 2-categorical context, fibered categories correspond to pseudo-functors from \mathcal{C}^{\mathrm{op}} to the 2-category of categories, where natural transformations serve as 2-morphisms, enabling a richer structure for handling equivalences and transformations between fibrations.^[18]^[19] Over a site (\mathcal{C}, \tau) equipped with a Grothendieck topology \tau, a stack is defined as a fibered category p: \mathcal{S} \to \mathcal{C} that satisfies the stack condition with respect to \tau: for every covering family \{U_i \to U\}_{i \in I} in \tau, the canonical functor \mathcal{S}_U \to \mathrm{DD}(\{U_i \to U\}) from the fiber over U to the category of descent data—consisting of objects X_i \in \mathcal{S}_{U_i}, isomorphisms \varphi_{ij}: \mathrm{pr}_1^* X_i \to \mathrm{pr}_0^* X_j over U_i \times_U U_j satisfying a cocycle condition on triple overlaps, and 2-isomorphisms ensuring coherence—is an equivalence of categories. This means descent data are effective: every such datum is isomorphic to the pullback of a unique (up to unique isomorphism) object in \mathcal{S}_U.^[20]^[18] The stack condition specifically requires effective descent for representable objects and isomorphisms. Representable objects in \mathcal{S} (those isomorphic to the fiber category of a representable presheaf on \mathcal{C}) descend if an effective epimorphism of representables (a covering in \tau) induces an equivalence on descent data, ensuring that local data glue uniquely to global objects. For isomorphisms, the condition extends to 2-descent, where morphisms between descent data (natural transformations satisfying cocycle conditions) glue to global morphisms in \mathcal{S}_U, reflecting the 2-categorical nature where vertical composition and whiskering ensure associativity and unitarity. This full equivalence captures both essential surjectivity (existence of gluing) and full faithfulness (uniqueness up to isomorphism).^[21]^[18] In contrast, a prestack is a fibered category over the site where the same canonical functor \mathcal{S}_U \to \mathrm{DD}(\{U_i \to U\}) is merely fully faithful for every covering, meaning that objects and morphisms in \mathcal{S}_U correspond bijectively to descent data that are "representable" or consistent locally, but without guaranteeing the existence of gluing for arbitrary descent data—only that local isomorphisms match uniquely. Thus, prestacks satisfy descent for representables and isomorphisms (ensuring no "phantom" automorphisms or inconsistencies) but fail effective descent in general, requiring a stackification process to adjoin missing gluings while preserving the fully faithful part.^[18]^[21] A basic example is the stack \mathrm{Sch}/S of schemes over a base scheme S, fibered over the big site of schemes over S: the fiber category over a test object U \to S consists of schemes X \to U as objects and Cartesian diagrams as morphisms, satisfying the stack condition via étale descent for schemes, though the general categorical setup here abstracts from topology specifics. Equivalently, stacks can be presented as categories fibered in groupoids, where fibers are groupoids and Cartesian morphisms are isomorphisms, providing a concrete 2-categorical model.^[18]^[22]

Groupoid fibered in groupoids

A category fibered in groupoids over a site \mathcal{C} is a fibered category p: \mathcal{S} \to \mathcal{C} such that for every object U \in \mathcal{C}, the fiber \mathcal{S}_U is a groupoid, meaning all morphisms in \mathcal{S}_U are isomorphisms. Additionally, every isomorphism in \mathcal{C} admits a strict Cartesian lift in \mathcal{S}, ensuring that the fibration preserves the groupoid structure precisely. This setup specializes the general framework of fibered categories by restricting to invertible morphisms in the fibers, which simplifies computations in algebraic geometry.^[23]^[18] Every stack, understood as a fibered category satisfying the sheaf and descent conditions over \mathcal{C}, is equivalent to a category fibered in groupoids. This equivalence arises via the nerve construction, which associates to any fibered category a functor to the (2,1)-category of groupoids, yielding a stack in groupoids that is 2-equivalent to the original. In practice, this means algebraic stacks can be realized concretely as quotients by groupoid actions without loss of generality.^[24]^[5] In this formulation, operations such as 2-fiber products are well-defined and exist in the 2-category of fibered categories. Specifically, for two categories fibered in groupoids \mathcal{F}, \mathcal{G}: \mathcal{C}^{op} \to \mathbf{Groupoids}, their 2-fiber product \mathcal{F} \times_\mathcal{C} \mathcal{G} is the category whose objects over U consist of pairs (x, y) with x \in \mathcal{F}(U), y \in \mathcal{G}(U), and an isomorphism \phi: f(x) \to g(y) for the structure functors f, g, with morphisms respecting this data. Equivalence relations on objects emerge naturally: over a base U, objects in the fiber are related via isomorphisms that generate an equivalence relation, allowing the stack to model moduli problems where families are identified up to isomorphism.^[25]^[18] The stack condition—that descent data are effective for every covering sieve—equates precisely to local equivalence relations being effective. To see this, consider a covering \{U_i \to U\}_{i \in I} in the topology of \mathcal{C}. Descent data on an object x \in \mathcal{S}_U consist of pullbacks x_i \in \mathcal{S}_{U_i}, isomorphisms \phi_{ij}: \text{pr}_1^* x_i \to \text{pr}_2^* x_j over U_{ij} = U_i \times_U U_j satisfying the cocycle condition on triple intersections, and an action of these on x. Effectiveness means the category of such descent data is equivalent to \mathcal{S}_U, so every datum glues uniquely to an object over U. Locally, this corresponds to the equivalence relation on the disjoint union \coprod_i x_i generated by the \phi_{ij} being effective, i.e., the quotient category is equivalent to the fiber over U. The proof proceeds by constructing the stackification, where ineffective relations are resolved via the nerve, ensuring the resulting fibered groupoid satisfies descent by iteratively gluing local data.^[21]^[18] This groupoid perspective offers computational advantages, particularly in algebraic geometry, by allowing atlases via representable groupoid presentations. A stack admits an atlas if it is presented as the quotient [U/R] by a representable groupoid object R \rightrightarrows U in the category of schemes over a base, where R defines an equivalence relation on U. Such presentations facilitate explicit geometric constructions and verify stack properties through the representability of U and R.^[5]

