On a Morphism of Compactifications of Moduli Schemes of Vector Bundes
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California State University, Northridge Torsion
CALIFORNIA STATE UNIVERSITY, NORTHRIDGE TORSION CLASSES OF COHERENT SHEAVES ON AN ELLIPTIC OR PRODUCT THREEFOLD A thesis submitted in partial fulfillment of the requirements for the degree of Master of Science in Mathematics By Jeremy Keat-Wah Khoo August 2020 The thesis of Jeremy Keat-Wah Khoo is approved: Katherine Stevenson, Ph.D. Date Jerry D. Rosen, Ph.D. Date Jason Lo, Ph.D., Chair Date California State University, Northridge ii Table of Contents Signature page ii Abstract iv 1 Introduction 1 1.1 Our Methods . .1 1.2 Main Results . .2 2 Background Concepts 3 2.1 Concepts from Homological Algebra and Category Theory . .3 2.2 Concepts from Algebraic Geometry . .8 2.3 Concepts from Scheme theory . 10 3 Main Definitions and “Axioms” 13 3.1 The Variety X ................................. 13 3.2 Supports of Coherent Sheaves . 13 3.3 Dimension Subcategories of AX ....................... 14 3.4 Torsion Pairs . 14 3.5 The Relative Fourier-Mukai Transforms Φ; Φ^ ................ 15 3.6 The Product Threefold and Chern Classes . 16 4 Preliminary Results 18 5 Properties Characterizing Tij 23 6 Generating More Torsion Classes 30 6.1 Generalizing Lemma 4.2 . 30 6.2 “Second Generation” Torsion Classes . 34 7 The Torsion Class hC00;C20i 39 References 42 iii ABSTRACT TORSION CLASSES OF COHERENT SHEAVES ON AN ELLIPTIC OR PRODUCT THREEFOLD By Jeremy Keat-Wah Khoo Master of Science in Mathematics Let X be an elliptic threefold admitting a Weierstrass elliptic fibration. We extend the main results of Angeles, Lo, and Van Der Linden in [1] by providing explicit properties charac- terizing the coherent sheaves contained in the torsion classes constructed there. -
X → S Be a Proper Morphism of Locally Noetherian Schemes and Let F Be a Coherent Sheaf on X That Is flat Over S (E.G., F Is Smooth and F Is a Vector Bundle)
COHOMOLOGY AND BASE CHANGE FOR ALGEBRAIC STACKS JACK HALL Abstract. 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, generaliz- ing results of Grothendieck, Brochard, Olsson, Lieblich, and Roth{Starr. To accomplish all of this, we prove that a wide class of relative Ext-functors in algebraic geometry are coherent (in the sense of M. Auslander). Introduction Let f : X ! S be a proper morphism of locally noetherian schemes and let F be a coherent sheaf on X that is flat over S (e.g., f is smooth and F is a vector bundle). If s 2 S is a point, then define Xs to be the fiber of f over s. If s has residue field κ(s), then for each integer q there is a natural base change morphism of κ(s)-vector spaces q q q b (s):(R f∗F) ⊗OS κ(s) ! H (Xs; FXs ): Cohomology and Base Change originally appeared in [EGA, III.7.7.5] in a quite sophisticated form. Mumford [Mum70, xII.5] and Hartshorne [Har77, xIII.12], how- ever, were responsible for popularizing a version similar to the following. Let s 2 S and let q be an integer. (1) The following are equivalent. (a) The morphism bq(s) is surjective. (b) There exists an open neighbourhood U of s such that bq(u) is an iso- morphism for all u 2 U. (c) There exists an open neighbourhood U of s, a coherent OU -module Q, and an isomorphism of functors: Rq+1(f ) (F ⊗ f ∗ I) =∼ Hom (Q; I); U ∗ XU OXU U OU where fU : XU ! U is the pullback of f along U ⊆ S. -
LOCAL PROPERTIES of GOOD MODULI SPACES We Address The
LOCAL PROPERTIES OF GOOD MODULI SPACES JAROD ALPER ABSTRACT. We study the local properties of Artin stacks and their good moduli spaces, if they exist. We show that near closed points with linearly reductive stabilizer, Artin stacks formally locally admit good moduli spaces. In particular, the geometric invariant theory is developed for actions of linearly reductive group schemes on formal affine schemes. We also give conditions for when the existence of good moduli spaces can be deduced from the existence of etale´ charts admitting good moduli spaces. 