DOCSLIB.ORG

Derived Algebraic Geometry VII: Spectral Schemes

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Derived Algebraic Geometry VII: Spectral Schemes Derived Algebraic Geometry VII: Spectral Schemes November 5, 2011 Contents 1 Sheaves of Spectra 4 2 Spectral Schemes 11 3 Coherent 1-Topoi 26 4 Deligne's Completeness Theorem 33 5 Flat Descent 42 6 Flat Descent for Modules 52 7 Digression: Henselian Rings 63 8 Spectral Deligne-Mumford Stacks 68 9 Comparison Results 85 1 Introduction Our goal in this paper is to introduce a variant of algebraic geometry, which we will refer to as spectral algebraic geometry. We will take as our starting point Grothendieck's theory of schemes. Recall that a scheme is a pair (X; OX ), where X is a topological space, OX is a sheaf of commutative rings on X, and the pair (X; OX ) is locally (with respect to the topology of X) isomorphic to the Zariski spectrum of a commutative ring. We can regard a commutative ring R as a set equipped with addition and multiplication maps a : R × R ! R m : R × R ! R which are required to satisfy certain identities. For certain applications (particularly in algebraic topology), it is useful to consider a variation, where R is equipped with a topology. Roughly speaking, a (connective) E1-ring is a space X equipped with continuous addition and multiplication maps a : X × X ! X m : X × X ! X which are required to satisfy the same identities up to (coherent) homotopy. The theory of E1-rings is a robust generalization of commutative algebra: in particular, the basic formal constructions needed to set up the theory of schemes (such as localization) make sense in the setting of E1-rings. We will use this observation to introduce the notion of a spectral scheme: a mathematical object which is obtained by \gluing together" a collection of (connective) E1-rings, just as a scheme is obtained by \gluing together" a collection of commutative rings. The collection of commutative rings can be organized into a category Ring. That is, to every pair of 0 0 0 commutative rings R and R , we can associate a set HomRing(R; R ) of ring homomorphisms from R to R . 0 The analogous statement for E1-rings is more complicated: to every pair of E1-rings R and R , we can associate a space Map(R; R0) of morphisms from R to R0. Moreover, these mapping spaces are equipped with a composition products Map(R; R0) × Map(R0;R00) ! Map(R; R00), which are associative (and unital) up to coherent homotopy. To adequately describe this type of structure, it is convenient to use the language of 1-categories developed in [40]. The collection of all E1-rings is naturally organized into an 1-category which we will denote by CAlg, which contains (the nerve of) the category Ring as a full subcategory. Let us now outline the contents of this paper. Recall first that the category of schemes can be realized as a subcategory of the category of ringed spaces, whose objects are pairs (X; OX ) where X is a topological space and OX is a sheaf of commutative rings on X. Our first goal will be to introduce a suitable 1-categorical version of this category. In x1, we will introduce the 1-category RingTop of spectrally ringed 1-topoi. The objects of RingTop are given by pairs (X; OX), where X is an 1-topos and OX is a sheaf of E1-rings on X. In x2, we will introduce the notion of a spectral scheme. The collection of spectral schemes is organized into an 1-category, which we regard as a subcategory of the 1-category RingTop of spectrally ringed 1- topoi. This subcategory admits a number of characterizations (see Definitions 2.2, 2.7, and 2.27) which we will show to be equivalent. We will also explain the relationship between our theory of spectral schemes and the classical theory of schemes (Proposition 2.37). Let (X; OX ) be a scheme. Recall that X is said to be quasi-compact if every open covering of X has a finite subcovering, and quasi-separated if the collection of quasi-compact open subsets of X is closed under pairwise intersections. In x3, we will generalize these conditions to our 1-categorical setting by introducing the notion of a coherent 1-topos. If X is the 1-topos of sheaves on a topological space X, then X is coherent if and only if X is quasi-compact and quasi-separated. For every spectral scheme (X; OX), the 1-topos X is locally coherent (because coherence is automatic in the affine case), and the coherence of X is an important hypothesis for almost any nontrivial application. Our theory of coherent 1-topoi is an adaptation of the classical theory of coherent topoi (see, for example, [28]). A theorem of Deligne asserts that every coherent topos X has enough points: that is, that there exists a collection of geometric morphisms ffα : Set ! Xg such that a morphism φ in X is invertible if and ∗ only if each pullback fα(φ) is a bijection of sets. In x4, we will prove an 1-categorical analogue of this statement (Theorem 4.1). As an application, we prove a connectivity result for the geometric realization of a hypercovering (Theorem 4.20) which is useful in x5. 