Definitions and Variants

Artin algebraic stacks

Artin algebraic stacks provide a broad framework for generalizing schemes and algebraic spaces to incorporate group actions and moduli problems that may not admit coarse moduli spaces. Introduced by Michael Artin, these stacks allow for presentations via smooth morphisms rather than stricter étale ones, enabling the study of more singular or non-separated geometric objects. The concept builds on algebraic spaces by allowing non-trivial automorphisms, often interpreted as "orbifolds" in the algebraic setting, where points may have stabilizer groups.^[1] A stack in groupoids \mathcal{X} over the big fppf site (\mathrm{Sch}/S)_{\mathrm{fppf}} of a base scheme S is an Artin algebraic stack if it satisfies two key axioms, originally formulated by Artin using deformation theory but now stated in terms of representability and presentations. The first axiom, representability of the diagonal, requires that the diagonal morphism \Delta: \mathcal{X} \to \mathcal{X} \times_S \mathcal{X} is representable by algebraic spaces. This means that for any scheme T over S, the 2-fiber product \mathcal{X} \times_{\mathcal{X} \times_S \mathcal{X}} T—which parametrizes pairs of objects in \mathcal{X} over T equipped with isomorphisms between their pullbacks—is an algebraic space. This condition ensures that morphisms into \mathcal{X} behave like morphisms to algebraic spaces, controlling the complexity of isomorphisms between objects. For instance, the morphism condition can be expressed as the functor associating to T the groupoid of objects over T being algebraic, with isomorphisms forming a relative algebraic space over the space of objects. The second axiom, existence of a smooth atlas, requires a smooth surjective morphism U \to \mathcal{X} from a scheme U over S. Here, smoothness means that for every point, the fiber is smooth of finite type, and surjectivity ensures every object in \mathcal{X} locally lifts to an object over U. This atlas provides a presentation of \mathcal{X} as the quotient stack [U / R], where R = U \times_{\mathcal{X}} U is the groupoid inertia, often an algebraic space. The use of smooth rather than étale morphisms allows for greater generality, accommodating stacks with infinite stabilizers or non-reduced structures. Technical conditions on the base S include being locally Noetherian, with local rings \mathcal{O}_{S,s} at finite-type points s being G-rings (geometrically regular after base change to algebraically closed fields).^[26] These ensure the stack's formal properties, such as finite-dimensional tangent and obstruction spaces, hold for versal deformations, as per Artin's original deformation-theoretic criteria. Smoothness over the base is crucial for the atlas to cover geometric fibers without excessive singularities. Deligne–Mumford stacks form a stricter subclass where the atlas is étale and the stabilizer groups are finite.^[1] Algebraic stacks generalize algebraic spaces, which correspond to the case where the atlas is an étale equivalence relation (i.e., the stack is a gerbe over an algebraic space with trivial inertia). This "orbifold" perspective views Artin stacks as spaces with stacky structure encoding symmetries, pivotal for moduli theory beyond rigid objects.

Deligne–Mumford stacks

Deligne–Mumford stacks constitute a distinguished subclass of Artin algebraic stacks, distinguished by their possession of étale presentations that ensure finite automorphism groups, making them particularly suitable for compactifying moduli problems with rigid geometric structures.^[27] A precise definition identifies an Artin algebraic stack \mathcal{X} over a base scheme S as Deligne–Mumford if there exists a scheme U together with a surjective and étale morphism U \to \mathcal{X}.^[27] This étale atlas condition implies that the stack locally resembles a scheme quotiented by finite group actions, with the diagonal morphism \Delta_{\mathcal{X}}: \mathcal{X} \to \mathcal{X} \times_S \mathcal{X} being representable by algebraic spaces and unramified, thereby guaranteeing finite stabilizers for objects in the stack.^[28] In contrast to general Artin algebraic stacks, which are covered by smooth morphisms from schemes and allow for potentially infinite or non-finite automorphism groups, Deligne–Mumford stacks enforce étale covers that restrict the inertia stack \mathcal{I}_{\mathcal{X}} = \mathcal{X} \times_{\Delta_{\mathcal{X}}, \mathcal{X} \times_S \mathcal{X}, \Delta_{\mathcal{X}}} \mathcal{X} to be finite and unramified over \mathcal{X}, reflecting a finite presentation of automorphisms essential for applications in compact moduli theory.^[27] This rigidity distinguishes them from the smoother, more flexible Artin stacks, where the diagonal is smooth rather than unramified.^[28] A fundamental theorem establishes the equivalence of this definition with alternative characterizations: for an algebraic stack \mathcal{X} with representable diagonal, \mathcal{X} is Deligne–Mumford if and only if its diagonal is unramified, or equivalently, if the geometric stabilizers of objects are finite groups.^[28] This equivalence underscores the role of the unramified diagonal in ensuring the existence of an étale atlas, as the finite nature of stabilizers locally trivializes the stack as a gerbe banded by finite groups over an algebraic space.^[27] A canonical example arises in the theory of curves: the moduli stack [\mathcal{M}_{g,n}] parametrizing stable curves of genus g with n marked points is a Deligne–Mumford stack, where the finite stabilizers correspond to the finite automorphism groups of stable curves, enabling a compact Deligne–Mumford compactification of the coarse moduli space.^[29]