1. INTRODUCTION We address the question of whether good moduli spaces for an Artin stack can be constructed “locally.” The main results of this paper are: (1) good moduli spaces ex- ist formally locally around points with linearly reductive stabilizer and (2) sufficient conditions are given for the Zariski-local existence of good moduli spaces given the etale-local´ existence of good moduli spaces. We envision that these results may be of use to construct moduli schemes of Artin stacks without the classical use of geometric invariant theory and semi-stability computations. The notion of a good moduli space was introduced in [1] to assign a scheme or algebraic space to Artin stacks with nice geometric properties reminiscent of Mumford’s good GIT quotients. While good moduli spaces cannot be expected to distinguish between all points of the stack, they do parameterize points up to orbit closure equivalence. See Section 2 for the precise definition of a good moduli space and for a summary of its properties. While the paper [1] systematically develops the properties of good moduli spaces, the existence was only proved in certain cases. -
The Scheme of Monogenic Generators and Its Twists
THE SCHEME OF MONOGENIC GENERATORS AND ITS TWISTS SARAH ARPIN, SEBASTIAN BOZLEE, LEO HERR, HANSON SMITH Abstract. Given an extension of algebras B/A, when is B generated by a M single element θ ∈ B over A? We show there is a scheme B/A parameterizing the choice of a generator θ ∈ B, a “moduli space” of generators. This scheme relates naturally to Hilbert schemes and configuration spaces. We give explicit equations and ample examples. M A choice of a generator θ is a point of the scheme B/A. This inspires a local-to-global study of monogeneity, piecing together monogenerators over points, completions, open sets, and so on. Local generators may not come from global ones, but they often glue to twisted monogenerators that we de- fine. We show a number ring has class number one if and only if each twisted monogenerator is in fact a global generator θ. The moduli spaces of various M twisted monogenerators are either a Proj or stack quotient of B/A by natural symmetries. The various moduli spaces defined can be used to apply cohomo- logical tools and other geometric methods for finding rational points to the classical problem of monogenic algebra extensions. Contents 1. Introduction 2 1.1. Twists 3 1.2. Summary of the paper 4 1.3. Guide to notions of “Monogeneity” 4 1.4. Summary of Previous Results 5 1.5. Acknowledgements 6 2. The Scheme of Monogenic Generators 7 2.1. Functoriality of MX 10 2.2. Relation to the Hilbert Scheme 12 M arXiv:2108.07185v1 [math.AG] 16 Aug 2021 3. -
Complete Cohomology and Gorensteinness of Schemes ✩
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Journal of Algebra 319 (2008) 2626–2651 www.elsevier.com/locate/jalgebra Complete cohomology and Gorensteinness of schemes ✩ J. Asadollahi a,b,∗, F. Jahanshahi c,Sh.Salarianc,b a Department of Mathematics, Shahre-Kord University, PO Box 115, Shahre-Kord, Iran b School of Mathematics, Institute for Studies in Theoretical Physics and Mathematics (IPM), PO Box 19395-5746, Tehran, Iran c Department of Mathematics, University of Isfahan, PO Box 81746-73441, Isfahan, Iran Received 20 April 2007 Available online 21 December 2007 Communicated by Steven Dale Cutkosky Abstract We develop and study Tate and complete cohomology theory in the category of sheaves of OX-modules. Different approaches are included. We study the properties of these theories and show their power in reflect- ing the Gorensteinness of the underlying scheme. The connection of these two theories will be discussed. © 2007 Elsevier Inc. All rights reserved. Keywords: Gorenstein scheme; Locally free sheaf; Cohomology of sheaves; Homological dimension; Complete cohomology Contents 1. Introduction . 2627 2. Totally reflexive sheaves . 2628 2.1. Examples and descriptions . 2629 2.2. Gorenstein homological dimension . 2632 3. Tate cohomology sheaves over noetherian schemes . 2634 ✩ This research was in part supported by a grant from IPM (Nos. 86130019 and 86130024). The first author thanks the Center of Excellence of Algebraic Methods and Applications of Isfahan University of Technology (IUT-CEAMA). The second and the third authors thank the Center of Excellence for Mathematics (University of Isfahan). * Corresponding author at: Department of Mathematics, Shahre-Kord University, PO Box 115, Shahre-Kord, Iran. -