2 To every connective E1-ring R, one can associate a spectral scheme SpecZ(R) 2 SpSch. Consequently, cn every spectral scheme X = (X; OX) represents a functor X on the 1-category CAlg of connective E1-rings, given by the formula X(R) = MapSpSch(SpecZ(R); X). In x5, we will show that if X is 0-localic (meaning that the underlying 1-topos of X can be realized as the category of sheaves on a topological space), then X is a sheaf with respect to the flat topology (Theorem 5.15). The proof makes use of the fact that the flat topology is subcanonical on the 1-category of E1-rings: that is, that every corepresentable functor is a sheaf with respect to the flat topology. We will deduce this subcanonicality from a more general result concerning descent for modules over E1-rings, which is proven in x6. In classical algebraic geometry, the category of schemes can be regarded as a full subcategory of a larger 2-category of Deligne-Mumford stacks. In x8, we will introduce the notion of a spectral Deligne-Mumford stack. Our definition involves the notion of a strictly Henselian sheaf of E1-rings, which is generalization of the classical theory of strictly Henselian rings; we include a brief review of the classical theory in x7. As with spectral schemes, we can think of spectral Deligne-Mumford stacks as mathematical objects obtained by \gluing together" connective E1-rings. The difference lies in the nature of the gluing: in the setting of spectral Deligne-Mumford stacks, we replace the Zariski topology by the (far more flexible) ´etaletopology on E1-rings. The collection of all spectral Deligne-Mumford stacks is organized into an 1-category Stk, and there is an evident functor SpSch ! Stk. In x9, we will show that this functor is fully faithful when restricted to the 1-category of 0-localic spectral schemes. Remark 0.1. Our theory of spectral algebraic geometry is closely related to the theory of homotopical algebraic geometry introduced by To¨enand Vezzosi, and there is substantial overlap between their work (see [68], [69], [70], and [71]) and the ideas treated in this paper. Perhaps the primary difference in our presentation is that we stick closely to the classical view of scheme as a kind of ringed space, while To¨enand Vezzosi make use of the \functor of points" philosophy which identifies an algebro-geometric object X with the underlying functor R 7! Hom(Spec R; X). Notation and Terminology This paper will make extensive use of the theory of 1-categories, as developed in [40]. We will also need the theory of structured ring spectra, which is presented from an 1-categorical point of view in [41]. Finally, we will make use of the theory of geometries developed in [42], and earlier paper in this series. For convenience, we will adopt the following reference conventions: (T ) We will indicate references to [40] using the letter T. (A) We will indicate references to [41] using the letter A. (V ) We will indicate references to [42] using the Roman numeral V. For example, Theorem T.6.1.0.6 refers to Theorem 6.1.0.6 of [40]. Let R be a commutative ring. We let SpecZ R denote the collection of all prime ideals in R. We will refer to SpecZ R as the Zariski spectrum of R. We regard SpecZ R as endowed with the Zariski topology: a set Z Z U ⊆ Spec R is open if and only if there exists an ideal I ⊆ R such that U = fp 2 Spec R : I * pg. This Z topology has a basis of open sets given by Ux = fp 2 Spec R : x2 = pg, where x ranges over the collection of elements of R. Z Z If R is an E1-ring, we let Spec R denote the Zariski spectrum Spec (π0R) of the commutative ring π0R. We will occasionally need the following result from commutative algebra: Proposition 0.2. Let f : R ! R0 be an ´etalemap of commutative rings. Then f induces an open map of topological spaces SpecZ(R0) ! SpecZ(R). 3 Let X be an 1-topos and let OX be a sheaf on X with values in an 1-category C (that is, a functor op X ! C which preserves small limits).
Load more

Recommended publications
  • De Rham Cohomology
    De Rham Cohomology 1. Definition of De Rham Cohomology Let X be an open subset of the plane. If we denote by C0(X) the set of smooth (i. e. infinitely differentiable functions) on X and C1(X), the smooth 1-forms on X (i. e. expressions of the form fdx + gdy where f; g 2 C0(X)), we have natural differntiation map d : C0(X) !C1(X) given by @f @f f 7! dx + dy; @x @y usually denoted by df. The kernel for this map (i. e. set of f with df = 0) is called the zeroth De Rham Cohomology of X and denoted by H0(X). It is clear that these are precisely the set of locally constant functions on X and it is a vector space over R, whose dimension is precisley the number of connected components of X. The image of d is called the set of exact forms on X. The set of pdx + qdy 2 C1(X) @p @q such that @y = @x are called closed forms. It is clear that exact forms and closed forms are vector spaces and any exact form is a closed form. The quotient vector space of closed forms modulo exact forms is called the first De Rham Cohomology and denoted by H1(X). A path for this discussion would mean piecewise smooth. That is, if γ : I ! X is a path (a continuous map), there exists a subdivision, 0 = t0 < t1 < ··· < tn = 1 and γ(t) is continuously differentiable in the open intervals (ti; ti+1) for all i.