Stacks over other sites

Algebraic stacks can be generalized beyond the standard big étale site to other Grothendieck topologies on the category of schemes, such as the fppf (faithfully flat and locally of finite presentation), flat, or Zariski sites.^[30] In the fppf topology, a stack in groupoids over the big fppf site (\mathit{Sch}/S)_{\text{fppf}} is algebraic if it satisfies the sheaf condition for fppf covers, its diagonal morphism is representable by algebraic spaces, and there exists a smooth surjective representable morphism from an algebraic space (often taken to be a scheme) serving as an fppf atlas.^[31] Similar conditions apply over the flat site, where covers are flat morphisms, though the Zariski site—generated by open immersions—yields coarser topologies and typically requires atlases that are Zariski-open covers, limiting its utility for capturing descent data in moduli problems.^[32] These generalizations facilitate applications in areas requiring broader descent properties, such as crystalline cohomology of algebraic stacks. In this context, the fppf or fpqc (faithfully flat and quasi-compact) topology ensures effective descent for crystalline sheaves and de Rham-Witt complexes on stacks, enabling the study of p-adic cohomology for moduli stacks like those of abelian varieties.^[33] Similarly, in rigid geometry over p-adic fields, algebraic stacks over the étale site of rigid analytic spaces or adic spaces extend classical notions to non-Archimedean settings, supporting p-adic Hodge theory and uniformization of Shimura varieties.^[34] A key result is that, for any base scheme S, the categories of algebraic stacks over the big étale site (\mathit{Sch}/S)_{\acute{e}tale} and the big fppf site (\mathit{Sch}/S)_{\text{fppf}} coincide: a stack satisfying the algebraicity conditions in the étale topology automatically does so in the fppf topology, and vice versa, via equivalences with quotient presentations by algebraic spaces.^[35] Deligne–Mumford stacks, which admit étale atlases with finite stabilizers, can likewise be defined over the fpqc site by requiring an fpqc presentation via a groupoid in schemes where the source and target maps are fpqc and the stabilizers are finite étale.^[36] This fpqc variant is particularly relevant in logarithmic geometry, where log moduli stacks parameterize families of curves with log structures encoding degenerations and facilitate compactifications of moduli spaces.^[37]

Key Properties

Atlases and presentations

An atlas for an algebraic stack \mathcal{X} over a base scheme S is a representable surjective morphism u: U \to \mathcal{X} from a scheme U (or more generally, an algebraic space) to \mathcal{X}.^[1] For Artin algebraic stacks, such an atlas is smooth, meaning u is smooth as a morphism of algebraic spaces (or schemes when U is a scheme).^[1] This smooth surjectivity ensures that \mathcal{X} locally resembles the scheme U, facilitating geometric interpretations and computations.^[38] A presentation of an algebraic stack \mathcal{X} arises from a smooth atlas U \to \mathcal{X} by forming the quotient stack [U/R], where R = U \times_{\mathcal{X}} U defines an equivalence relation on U via the source and target maps s, t: R \to U, which are themselves smooth.^[38] The canonical morphism [U/R] \to \mathcal{X} is then an equivalence of stacks, presenting \mathcal{X} as a quotient by this groupoid object in schemes (or algebraic spaces).^[38] Such presentations are banded gerbes over the coarse space when the stabilizer groupoids are appropriate, but in general, they capture the stack's structure via the atlas.^[2] By definition, every Artin algebraic stack admits a smooth atlas, as the existence of a smooth surjective representable morphism from a scheme is part of the defining conditions alongside a diagonal representable by algebraic spaces.^[1] For Deligne–Mumford stacks, which are Artin stacks with an unramified diagonal, there exists an étale atlas: a surjective étale morphism from a scheme U to the stack.^[27] This follows from the unramified diagonal allowing refinement of any smooth atlas to an étale one via étale descent.^[27] Smoothness criteria for atlases or morphisms to algebraic stacks can be assessed using the cotangent complex L_{U/\mathcal{X}}, where the stack is smooth relative to the base if this complex is perfect and concentrated in degree zero (quasi-isomorphic to a vector bundle).^[39] Obstructions to lifting or smoothness often lie in H^1 or higher cohomology of the cotangent complex, providing a derived perspective on infinitesimal deformations.^[39] The structure sheaf on \mathcal{X} may be pulled back from the atlas U to define these complexes compatibly.

Structure sheaf and descent

The structure sheaf \mathcal{O}_\mathcal{X} on an algebraic stack \mathcal{X} over a base scheme S is the inverse image sheaf p^{-1}\mathcal{O} under the structure morphism p: \mathcal{X} \to (\mathrm{Sch}/S)_{\mathrm{fppf}} to the big fppf site, where \mathcal{O} denotes the structure sheaf on (\mathrm{Sch}/S)_{\mathrm{fppf}}. This equips \mathcal{X} with the structure of a ringed site (\mathcal{X}, \mathcal{O}_\mathcal{X}), where \mathcal{O}_\mathcal{X} is a sheaf of \mathcal{O}_S-algebras, and for any object x of \mathcal{X} over a scheme U, the sections are \mathcal{O}_\mathcal{X}(x) = \Gamma(U, \mathcal{O}_U).^[40] For a smooth presentation or atlas U \to \mathcal{X} by a scheme U, \mathcal{O}_\mathcal{X} is constructed by pulling back \mathcal{O}_U along the atlas and endowing the pullback with a suitable descent datum to ensure it glues coherently over \mathcal{X}.^[41] This construction aligns with the stack property in the fppf topology, making \mathcal{O}_\mathcal{X} functorial under morphisms of algebraic stacks. Descent theory provides the mechanism to glue local algebraic structures on an algebraic stack to global ones. In particular, effective descent holds for quasi-coherent sheaves along fpqc covers: given an fpqc covering \{U_i \to \mathcal{X}\}_{i \in I} of \mathcal{X} and a descent datum consisting of quasi-coherent sheaves \mathcal{F}_i on each U_i together with isomorphisms \varphi_{ij}: \mathrm{pr}_1^*\mathcal{F}_i \to \mathrm{pr}_2^*\mathcal{F}_j over U_i \times_\mathcal{X} U_j satisfying the cocycle condition over triple products U_i \times_\mathcal{X} U_j \times_\mathcal{X} U_k, there exists a unique quasi-coherent sheaf \mathcal{F} on \mathcal{X} such that \mathcal{F}|_{U_i} \cong \mathcal{F}_i compatibly with the \varphi_{ij}.^[42] The functor from quasi-coherent \mathcal{O}_\mathcal{X}-modules to such descent data is fully faithful, ensuring that global quasi-coherent sheaves are precisely those arising from effective local data.^[43] For quasi-coherent modules, the descent object corresponding to a cover U \to \mathcal{X} is captured by a descent module M, which consists of a quasi-coherent \mathcal{O}_U-module P on U equipped with an isomorphism over U \times_\mathcal{X} U between the two pullbacks of P, satisfying a cocycle condition; explicitly, sections of M over \mathcal{X} form the equalizer