Moduli of Curves 1
THE MODULI SPACE OF CURVES In this section, we will give a sketch of the construction of the moduli space Mg of curves of genus g and the closely related moduli space Mg;n of n-pointed curves of genus g using two different approaches. Throughout this section we always assume that 2g − 2 + n > 0. In the first approach, we will embed curves in a projective space PN using a sufficiently high power n of their dualizing sheaf. Then a locus in the Hilbert scheme parameterizes n-canonically embedded curves of genus g. The automorphism group PGL(N + 1) acts on the Hilbert scheme. Using Mumford's Geometric Invariant Theory, one can take the quotient of the appropriate locus in the Hilbert scheme by the action of PGL(N + 1) to construct Mg. In the second approach, one first constructs the moduli space of curves as a Deligne-Mumford stack. One then exhibits an ample line bundle on the coarse moduli scheme of this stack. 1. Basics about curves We begin by collecting basic facts and definitions about stable curves. A curve singularity (C; p) is called a node if locally analytically the singularity is isomophic to the plane curve singularity xy = 0. A curve C is called at-worst-nodal or more simply nodal if the only singularities of C are nodes. The dualizing sheaf !C of an at-worst-nodal curve C is an invertible sheaf that has a simple description. Let ν : Cν ! C be the nor- malization of the curve C. Let p1; : : : ; pδ be the nodes of C and let −1 fri; sig = ν (pi). -
Cotilting Sheaves on Noetherian Schemes
COTILTING SHEAVES ON NOETHERIAN SCHEMES PAVEL COUPEKˇ AND JAN SˇTOVˇ ´ICEKˇ Abstract. We develop theory of (possibly large) cotilting objects of injective dimension at most one in general Grothendieck categories. We show that such cotilting objects are always pure-injective and that they characterize the situation where the Grothendieck category is tilted using a torsion pair to another Grothendieck category. We prove that for Noetherian schemes with an ample family of line bundles a cotilting class of quasi-coherent sheaves is closed under injective envelopes if and only if it is invariant under twists by line bundles, and that such cotilting classes are parametrized by specialization closed subsets disjoint from the associated points of the scheme. Finally, we compute the cotilting sheaves of the latter type explicitly for curves as products of direct images of indecomposable injective modules or completed canonical modules at stalks. Contents 1. Introduction 1 2. Cotilting objects in Grothendieck categories 3 3. Pure-injectivity of cotilting objects 9 4. Derived equivalences 16 5. Torsion pairs in categories of sheaves 18 6. Classification of cotilting sheaves 23 AppendixA. Ext-functorsandproductsinabeliancategories 28 Appendix B. Quasi-coherent sheaves on locally Noetherian schemes 30 B.1. Injective sheaves on locally Noetherian schemes 30 B.2. Supports and associated points 32 B.3. Theclosedmonoidalstructureonsheaves 36 References 37 arXiv:1707.01677v2 [math.AG] 16 Apr 2019 1. Introduction Tilting theory is a collection of well established methods for studying equiv- alences between triangulated categories in homological algebra. Although it has many facets (see [AHHK07]), in its basic form [Hap87, Ric89] it struggles to an- swer the following question: Given two abelian categories A, H, which may not be Keywords and phrases: Grothendieck category, cotilting objects, pure-injective objects, Noe- therian scheme, classification. -
The Moduli Stack Mg