    [Show full text]
  • Mathematics 1
    Mathematics 1 MAA 4103 Introduction to Advanced Calculus for Engineers and Physical MATHEMATICS Scientists 2 3 Credits Grading Scheme: Letter Grade Not all courses are offered every semester. Refer to the schedule of Continues the advanced calculus for engineers and physical scientists courses for each term's specific offerings. sequence. Theory of integration, transcendental functions and infinite More Info (http://registrar.ufl.edu/soc/) series. MAA 4102 is not recommended for those who plan to do graduate work in mathematics; these students should take MAA 4212. Credit will Unless otherwise indicated in the course description, all courses at the be given for, at most, one of MAA 4103, MAA 4212 and MAA 5105. University of Florida are taught in English, with the exception of specific Prerequisite: MAA 4102 with minimum grade of C. foreign language courses. MAA 4211 Advanced Calculus 1 3 Credits Department Information Grading Scheme: Letter Grade Advanced treatment of limits, differentiation, integration and series. Graduates from the Department of Mathematics might take a job Includes calculus of functions of several variables. Credit will be given for, that uses their math major in an area like statistics, biomathematics, at most, one of MAA 4211, MAA 4102 and MAA 5104. operations research, actuarial science, mathematical modeling, Prerequisite: MAS 4105 with minimum grade of C. cryptography, or mathematics education. Or they might continue into graduate school leading to a research career. Professional schools in MAA 4212 Advanced Calculus 2 3 Credits business, law, and medicine appreciate mathematics majors because of Grading Scheme: Letter Grade the analytical and problem solving skills developed in the math courses.
    [Show full text]
  • The Jouanolou-Thomason Homotopy Lemma
    The Jouanolou-Thomason homotopy lemma Aravind Asok February 9, 2009 1 Introduction The goal of this note is to prove what is now known as the Jouanolou-Thomason homotopy lemma or simply \Jouanolou's trick." Our main reason for discussing this here is that i) most statements (that I have seen) assume unncessary quasi-projectivity hypotheses, and ii) most applications of the result that I know (e.g., in homotopy K-theory) appeal to the result as merely a \black box," while the proof indicates that the construction is quite geometric and relatively explicit. For simplicity, throughout the word scheme means separated Noetherian scheme. Theorem 1.1 (Jouanolou-Thomason homotopy lemma). Given a smooth scheme X over a regular Noetherian base ring k, there exists a pair (X;~ π), where X~ is an affine scheme, smooth over k, and π : X~ ! X is a Zariski locally trivial smooth morphism with fibers isomorphic to affine spaces. 1 Remark 1.2. In terms of an A -homotopy category of smooth schemes over k (e.g., H(k) or H´et(k); see [MV99, x3]), the map π is an A1-weak equivalence (use [MV99, x3 Example 2.4]. Thus, up to A1-weak equivalence, any smooth k-scheme is an affine scheme smooth over k. 2 An explicit algebraic form Let An denote affine space over Spec Z. Let An n 0 denote the scheme quasi-affine and smooth over 2m Spec Z obtained by removing the fiber over 0. Let Q2m−1 denote the closed subscheme of A (with coordinates x1; : : : ; x2m) defined by the equation X xixm+i = 1: i Consider the following simple situation.