M(\mathcal{X}) = \eq\left( \Gamma(U, P) \rightrightarrows \Gamma(U \times_\mathcal{X} U, P) \right),

where the two maps are induced by the projections \mathrm{pr}_1, \mathrm{pr}_2: U \times_\mathcal{X} U \to U.^[44] Algebraic stacks in the sense of Artin satisfy affine descent for modules: under the axioms including representability of the diagonal by algebraic spaces, the stack property in the étale topology, and limit preservation, quasi-coherent modules descend effectively along faithful flat covers by affine schemes, with the full conditions ensuring compatibility with infinitesimal thickenings and versal deformations. This affine descent underpins the algebraic structure, allowing reduction to scheme-theoretic computations. The inertia stack I_\mathcal{X} \to \mathcal{X} of an algebraic stack \mathcal{X}, which parametrizes pairs (x, \alpha) where x \in \mathcal{X}(T) for a scheme T and \alpha: x \to x is an automorphism, plays a key role in the theory of equivariant sheaves. For a quotient presentation \mathcal{X} = [U/R] by a groupoid (U, R) in algebraic spaces, I_\mathcal{X} is the quotient [G / R'], where G \to U is the stabilizer group algebraic space (the equalizer of source and target on R) and R' = R \times_{U} G via the source map s: R \to U and the structure map G \to U, equipped with suitable source, target, and composition maps forming a groupoid.^[45] Equivariant quasi-coherent sheaves on \mathcal{X} are pairs (\mathcal{F}, \phi) where \mathcal{F} is quasi-coherent on U and \phi: \mathrm{pr}_1^*\mathcal{F} \to \mathrm{pr}_2^*\mathcal{F} is an isomorphism over R satisfying the cocycle condition with respect to the groupoid multiplication and unit over R \times_U R, making them representations of the groupoid (or modules over the associated category).^[46] For a representation \rho of the stabilizer groups (e.g., a character or linear representation), the associated eigensheaf is the subsheaf of \mathcal{F} on which the inertia action factors through \rho, consisting of sections s such that the action of g \in G satisfies \phi(g \cdot s) = \rho(g) s; these eigensheaves decompose equivariant sheaves into weight components, facilitating computations in representation-theoretic contexts.^[47]

Coarse and fine moduli spaces

In algebraic geometry, a coarse moduli space for an algebraic stack \mathcal{X} locally of finite type over a scheme S is an algebraic space M together with a morphism \pi: \mathcal{X} \to M such that \pi induces a bijection on geometric points (i.e., isomorphism classes of objects over algebraically closed fields), and \pi is universal: any morphism \mathcal{X} \to N to another algebraic space N factors uniquely through \pi.^[48] Moreover, \pi is initial among such maps, and for Deligne-Mumford stacks, the fibers of \pi over closed points with non-trivial stabilizers are classifying stacks of the finite stabilizer groups (gerbes banded by finite étale group schemes corresponding to the stabilizer groups).^[48] This construction forgets the stacky structure (automorphisms and families) while preserving the underlying moduli problem up to isomorphism.^[5] A fine moduli space arises in the special case where \pi is an isomorphism, meaning \mathcal{X} itself is representable by a scheme M; this requires the moduli functor to be representable, which typically demands rigid families with no non-trivial automorphisms.^[5] Such situations are rare for stacks parametrizing objects with symmetries, but they occur when a universal family exists without obstructions from automorphisms, often ensured by the presence of a sufficiently ample line bundle on the parameter space that rigidifies the objects.^[5] For Deligne-Mumford stacks, the existence of a coarse moduli space is guaranteed by the Keel-Mori theorem: every separated Deligne-Mumford stack of finite type over a scheme admits a coarse moduli space, constructed via methods analogous to geometric invariant theory for groupoid quotients.^[49] The theorem applies under the condition of finite inertia (finite stabilizers), which holds for Deligne-Mumford stacks, yielding a separated algebraic space as the coarse space.^[49] A representative example is the Deligne-Mumford stack \overline{\mathcal{M}}_g of stable curves of genus g \geq 2, whose coarse moduli space is the scheme \bar{M}_g, a projective variety parametrizing isomorphism classes of stable curves while the stack encodes their automorphisms.^[48] The map \overline{\mathcal{M}}_g \to \bar{M}_g has fibers that are classifying stacks over points with non-trivial stabilizers, such as curves with extra automorphisms.^[48]