The moduli stack Mg Giulio Orecchia 24 September 2015 This document contains the notes for the first talk of the seminar on Moduli stacks of curves, held in Leiden in Autumn 2015. Thanks to Bas Edixhoven, David Holmes and Jinbi Jin for corrections/annotations. 1 Introduction Many interesting types of objects in algebraic geometry are parametrized in a natural way by geometric objects, called \moduli schemes" or \moduli spaces". Here we are interested in the case of smooth, proper curves of genus g. For these we have a notion of \coarse moduli scheme", usually denoted Mg, which does not have the good properties that a “fine moduli scheme" would have. For example, to a k-point of Mg one cannot always associate a curve defined over k, but rather sep a curve defined over k and isomorphic to its Galois conjugates. Hence, M0(R) consists of only one point although there exists more than one isomorphism class of curves of genus zero defined over . For instance, the conic V (x2 +y2 +z2) ⊂ 2 R PR is not isomorphic to 1 . PR On the other hand, by definition, the k-points of a fine moduli scheme corre- spond bijectively and functorially to (isomorphism classes of) objects defined over k. However, it turns out there cannot exist a fine moduli scheme for smooth curves. This happens more in general when the objects we consider have non-trivial auto- morphisms, as in the case of curves. Nonetheless, if we enlarge the category where we would like our moduli space to live (to the point where it is not a category any- more, but a 2-category), and introduce stacks, we can still find objects enjoying all the properties that we expect from a fine moduli space. -
DEFORMATIONS of FORMAL EMBEDDINGS of Schemesi1 )
TRANSACTIONSOF THE AMERICAN MATHEMATICALSOCIETY Volume 221, Number 2, 1976 DEFORMATIONSOF FORMALEMBEDDINGS OF SCHEMESi1) BY MIRIAM P. HALPERIN ABSTRACT. A family of isolated singularities of k-varieties will be here called equisingular if it can be simultaneously resolved to a family of hypersur- faces embedded in nonsingular spaces which induce only locally trivial deforma- tions of pairs of schemes over local artin Ac-algebras. The functor of locally tri- vial deformations of the formal embedding of an exceptional set has a versal object in the sense of Schlessinger. When the exceptional set Xq is a collection of nonsingular curves meeting normally in a nonsingular surface X, the moduli correspond to Laufer's moduli of thick curves. When X is a nonsingular scheme of finite type over an algebraically closed field k and X0 is a reduced closed subscheme of X, every deformation of (X, Xq) to k[e] such that the deformation of X0 is locally trivial, is in fact a locally trivial deformation of pairs. 1. Introduction. Much progress has been made recently in the classification of normal singularitiesof complex analytic surfaces by considering their resolu- tions (see [3], [7], [11], [12], [13]). The present paper investigates deforma- tions of formally embedded schemes with the aim of eventually using these ob- jects in the algebraic category to classify singularities of dimension two and high- er. The present work suggests the following notion of equisingularity: A family of (isolated) singularities is equisingular if it can be resolved simultaneously to a family of embeddings in nonsingular spaces which induce only locally trivial de- formations of pairs of schemes over any local artin fc-algebra. -
Étale Cohomology
CHAPTER 1 Etale´ cohomology This chapter summarizes the theory of the ´etaletopology on schemes, culmi- nating in the results on `-adic cohomology that are needed in the construction of Galois representations and in the proof of the Ramanujan–Petersson conjecture. In §1.1 we discuss the basic properties of the ´etale topology on a scheme, includ- ing the concept of a constructible sheaf of sets. The ´etalefundamental group and cohomological functors are introduced in §1.2, and we use Cechˇ methods to com- 1 pute some H ’s in terms of π1’s, as in topology. These calculations provide the starting point for the proof of the ´etale analogue of the topological proper base change theorem. This theorem is discussed in §1.3, where we also explain the ´etale analogue of homotopy-invariance for the cohomology of local systems and we intro- duce the vanishing-cycles spectral sequence, Poincar´eduality, the K¨unneth formula, and the