    [Show full text]
  • 1.1.5 Hausdorff Spaces
    1. Preliminaries Proof. Let xn n N be a sequence of points in X convergent to a point x X { } 2 2 and let U (f(x)) in Y . It is clear from Definition 1.1.32 and Definition 1.1.5 1 2F that f − (U) (x). Since xn n N converges to x,thereexistsN N s.t. 1 2F { } 2 2 xn f − (U) for all n N.Thenf(xn) U for all n N. Hence, f(xn) n N 2 ≥ 2 ≥ { } 2 converges to f(x). 1.1.5 Hausdor↵spaces Definition 1.1.40. A topological space X is said to be Hausdor↵ (or sepa- rated) if any two distinct points of X have neighbourhoods without common points; or equivalently if: (T2) two distinct points always lie in disjoint open sets. In literature, the Hausdor↵space are often called T2-spaces and the axiom (T2) is said to be the separation axiom. Proposition 1.1.41. In a Hausdor↵space the intersection of all closed neigh- bourhoods of a point contains the point alone. Hence, the singletons are closed. Proof. Let us fix a point x X,whereX is a Hausdor↵space. Denote 2 by C the intersection of all closed neighbourhoods of x. Suppose that there exists y C with y = x. By definition of Hausdor↵space, there exist a 2 6 neighbourhood U(x) of x and a neighbourhood V (y) of y s.t. U(x) V (y)= . \ ; Therefore, y/U(x) because otherwise any neighbourhood of y (in particular 2 V (y)) should have non-empty intersection with U(x).
    [Show full text]
  • The Non-Hausdorff Number of a Topological Space
    THE NON-HAUSDORFF NUMBER OF A TOPOLOGICAL SPACE IVAN S. GOTCHEV Abstract. We call a non-empty subset A of a topological space X finitely non-Hausdorff if for every non-empty finite subset F of A and every family fUx : x 2 F g of open neighborhoods Ux of x 2 F , \fUx : x 2 F g 6= ; and we define the non-Hausdorff number nh(X) of X as follows: nh(X) := 1 + supfjAj : A is a finitely non-Hausdorff subset of Xg. Using this new cardinal function we show that the following three inequalities are true for every topological space X d(X) (i) jXj ≤ 22 · nh(X); 2d(X) (ii) w(X) ≤ 2(2 ·nh(X)); (iii) jXj ≤ d(X)χ(X) · nh(X) and (iv) jXj ≤ nh(X)χ(X)L(X) is true for every T1-topological space X, where d(X) is the density, w(X) is the weight, χ(X) is the character and L(X) is the Lindel¨ofdegree of the space X. The first three inequalities extend to the class of all topological spaces d(X) Pospiˇsil'sinequalities that for every Hausdorff space X, jXj ≤ 22 , 2d(X) w(X) ≤ 22 and jXj ≤ d(X)χ(X). The fourth inequality generalizes 0 to the class of all T1-spaces Arhangel ski˘ı’s inequality that for every Hausdorff space X, jXj ≤ 2χ(X)L(X). It is still an open question if 0 Arhangel ski˘ı’sinequality is true for all T1-spaces. It follows from (iv) that the answer of this question is in the afirmative for all T1-spaces with nh(X) not greater than the cardianilty of the continuum.
    [Show full text]
  • Partial Mal'tsevness and Partial Protomodularity
    PARTIAL MAL’TSEVNESS AND PARTIAL PROTOMODULARITY DOMINIQUE BOURN Abstract. We introduce the notion of Mal’tsev reflection which allows us to set up a partial notion of Mal’tsevness with respect to a class Σ of split epimorphisms stable under pullback and containing the isomorphisms, and we investigate what is remaining of the properties of the global Mal’tsev context. We introduce also the notion of partial protomodularity in the non-pointed context. Introduction A Mal’tsev category is a category in which any reflexive relation is an equivalence relation, see [9] and [10]. The categories Gp of groups and K-Lie of Lie K-algebras are major examples of Mal’tsev categories. The terminology comes from the pi- oneering work of Mal’tsev in the varietal context [16] which was later on widely developped in [18]. In [3], Mal’tsev categories were characterized in terms of split epimorphisms: a finitely complete category D is a Mal’tsev one if and only if any pullback of split epimorphisms in D: t¯ X′ o X O g¯ / O ′ ′ f s f s t Y ′ o Y g / is such that the pair (s′, t¯) is jointly extremally epic. More recently the same kind of property was observed, but only for a certain class Σ of split epimorphisms (f,s) which is stable under pullback and contains arXiv:1507.02886v1 [math.CT] 10 Jul 2015 isomorphisms. By the classes of Schreier or homogeneous split epimorphisms in the categories Mon of monoids and SRng of semi-rings [8], by the classes of puncturing or acupuncturing split epimorphims in the category of quandles [6], by the class of split epimorphic functors with fibrant splittings in the category CatY of categories with a fixed set of objects Y [5].