Examples

Classifying stacks

The classifying stack BG for a finite group G over a base field k serves as a prototype for algebraic stacks with constant fibers, parameterizing principal G-torsors over test schemes. It is defined as the fibered category in groupoids over the big étale site of schemes, where an object over a scheme S is a G-torsor P \to S (a principal homogeneous space under G, locally trivial in the étale topology), and morphisms are G-equivariant isomorphisms of torsors. Equivalently, BG can be presented as the quotient stack [\Spec k / G], with G acting trivially on \Spec k. An atlas for BG is given by the universal G-torsor, realized as the scheme G (viewed over \Spec k) equipped with the free left G-action by multiplication, mapping étale-surjectively to BG via the associated quotient; this covers the stack since every G-torsor is locally isomorphic to this universal one.^[2]^[50] The stack BG is universal for principal G-bundles in the sense that, given any G-torsor P \to T over a scheme T, there exists a unique morphism T \to BG in the 2-category of stacks such that P is the pullback of the universal torsor along this map. A defining property is the structure of its inertia stack, which captures automorphisms of objects: the 2-fiber product BG \times_{BG} BG is isomorphic to the stack classifying pairs consisting of a G-torsor equipped with an automorphism, and over a base scheme S it takes the form BG \times_S BG \cong [S/G \times G], where the product reflects the conjugation action of G on itself. This inertia encodes the constant automorphism group G for the trivial torsor and highlights the stack's rigid fiber structure.^[2]^[51] As an algebraic stack, BG is of Artin type when G is a finite group scheme, possessing a representable diagonal and admitting a smooth surjective morphism from an affine scheme. If G is instead a finite étale group scheme, then BG is a Deligne-Mumford stack, with an étale atlas and finite diagonal, ensuring good behavior under base change and descent. These classifications underscore BG's foundational role in stack theory.^[52]^[2] Cohomological computations on BG reduce to classical group cohomology: for the constant sheaf \mathbb{Z}/n\mathbb{Z}, the étale cohomology satisfies H^i(BG, \mathbb{Z}/n\mathbb{Z}) \cong H^i(G, \mathbb{Z}/n\mathbb{Z}), the group cohomology of G.

Quotient stacks

Quotient stacks arise as a fundamental construction in algebraic geometry, generalizing the notion of geometric quotients by group actions to the stacky setting. Given an algebraic group scheme G acting on a scheme X, the quotient stack [X/G] is defined as the stack over the site of schemes that assigns to each test scheme S the groupoid whose objects consist of principal G-torsors P \to S equipped with G-equivariant morphisms P \to X, and whose morphisms are G-equivariant isomorphisms over S. This construction captures the G-equivariant geometry of X, where the natural projection X \to [X/G] serves as an atlas, being a representable, surjective, and smooth morphism.^[53] For [X/G] to be an algebraic stack, suitable finiteness conditions are required on both X and G. Specifically, if X is a scheme locally of finite presentation over a base scheme and G is a group scheme locally of finite presentation acting on X, then [X/G] is an algebraic stack in the sense of Artin. In the more restrictive setting over a field, if X is of finite type and G is an algebraic group of finite type, the quotient stack inherits algebraic properties, including a representable diagonal and local finite presentation. These conditions ensure that [X/G] satisfies the necessary criteria for algebraicity, such as being covered by quotients of affine schemes.^[53]^[2] Rigidification plays a key role in simplifying the study of quotient stacks by quotienting out trivial components of the automorphism groups. For instance, root stacks provide a mechanism for rigidifying along roots of line bundles: given a scheme X, a line bundle L on X, and an integer n \geq 1, the n-th root stack \sqrt{L}/X is the quotient stack [Y / \mu_n], where Y \to X is obtained by adjoining an n-th root section of L (parametrizing pairs (M, \phi) with M^{\otimes n} \cong L and a compatible isomorphism), and \mu_n acts by scaling the root. This construction yields a smooth gerbe over X banded by \mu_n, effectively rigidifying the stack by incorporating root data while preserving the coarse geometry. Root stacks are algebraic when X and L satisfy finite presentation conditions, and they are often used to resolve singularities or add cyclic covers in moduli problems.^[54]^[5] A significant result characterizes when quotient stacks are Deligne-Mumford: if G is a finite group scheme acting on a scheme X, then [X/G] is a Deligne-Mumford stack, as the stabilizers are finite and the inertia stack is finite over [X/G]. Moreover, if the action is free (i.e., stabilizers are trivial), the quotient stack [X/G] is in fact an algebraic space, hence representable and Deligne-Mumford with trivial automorphisms. This follows from the general fact that separated Deligne-Mumford stacks are étale-locally quotients by finite groups, but the global quotient structure simplifies computations of cohomology and moduli.^[53]

Moduli stacks of curves

The moduli stack of stable curves, denoted \overline{\mathcal{M}}_{g,n}, parametrizes families of stable n-pointed curves of genus g, where a stable curve is a connected, projective curve with at worst nodal singularities and finite automorphism group, and the marked points impose no infinitesimal automorphisms. This stack was introduced by Deligne and Mumford as a Deligne–Mumford stack to resolve the irreducibility issues of the coarse moduli space \mathcal{M}_g, and it was extended to the pointed case by Knudsen using similar stack-theoretic methods. The stack \overline{\mathcal{M}}_{g,n} is proper and smooth over \mathbb{Z}, with dimension $3g - 3 + n.^[55] A key feature is the universal curve fibration \pi: \overline{\mathcal{M}}_{g,n+1} \to \overline{\mathcal{M}}_{g,n}, which forgets the last marked point and contracts unstable rational components, providing a universal family of stable pointed curves over the base stack. This fibration is representable and proper, ensuring that \overline{\mathcal{M}}_{g,n+1} serves as the universal curve over \overline{\mathcal{M}}_{g,n}. For stable points in \overline{\mathcal{M}}_{g,n}, the automorphism groups are finite by the stability condition, which endows the stack with an orbifold structure, where stabilizers are finite groups acting on etale neighborhoods. A variant is the moduli stack of weighted pointed stable curves, \overline{\mathcal{M}}_{g,n}(w), where w = (w_1, \dots, w_n) is a weight vector with $0 < w_i \leq 1 and \sum w_i > 2, parametrizing weighted pointed curves in which marked points may coincide if their weights sum to at most 1, and stability requires that the weighted points impose finite automorphisms.^[56] These stacks are Deligne–Mumford, proper over \mathbb{Z}, and connected, constructed via the log minimal model program; they form a family parametrized by weight data, with morphisms \overline{\mathcal{M}}_{g,n}(w') \to \overline{\mathcal{M}}_{g,n}(w) for w' \leq w that are isomorphisms away from strata where weights allow new coincidences. The space \overline{\mathcal{M}}_{g,n}(w) can be realized as a root stack variant of \overline{\mathcal{M}}_{g,n}, obtained by taking roots along the boundary divisors corresponding to marked points, adjusted by the weights to incorporate ramification data at collision loci. In enumerative geometry, the cohomology of \overline{\mathcal{M}}_{g,n} features virtual classes that enable counts of curve configurations, such as intersections in tautological rings or genus-zero Gromov–Witten invariants, via perfect obstruction theories on the stack. Behrend and Fantechi constructed these virtual fundamental classes for Deligne–Mumford stacks using the intrinsic normal cone, providing a rigorous tool for such enumerative problems without relying on explicit resolutions.

