GAGA THEOREMS 1. Introduction Let X Be a Noetherian Scheme
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
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. -
Arxiv:1810.04493V2 [Math.AG]
MACAULAYFICATION OF NOETHERIAN SCHEMES KĘSTUTIS ČESNAVIČIUS Abstract. To reduce to resolving Cohen–Macaulay singularities, Faltings initiated the program of “Macaulayfying” a given Noetherian scheme X. For a wide class of X, Kawasaki built the sought Cohen–Macaulay modifications, with a crucial drawback that his blowing ups did not preserve the locus CMpXq Ă X where X is already Cohen–Macaulay. We extend Kawasaki’s methods to show that every quasi-excellent, Noetherian scheme X has a Cohen–Macaulay X with a proper map X Ñ X that is an isomorphism over CMpXq. This completes Faltings’ program, reduces the conjectural resolution of singularities to the Cohen–Macaulay case, and implies thatr every proper, smoothr scheme over a number field has a proper, flat, Cohen–Macaulay model over the ring of integers. 1. Macaulayfication as a weak form of resolution of singularities .................. 1 Acknowledgements ................................... ................................ 5 2. (S2)-ification of coherent modules ................................................. 5 3. Ubiquity of Cohen–Macaulay blowing ups ........................................ 9 4. Macaulayfication in the local case ................................................. 18 5. Macaulayfication in the global case ................................................ 20 References ................................................... ........................... 22 1. Macaulayfication as a weak form of resolution of singularities The resolution of singularities, in Grothendieck’s formulation, predicts the following. Conjecture 1.1. For a quasi-excellent,1 reduced, Noetherian scheme X, there are a regular scheme X and a proper morphism π : X Ñ X that is an isomorphism over the regular locus RegpXq of X. arXiv:1810.04493v2 [math.AG] 3 Sep 2020 r CNRS, UMR 8628, Laboratoirer de Mathématiques d’Orsay, Université Paris-Saclay, 91405 Orsay, France E-mail address: [email protected]. -
Special Sheaves of Algebras
Special Sheaves of Algebras Daniel Murfet October 5, 2006 Contents 1 Introduction 1 2 Sheaves of Tensor Algebras 1 3 Sheaves of Symmetric Algebras 6 4 Sheaves of Exterior Algebras 9 5 Sheaves of Polynomial Algebras 17 6 Sheaves of Ideal Products 21 1 Introduction In this note “ring” means a not necessarily commutative ring. If A is a commutative ring then an A-algebra is a ring morphism A −→ B whose image is contained in the center of B. We allow noncommutative sheaves of rings, but if we say (X, OX ) is a ringed space then we mean OX is a sheaf of commutative rings. Throughout this note (X, OX ) is a ringed space. Associated to this ringed space are the following categories: Mod(X), GrMod(X), Alg(X), nAlg(X), GrAlg(X), GrnAlg(X) We show that the forgetful functors Alg(X) −→ Mod(X) and nAlg(X) −→ Mod(X) have left adjoints. If A is a nonzero commutative ring, the forgetful functors AAlg −→ AMod and AnAlg −→ AMod have left adjoints given by the symmetric algebra and tensor algebra con- structions respectively. 2 Sheaves of Tensor Algebras Let F be a sheaf of OX -modules, and for an open set U let P (U) be the OX (U)-algebra given by the tensor algebra T (F (U)). That is, ⊗2 P (U) = OX (U) ⊕ F (U) ⊕ F (U) ⊕ · · · For an inclusion V ⊆ U let ρ : OX (U) −→ OX (V ) and η : F (U) −→ F (V ) be the morphisms of abelian groups given by restriction. For n ≥ 2 we define a multilinear map F (U) × · · · × F (U) −→ F (V ) ⊗ · · · ⊗ F (V ) (m1, . -
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. -
INTRODUCTION to ALGEBRAIC GEOMETRY, CLASS 3 Contents 1