    [Show full text]
  • Math 601 Algebraic Topology Hw 4 Selected Solutions Sketch/Hint
    MATH 601 ALGEBRAIC TOPOLOGY HW 4 SELECTED SOLUTIONS SKETCH/HINT QINGYUN ZENG 1. The Seifert-van Kampen theorem 1.1. A refinement of the Seifert-van Kampen theorem. We are going to make a refinement of the theorem so that we don't have to worry about that openness problem. We first start with a definition. Definition 1.1 (Neighbourhood deformation retract). A subset A ⊆ X is a neighbourhood defor- mation retract if there is an open set A ⊂ U ⊂ X such that A is a strong deformation retract of U, i.e. there exists a retraction r : U ! A and r ' IdU relA. This is something that is true most of the time, in sufficiently sane spaces. Example 1.2. If Y is a subcomplex of a cell complex, then Y is a neighbourhood deformation retract. Theorem 1.3. Let X be a space, A; B ⊆ X closed subspaces. Suppose that A, B and A \ B are path connected, and A \ B is a neighbourhood deformation retract of A and B. Then for any x0 2 A \ B. π1(X; x0) = π1(A; x0) ∗ π1(B; x0): π1(A\B;x0) This is just like Seifert-van Kampen theorem, but usually easier to apply, since we no longer have to \fatten up" our A and B to make them open. If you know some sheaf theory, then what Seifert-van Kampen theorem really says is that the fundamental groupoid Π1(X) is a cosheaf on X. Here Π1(X) is a category with object pints in X and morphisms as homotopy classes of path in X, which can be regard as a global version of π1(X).
    [Show full text]
  • Open Sets in Topological Spaces
    International Mathematical Forum, Vol. 14, 2019, no. 1, 41 - 48 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/imf.2019.913 ii – Open Sets in Topological Spaces Amir A. Mohammed and Beyda S. Abdullah Department of Mathematics College of Education University of Mosul, Mosul, Iraq Copyright © 2019 Amir A. Mohammed and Beyda S. Abdullah. This article is distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract In this paper, we introduce a new class of open sets in a topological space called 푖푖 − open sets. We study some properties and several characterizations of this class, also we explain the relation of 푖푖 − open sets with many other classes of open sets. Furthermore, we define 푖푤 − closed sets and 푖푖푤 − closed sets and we give some fundamental properties and relations between these classes and other classes such as 푤 − closed and 훼푤 − closed sets. Keywords: 훼 − open set, 푤 − closed set, 푖 − open set 1. Introduction Throughout this paper we introduce and study the concept of 푖푖 − open sets in topological space (푋, 휏). The 푖푖 − open set is defined as follows: A subset 퐴 of a topological space (푋, 휏) is said to be 푖푖 − open if there exist an open set 퐺 in the topology 휏 of X, such that i. 퐺 ≠ ∅,푋 ii. A is contained in the closure of (A∩ 퐺) iii. interior points of A equal G. One of the classes of open sets that produce a topological space is 훼 − open.
    [Show full text]
  • MTH 304: General Topology Semester 2, 2017-2018
    MTH 304: General Topology Semester 2, 2017-2018 Dr. Prahlad Vaidyanathan Contents I. Continuous Functions3 1. First Definitions................................3 2. Open Sets...................................4 3. Continuity by Open Sets...........................6 II. Topological Spaces8 1. Definition and Examples...........................8 2. Metric Spaces................................. 11 3. Basis for a topology.............................. 16 4. The Product Topology on X × Y ...................... 18 Q 5. The Product Topology on Xα ....................... 20 6. Closed Sets.................................. 22 7. Continuous Functions............................. 27 8. The Quotient Topology............................ 30 III.Properties of Topological Spaces 36 1. The Hausdorff property............................ 36 2. Connectedness................................. 37 3. Path Connectedness............................. 41 4. Local Connectedness............................. 44 5. Compactness................................. 46 6. Compact Subsets of Rn ............................ 50 7. Continuous Functions on Compact Sets................... 52 8. Compactness in Metric Spaces........................ 56 9. Local Compactness.............................. 59 IV.Separation Axioms 62 1. Regular Spaces................................ 62 2. Normal Spaces................................ 64 3. Tietze's extension Theorem......................... 67 4. Urysohn Metrization Theorem........................ 71 5. Imbedding of Manifolds..........................