Applications

In deformation theory

Algebraic stacks formalize deformation problems by representing functors that classify families of geometric objects up to isomorphism, extending the classical theory for schemes. A deformation functor for an algebraic stack X over a base scheme S associates to each S-scheme T the groupoid of deformations of X over T, capturing both the underlying space and its automorphisms. In this setting, the versal deformation space of X is a pro-representable stack, meaning it is locally represented by a pro-system of affine schemes that universally controls small deformations. This pro-representability ensures that the deformation functor satisfies Schlessinger's conditions, allowing for effective computation of the local structure around points of X.^[57] The cotangent complex L_{X/S} of a morphism of algebraic stacks X \to S governs the infinitesimal deformations and obstructions. Defined in the derived category of quasi-coherent sheaves on X, it measures the linear approximations to deformations via its cohomology. Specifically, for a flat family X \to S and an infinitesimal thickening S' \to S with square-zero ideal sheaf I, an obstruction to lifting the family lies in \Ext^2(L_{X/S}, x^* I), where x: S \to X is a point. If this obstruction vanishes, the isomorphism classes of liftings form a torsor over \Ext^1(L_{X/S}, x^* I), while \Ext^0(L_{X/S}, x^* I) parametrizes automorphisms of the lifted family. These Ext groups arise from the 2-categorical structure of stack deformations, providing a precise control mechanism analogous to the tangent and obstruction spaces for schemes.^[57] Deformations of coherent sheaves on algebraic stacks differ from those on schemes by incorporating the stack's automorphisms, often realized through equivariant structures on presentations. For instance, on a quotient stack [Y/G], a sheaf \mathcal{F} deforms to a G-equivariant family over a base, where the tangent dg-Lie algebra includes derivations respecting the group action, leading to cohomology groups H^i(G, \Der(\mathcal{F}, \mathcal{F})) that control infinitesimal extensions. This contrasts with scheme deformations, which lack such group actions and rely solely on the cotangent sheaf; on stacks, the equivariant condition ensures compatibility with descent data, allowing sheaves to be reconstructed globally from local equivariant pieces. Computations can be simplified using atlases, reducing stack deformations to equivariant ones on schemes. A key result in this context is the Artin approximation theorem adapted to algebraic stacks, which bridges formal and algebraic solutions to deformation problems. For a proper algebraic stack over a field or excellent ring, any formal versal deformation—given by a solution over a complete local ring—can be approximated to arbitrary order by an algebraic deformation over a finite-type scheme, with the completed local rings isomorphic. This algebraization ensures that formal power series solutions in the deformation functor correspond to actual algebraic families, facilitating the study of moduli stacks by guaranteeing the existence of geometric models.^[58]

Connections to derived geometry

Algebraic stacks find a natural extension in derived algebraic geometry, where classical notions are enhanced to incorporate homotopical data, particularly through the framework of derived stacks. Derived stacks are defined as functors from the category of simplicial commutative rings (or more generally, from derived affine schemes) to spaces, satisfying hyperdescent conditions in an ∞-topos, thus generalizing the étale or fppf presentations of classical algebraic stacks.^[59] This homotopical enhancement allows derived stacks to model geometric objects with nontrivial higher homotopy, such as intersections with excess dimension or singularities that classical stacks cannot resolve without losing information.^[59] A key aspect of this enhancement is the replacement of the classical structure sheaf on a ringed topos with a sheaf of simplicial commutative rings (or E_∞-ring spectra in the spectral setting), whose associated derived category of quasi-coherent complexes captures infinitesimal thickenings.^[60] Specifically, the higher homotopy groups π_i(O_X) for i > 0 encode nilpotent extensions and higher-order obstructions, enabling the treatment of derived intersections and cotangent complexes in a geometrically meaningful way.^[60] The category of quasi-coherent complexes on a derived stack is an ∞-category, often presented via DG-categories of modules over the derived structure sheaf, which provide a model for computing derived functors like tensor products and Hom-spaces.^[59] Classical algebraic stacks arise as the underived truncations τ_{\leq 0} of these derived stacks, where higher homotopy groups vanish, recovering the ordinary structure sheaf and category of quasi-coherent sheaves.^[60] Derived atlases play a central role in presenting these objects, consisting of smooth derived schemes or derived groupoids that locally model the stack via étale or smooth morphisms, ensuring that properties like smoothness and properness extend from the atlas to the stack.^[59] A prominent example is the derived moduli stack of perfect complexes on a scheme X, denoted RPerf_X, which is a derived Artin stack representable in this ∞-categorical sense and carries a canonical 2-shifted symplectic structure arising from the trace map on Hochschild homology.^[61] This derived enhancement is crucial for moduli problems involving coherent sheaves, as it resolves classical obstructions through higher Ext groups.^[59] In the context of mirror symmetry, derived stacks provide a bridge between algebraic and symplectic geometry. The derived Fukaya category of a symplectic manifold, enhanced with higher categorical structures, is conjectured to be equivalent to the derived category of coherent sheaves on its mirror algebraic variety, with derived stacks of Lagrangians carrying shifted symplectic or Poisson structures that encode the necessary homotopical data for this duality.^[61] For instance, the moduli stack of derived Lagrangians in a Calabi-Yau manifold admits a (-1)-shifted symplectic structure, mirroring the geometry of coherent sheaves on the dual side.^[61]