INTRODUCTION TO ALGEBRAIC GEOMETRY, CLASS 3 RAVI VAKIL Contents 1. Where we are 1 2. Noetherian rings and the Hilbert basis theorem 2 3. Fundamental definitions: Zariski topology, irreducible, affine variety, dimension, component, etc. 4 (Before class started, I showed that (finite) Chomp is a first-player win, without showing what the winning strategy is.) If you’ve seen a lot of this before, try to solve: “Fun problem” 2. Suppose f1(x1,...,xn)=0,... , fr(x1,...,xn)=0isa system of r polynomial equations in n unknowns, with integral coefficients, and suppose this system has a finite number of complex solutions. Show that each solution is algebraic, i.e. if (x1,...,xn) is a solution, then xi ∈ Q for all i. 1. Where we are We now have seen, in gory detail, the correspondence between radical ideals in n k[x1,...,xn], and algebraic subsets of A (k). Inclusions are reversed; in particular, maximal proper ideals correspond to minimal non-empty subsets, i.e. points. A key part of this correspondence involves the Nullstellensatz. Explicitly, if X and Y are two algebraic sets corresponding to ideals IX and IY (so IX = I(X)andIY =I(Y),p and V (IX )=X,andV(IY)=Y), then I(X ∪ Y )=IX ∩IY,andI(X∩Y)= (IX,IY ). That “root” is necessary. Some of these links required the following theorem, which I promised you I would prove later: Theorem. Each algebraic set in An(k) is cut out by a finite number of equations. This leads us to our next topic: Date: September 16, 1999. -
Emmy Noether, Greatest Woman Mathematician Clark Kimberling
Emmy Noether, Greatest Woman Mathematician Clark Kimberling Mathematics Teacher, March 1982, Volume 84, Number 3, pp. 246–249. Mathematics Teacher is a publication of the National Council of Teachers of Mathematics (NCTM). With more than 100,000 members, NCTM is the largest organization dedicated to the improvement of mathematics education and to the needs of teachers of mathematics. Founded in 1920 as a not-for-profit professional and educational association, NCTM has opened doors to vast sources of publications, products, and services to help teachers do a better job in the classroom. For more information on membership in the NCTM, call or write: NCTM Headquarters Office 1906 Association Drive Reston, Virginia 20191-9988 Phone: (703) 620-9840 Fax: (703) 476-2970 Internet: http://www.nctm.org E-mail: [email protected] Article reprinted with permission from Mathematics Teacher, copyright March 1982 by the National Council of Teachers of Mathematics. All rights reserved. mmy Noether was born over one hundred years ago in the German university town of Erlangen, where her father, Max Noether, was a professor of Emathematics. At that time it was very unusual for a woman to seek a university education. In fact, a leading historian of the day wrote that talk of “surrendering our universities to the invasion of women . is a shameful display of moral weakness.”1 At the University of Erlangen, the Academic Senate in 1898 declared that the admission of women students would “overthrow all academic order.”2 In spite of all this, Emmy Noether was able to attend lectures at Erlangen in 1900 and to matriculate there officially in 1904. -
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. -
Relative Affine Schemes
Relative Affine Schemes Daniel Murfet October 5, 2006 The fundamental Spec(−) construction associates an affine scheme to any ring. In this note we study the relative version of this construction, which associates a scheme to any sheaf of algebras. The contents of this note are roughly EGA II §1.2, §1.3. Contents 1 Affine Morphisms 1 2 The Spec Construction 4 3 The Sheaf Associated to a Module 8 1 Affine Morphisms Definition 1. Let f : X −→ Y be a morphism of schemes. Then we say f is an affine morphism or that X is affine over Y , if there is a nonempty open cover {Vα}α∈Λ of Y by open affine subsets −1 Vα such that for every α, f Vα is also affine. If X is empty (in particular if Y is empty) then f is affine. Any morphism of affine schemes is affine. Any isomorphism is affine, and the affine property is stable under composition with isomorphisms on either end. Example 1. Any closed immersion X −→ Y is an affine morphism by our solution to (Ex 4.3). Remark 1. A scheme X affine over S is not necessarily affine (for example X = S) and if an affine scheme X is an S-scheme, it is not necessarily affine over S. However, if S is separated then an S-scheme X which is affine is affine over S. Lemma 1. An affine morphism is quasi-compact and separated. Any finite morphism is affine. Proof. Let f : X −→ Y be affine. Then f is separated since any morphism of affine schemes is separated, and the separatedness condition is local. -
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.