comparison isomorphism with topological cohomology over C (for torsion coefficients). The adic formalism is developed in §1.4, and it is used to define ´etalecoho- mology with `-adic coefficients; we discuss the K¨unneth isomorphism and Poincar´e duality with Q`-coefficients, and extend the comparison isomorphism with topo- logical cohomology to the `-adic case. We conclude in §1.5 by discussing ´etale cohomology over finite fields, L-functions of `-adic sheaves, and Deligne’s purity theorems for the cohomology of `-adic sheaves. Our aim is to provide an overview of the main constructions and some useful techniques of proof, not to give a complete account of the theory. -
THE FUNDAMENTAL THEOREMS of HIGHER K-THEORY We Now Restrict Our Attention to Exact Categories and Waldhausen Categories, Where T
CHAPTER V THE FUNDAMENTAL THEOREMS OF HIGHER K-THEORY We now restrict our attention to exact categories and Waldhausen categories, where the extra structure enables us to use the following types of comparison the- orems: Additivity (1.2), Cofinality (2.3), Approximation (2.4), Resolution (3.1), Devissage (4.1), and Localization (2.1, 2.5, 5.1 and 7.3). These are the extensions to higher K-theory of the corresponding theorems of chapter II. The highlight of this chapter is the so-called “Fundamental Theorem” of K-theory (6.3 and 8.2), comparing K(R) to K(R[t]) and K(R[t,t−1]), and its analogue (6.13.2 and 8.3) for schemes. §1. The Additivity theorem If F ′ → F → F ′′ is a sequence of exact functors F ′,F,F ′′ : B→C between two exact categories (or Waldhausen categories), the Additivity Theorem tells us when ′ ′′ the induced maps K(B) → K(C) satisfy F∗ = F∗ + F∗ . To state it, we need to introduce the notion of a short exact sequence of functors, which was mentioned briefly in II(9.1.8). Definition 1.1. (a) If B and C are exact categories, we say that a sequence F ′ → F → F ′′ of exact functors and natural transformations from B to C is a short exact sequence of exact functors, and write F ′ F ։ F ′′, if 0 → F ′(B) → F (B) → F ′′(B) → 0 is an exact sequence in C for every B ∈B. (b) If B and C are Waldhausen categories, we say that F ′ F ։ F ′′ is a short exact sequence, or a cofibration sequence of exact functors if each F ′(B) F (B) ։ F ′′(B) is a cofibration sequence and if for every cofibration A B in B, the evident ′ map F (A) ∪F ′(A) F (B) F (B) is a cofibration in C. -
Handbook of Moduli? the Reason Is That Shimura Varieties Are Often Moduli Spaces of Abelian Varieties with Additional Structure
Local models of Shimura varieties, I. Geometry and combinatorics Georgios Pappas*, Michael Rapoport, and Brian Smithling Abstract. We survey the theory of local models of Shimura varieties. In particular, we discuss their definition and illustrate it by examples. We give an overview of the results on their geometry and combinatorics obtained in the last 15 years. We also exhibit their connections to other classes of algebraic varieties. Introduction Local models of Shimura varieties are projective algebraic varieties over the spectrum of a discrete valuation ring. Their singularities are supposed to model the singularities that arise in the reduction modulo p of Shimura varieties, in the cases where the level structure at p is of parahoric type. The simplest case occurs for the modular curve with Γ0(p)-level structure. In this example the local model 1 is obtained by blowing up the projective line over Spec p at the origin 0 of PZp Z 1 1 the special fiber = ×Spec Spec p. Local models for Shimura varieties are PFp PZp Zp F defined in terms of linear algebra data inside the product of Grassmann varieties, at least as far as type A, or C, or some cases of type D are concerned. Another version of these varieties arises as closures of Schubert varieties inside the Beilinson- Drinfeld-Gaitsgory deformation of affine flag varieties. It is the aim of this survey to discuss local models from various points of view, exhibit their connections to other classes of algebraic varieties, and give an overview of the results on them obtained in the last 15 years.