References

[1]
Section 94.12 (026N): Algebraic stacks—The Stacks project
94.12 Algebraic stacks. Here is the definition of an algebraic stack. We remark that condition (2) implies we can make sense out of the condition in part ...
[2]
[PDF] Algebraic stacks - arXiv
3. Definition 2.10 (Stack) A stack is a sheaf of groupoids, i.e. a 2-functor (presheaf) that satisfies the following sheaf ...
[3]
[PDF] The irreducibility of the space of curves of given genus - Numdam
In this paper, we will only give definitions and state without proof the general theorems which we apply. Using the method of algebraic stacks, we can prove not ...
[4]
Section 106.13 (0DUK): The Keel-Mori theorem—The Stacks project
106.13 The Keel-Mori theorem. In this section we start discussing the theorem of Keel and Mori in the setting of algebraic stacks.
[5]
[PDF] Introduction to Algebraic Stacks - UBC Mathematics
Dec 17, 2012 · Essentially, a category fibered in groupoids is an algebraic stack, if it is equivalent to the stack of torsors for an algebraic groupoid.
[6]
[PDF] Introduction to algebraic stacks - John Voight
In these notes, we give an introduction to stacks with an eye toward moduli spaces of elliptic curves. The goal is to give a full definition of a Deligne- ...
[7]
[PDF] lectures on moduli spaces of elliptic curves
The goal of these notes is to introduce and motivate basic concepts and constructions (such as orbifolds and stacks) important in the study of moduli spaces of ...
[8]
[PDF] Quot and Hilbert Spaces - Stacks Project
Some of these papers deal with the more general case of the stack of coherent sheaves on an algebraic stack over an algebraic stack and others deal with similar ...
[9]
[PDF] Introducing Algebraic Stacks
This document provides an informal introduction to algebraic stacks, aiming to provide a simple language for thinking about local and global properties of ...
[10]
[PDF] Picard Groups of Moduli - Applied Mathematics
In the fifth section, I describe precisely in two different ways the Picard groups associated to the moduli problem. In the last two sections, for g = 1, we ...Missing: stacks | Show results with:stacks
[11]
[PDF] The irreducibility of the space of curves of given genus
Using the method of algebraic stacks, we can prove not only the irreducibility of Mg itself, but of all higher level moduli spaces of curves too (cf. § 5 below) ...
[12]
Versal deformations and algebraic stacks | Inventiones mathematicae
Papers presented at the Bombay Colloquium, pp. 13–34. Bombay-Oxford: 1969. Artin, M.: Algebraic approximation of structures over complete local rings. Pub ...
[13]
[math/0201021] Fundamental Groups of Algebraic Stacks - arXiv
Jan 4, 2002 · We study fundamental groups of algebraic stacks. We show that these fundamental groups carry an additional structure coming from the inertia groups.Missing: Behrend | Show results with:Behrend
[14]
Notes on Grothendieck topologies, fibered categories and descent ...
Dec 28, 2004 · This is an introduction to Grothendieck's descent theory, with some stress on the general machinery of fibered categories and stacks.Missing: Tohoku SGA
[15]
[PDF] Notes on Grothendieck topologies, fibered categories and descent ...
Oct 2, 2008 · Algébrique du Bois-Marie 1963–1964 (SGA 4), Dirigé par M. Artin, A. Grothendieck et J. L.. Verdier. Avec la collaboration de N. Bourbaki, P ...Missing: Tohoku | Show results with:Tohoku
[16]
Section 4.33 (02XJ): Fibred categories—The Stacks project
4.33 Fibred categories. A very brief discussion of fibred categories is warranted. Let p : \mathcal{S} \to \mathcal{C} be a category over \mathcal{C}.
[17]
8.3 Descent data in fibred categories - Stacks Project
8.3 Descent data in fibred categories. In this section we define the notion of a descent datum in the abstract setting of a fibred category.
[18]
8.8 Stackification of fibred categories - Stacks Project
The result of the procedure in the following lemma will be called the stackification of a fibred category over a site.
[19]
4.35 Categories fibred in groupoids - Stacks project
In this section we explain how to think about categories fibred in groupoids and we see how they are basically the same as functors with values in the (2, 1)- ...
[20]
https://stacks.math.columbia.edu/tag/02ZC
[21]
https://stacks.math.columbia.edu/tag/02ZM
[22]
https://stacks.math.columbia.edu/tag/003S
[23]
[PDF] Artin's Axioms - Stacks Project
If the category fibred in groupoids is an algebraic stack, then every formal object is effective as follows from the next lemma. Lemma 9.5. 07X8. Let S be a ...Missing: prestack fibered
[24]
Definition 94.12.2 (03YO)—The Stacks project
The Stacks project · bibliography · blog · Table of contents; Part 7 ... We say \mathcal{X} is a Deligne-Mumford stack if there exists a scheme U and ...
[25]
[PDF] Deligne–Mumford stacks
Deligne and Mumford identified a class of stacks as algebraic stacks. These are known now as Deligne–Mumford stacks. They are all isomorphic to stacks of the ...
[26]
[PDF] The irreducibility of the space of curves of given genus
Using the method of algebraic stacks, we can prove not only the irreducibility of Mg itself, but of all higher level moduli spaces of curves too (cf. w 5 below) ...
[27]
97.19 Algebraic stacks in the étale topology - Stacks Project
Let S be a scheme. Instead of working with stacks in groupoids over the big fppf site (\mathit{Sch}/S)_{fppf} we could work with stacks in groupoids over the ...
[28]
https://math.colorado.edu/~casa/seminars/reading/stack_of_curves_21/papers/fultonetalstacks/5fultonDMStacks.pdf
[29]
Stacks in the Zariski topology? - ag.algebraic geometry - MathOverflow
Apr 15, 2010 · It's possible to define stacks on ANY category equipped with a Grothendieck topology (such a category with a topology is called a site).What is the Zariski topology good/bad for?Clarifying an interpretation of algebraic spacesMore results from mathoverflow.net