    [Show full text]
  • NOTES on CARTIER and WEIL DIVISORS Recall: Definition 0.1. A
    NOTES ON CARTIER AND WEIL DIVISORS AKHIL MATHEW Abstract. These are notes on divisors from Ravi Vakil's book [2] on scheme theory that I prepared for the Foundations of Algebraic Geometry seminar at Harvard. Most of it is a rewrite of chapter 15 in Vakil's book, and the originality of these notes lies in the mistakes. I learned some of this from [1] though. Recall: Definition 0.1. A line bundle on a ringed space X (e.g. a scheme) is a locally free sheaf of rank one. The group of isomorphism classes of line bundles is called the Picard group and is denoted Pic(X). Here is a standard source of line bundles. 1. The twisting sheaf 1.1. Twisting in general. Let R be a graded ring, R = R0 ⊕ R1 ⊕ ::: . We have discussed the construction of the scheme ProjR. Let us now briefly explain the following additional construction (which will be covered in more detail tomorrow). L Let M = Mn be a graded R-module. Definition 1.1. We define the sheaf Mf on ProjR as follows. On the basic open set D(f) = SpecR(f) ⊂ ProjR, we consider the sheaf associated to the R(f)-module M(f). It can be checked easily that these sheaves glue on D(f) \ D(g) = D(fg) and become a quasi-coherent sheaf Mf on ProjR. Clearly, the association M ! Mf is a functor from graded R-modules to quasi- coherent sheaves on ProjR. (For R reasonable, it is in fact essentially an equiva- lence, though we shall not need this.) We now set a bit of notation.
    [Show full text]
  • Arxiv:1307.5568V2 [Math.AG]
    PARTIAL POSITIVITY: GEOMETRY AND COHOMOLOGY OF q-AMPLE LINE BUNDLES DANIEL GREB AND ALEX KURONYA¨ To Rob Lazarsfeld on the occasion of his 60th birthday Abstract. We give an overview of partial positivity conditions for line bundles, mostly from a cohomological point of view. Although the current work is to a large extent of expository nature, we present some minor improvements over the existing literature and a new result: a Kodaira-type vanishing theorem for effective q-ample Du Bois divisors and log canonical pairs. Contents 1. Introduction 1 2. Overview of the theory of q-ample line bundles 4 2.1. Vanishing of cohomology groups and partial ampleness 4 2.2. Basic properties of q-ampleness 7 2.3. Sommese’s geometric q-ampleness 15 2.4. Ample subschemes, and a Lefschetz hyperplane theorem for q-ample divisors 17 3. q-Kodaira vanishing for Du Bois divisors and log canonical pairs 19 References 23 1. Introduction Ampleness is one of the central notions of algebraic geometry, possessing the extremely useful feature that it has geometric, numerical, and cohomological characterizations. Here we will concentrate on its cohomological side. The fundamental result in this direction is the theorem of Cartan–Serre–Grothendieck (see [Laz04, Theorem 1.2.6]): for a complete arXiv:1307.5568v2 [math.AG] 23 Jan 2014 projective scheme X, and a line bundle L on X, the following are equivalent to L being ample: ⊗m (1) There exists a positive integer m0 = m0(X, L) such that L is very ample for all m ≥ m0. (2) For every coherent sheaf F on X, there exists a positive integer m1 = m1(X, F, L) ⊗m for which F ⊗ L is globally generated for all m ≥ m1.
    [Show full text]
  • 4. Coherent Sheaves Definition 4.1. If (X,O X) Is a Locally Ringed Space
    4. Coherent Sheaves Definition 4.1. If (X; OX ) is a locally ringed space, then we say that an OX -module F is locally free if there is an open affine cover fUig of X such that FjUi is isomorphic to a direct sum of copies of OUi . If the number of copies r is finite and constant, then F is called locally free of rank r (aka a vector bundle). If F is locally free of rank one then we way say that F is invertible (aka a line bundle). The group of all invertible sheaves under tensor product, denoted Pic(X), is called the Picard group of X. A sheaf of ideals I is any OX -submodule of OX . Definition 4.2. Let X = Spec A be an affine scheme and let M be an A-module. M~ is the sheaf which assigns to every open subset U ⊂ X, the set of functions a s: U −! Mp; p2U which can be locally represented at p as a=g, a 2 M, g 2 R, p 2= Ug ⊂ U. Lemma 4.3. Let A be a ring and let M be an A-module. Let X = Spec A. ~ (1) M is a OX -module. ~ (2) If p 2 X then Mp is isomorphic to Mp. ~ (3) If f 2 A then M(Uf ) is isomorphic to Mf . Proof. (1) is clear and the rest is proved mutatis mutandis as for the structure sheaf. Definition 4.4. An OX -module F on a scheme X is called quasi- coherent if there is an open cover fUi = Spec Aig by affines and ~ isomorphisms FjUi ' Mi, where Mi is an Ai-module.
    [Show full text]