[30]
[PDF] Crystalline cohomology of algebraic stacks and Hyodo-Kato ...
Apr 4, 2017 · We develop a general theory of crystalline cohomology and de Rham-Witt complexes for algebraic stacks, and apply it to the construction and ...
[31]
[2401.07738] The analytic de Rham stack in rigid geometry - arXiv
Jan 15, 2024 · The paper introduces the analytic de Rham stack in rigid geometry, extending D-cap-modules to analytic D-modules, and proving a six functor ...
[32]
Lemma 97.19.1 (076V)—The Stacks project
### Summary of Lemma 97.19.1 and Conditions on Base Scheme S
[33]
[2504.13642] Descent for algebraic stacks - arXiv
Apr 18, 2025 · We prove that algebraic stacks satisfy 2-descent for fppf coverings. We generalize Galois descent for schemes to stacks.
[34]
[1404.0157] Equivalence of two notions of log moduli stacks - arXiv
Apr 1, 2014 · As an application, we obtain several fundamental results of algebraic log stacks which resemble to those in algebraic stacks. Comments: 27 pages.
[35]
Lemma 94.16.2 (04T5)—The Stacks project
Section 94.16: From an algebraic stack to a presentation; Lemma 94.16.2 (cite) ... I suggest the following for readability: "Take a smooth atlas .
[36]
[PDF] a modern introduction to algebraic stacks - Adeel A. Khan
The idea of moduli theory is to transform questions about gadgets into questions about the moduli space, which we may then try to tackle via,.
[37]
Section 96.6 (06TU): The structure sheaf—The Stacks project
The structure sheaf of \mathcal{X} is the sheaf of rings \mathcal{O}_\ ... algebraic stacks is typically not flat (as a morphism of algebraic stacks).
[38]
96.14 Quasi-coherent sheaves and presentations - Stacks Project
The following (formal) proposition tells us that we can study quasi-coherent sheaves on quotient stacks in terms of quasi-coherent modules on presentations.
[39]
https://www.preschema.com/lecture-notes/2022-stacks/stacksncts.pdf
[40]
74.4 Fpqc descent of quasi-coherent sheaves - Stacks Project
The main application of flat descent for modules is the corresponding descent statement for quasi-coherent sheaves with respect to fpqc-coverings. Proposition ...
[41]
74.3 Descent data for quasi-coherent sheaves - Stacks Project
A descent datum (\mathcal{F}_ i, \varphi _{ij}) for quasi-coherent sheaves with respect to the given family is said to be effective if there exists a quasi- ...
[42]
https://stacks.math.columbia.edu/tag/04W8
[43]
https://stacks.math.columbia.edu/tag/04W7
[44]
[PDF] Groupoids in Algebraic Spaces - Stacks Project
of global sections of the structure sheaf. This is representable by the group algebraic space. Ga,B = B ×S Ga,S over B. Here Ga,S is the additive group ...<|control11|><|separator|>
[45]
Subsection 112.5.2 (04UX): Coarse moduli spaces—The Stacks ...
A general coarse moduli space for an Artin stack with finite inertia will only commute with flat base change.
[46]
[alg-geom/9508012] Quotients by Groupoids - arXiv
Aug 25, 1995 · Title:Quotients by Groupoids. Authors:Sean Keel, Shigefumi Mori. View a PDF of the paper titled Quotients by Groupoids, by Sean Keel and 1 other ...
[47]
[PDF] MATH 245C (AN INTRODUCTION TO ALGEBRAIC STACKS)
May 17, 2022 · Algebraic stacks. 40. DEFINITION: ORBIFOLDS, DM STACKS, ALGEBRAIC STACKS, COMPLEX ALGEBRAIC. STACKS. 40.1. Definition. A [blank] is a locally ...<|control11|><|separator|>
[48]
[PDF] Gromov-Witten theory of product stacks - International Press of Boston
The inertia stack of BG admits the following decomposition. I(BG) = (g):conjugacy class. BCG(g), where CG(g) ⊂ G is the centralizer subgroup of g in G. This ...
[49]
Example 100.12.5 (0AFR)—The Stacks project
In particular, the dimension of the classifying stack BG=[\mathop{\mathrm{Spec}}(k)/G] is -\dim (G). Thus the dimension of an algebraic stack can be a negative ...
[50]
[PDF] Hodge theory of classifying stacks - UCLA Mathematics
Let G be a discrete group, considered as a group scheme over a field k. Then the Hodge cohomology of the algebraic stack BG is the group cohomology of G: Hi(BG, ...
[51]
Subsection 112.5.4 (04UZ): Quotient stacks—The Stacks project
The Stacks project · bibliography · blog · Table of contents; Part 9: Miscellany ... This motivates a definition that a Deligne-Mumford stack is projective if ...<|control11|><|separator|>
[52]
[PDF] arXiv:0802.0635v2 [math.AG] 19 Sep 2008
Sep 19, 2008 · Then there is an algebraic stack XH (called the rigidification of X along. H) together with a smooth morphism of algebraic stacks φ : X → XH ...
[53]
109.25 Properties of the stack of stable curves - Stacks Project
The algebraic stack \overline{\mathcal{M}}_ g is a Deligne-Mumford stack, proper and smooth over \mathop{\mathrm{Spec}}(\mathbf{Z}). Moreover, the locus \ ...
[54]
[math/0205009] Moduli spaces of weighted pointed stable curves
May 1, 2002 · A weighted pointed curve consists of a nodal curve and a sequence of marked smooth points, each assigned a number between zero and one. A subset ...
[55]
https://stacks.math.columbia.edu/tag/0E99
[56]
Homotopical Algebraic Geometry II: geometric stacks and applications
Apr 21, 2004 · We give several examples of derived version of classical moduli stacks, as for example the derived stack of local systems on a space, of algebra ...
[57]
[PDF] Derived algebraic geometry
Derived algebraic geometry is an extension of algebraic geometry whose main purpose is to propose a setting to treat geometrically special situations ...<|control11|><|separator|>
[58]
[1111.3209] Shifted Symplectic Structures - arXiv
Nov 14, 2011 · We prove that classifying stacks of reductive groups, as well as the derived stack of perfect complexes, carry canonical 2-shifted symplectic structures.