Introduction to Etale Cohomology
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Bézout's Theorem
Bézout's theorem Toni Annala Contents 1 Introduction 2 2 Bézout's theorem 4 2.1 Ane plane curves . .4 2.2 Projective plane curves . .6 2.3 An elementary proof for the upper bound . 10 2.4 Intersection numbers . 12 2.5 Proof of Bézout's theorem . 16 3 Intersections in Pn 20 3.1 The Hilbert series and the Hilbert polynomial . 20 3.2 A brief overview of the modern theory . 24 3.3 Intersections of hypersurfaces in Pn ...................... 26 3.4 Intersections of subvarieties in Pn ....................... 32 3.5 Serre's Tor-formula . 36 4 Appendix 42 4.1 Homogeneous Noether normalization . 42 4.2 Primes associated to a graded module . 43 1 Chapter 1 Introduction The topic of this thesis is Bézout's theorem. The classical version considers the number of points in the intersection of two algebraic plane curves. It states that if we have two algebraic plane curves, dened over an algebraically closed eld and cut out by multi- variate polynomials of degrees n and m, then the number of points where these curves intersect is exactly nm if we count multiple intersections and intersections at innity. In Chapter 2 we dene the necessary terminology to state this theorem rigorously, give an elementary proof for the upper bound using resultants, and later prove the full Bézout's theorem. A very modest background should suce for understanding this chapter, for example the course Algebra II given at the University of Helsinki should cover most, if not all, of the prerequisites. In Chapter 3, we generalize Bézout's theorem to higher dimensional projective spaces. -
Projective, Flat and Multiplication Modules
NEW ZEALAND JOURNAL OF MATHEMATICS Volume 31 (2002), 115-129 PROJECTIVE, FLAT AND MULTIPLICATION MODULES M a j i d M . A l i a n d D a v i d J. S m i t h (Received December 2001) Abstract. In this note all rings are commutative rings with identity and all modules are unital. We consider the behaviour of projective, flat and multipli cation modules under sums and tensor products. In particular, we prove that the tensor product of two multiplication modules is a multiplication module and under a certain condition the tensor product of a multiplication mod ule with a projective (resp. flat) module is a projective (resp. flat) module. W e investigate a theorem of P.F. Smith concerning the sum of multiplication modules and give a sufficient condition on a sum of a collection of modules to ensure that all these submodules are multiplication. W e then apply our result to give an alternative proof of a result of E l-B ast and Smith on external direct sums of multiplication modules. Introduction Let R be a commutative ring with identity and M a unital i?-module. Then M is called a multiplication module if every submodule N of M has the form IM for some ideal I of R [5]. Recall that an ideal A of R is called a multiplication ideal if for each ideal B of R with B C A there exists an ideal C of R such that B — AC. Thus multiplication ideals are multiplication modules. In particular, invertible ideals of R are multiplication modules. -
Modules Embedded in a Flat Module and Their Approximations
International Journal of Recent Development in Engineering and Technology Website: www.ijrdet.com (ISSN 2347 - 6435 (Online)) Volume 2, Issue 3, March 2014) Modules Embedded in a Flat Module and their Approximations Asma Zaffar1, Muhammad Rashid Kamal Ansari2 Department of Mathematics Sir Syed University of Engineering and Technology Karachi-75300S Abstract--An F- module is a module which is embedded in a Finally, the theory developed above is applied to flat flat module F. Modules embedded in a flat module approximations particularly the I (F)-flat approximations. demonstrate special generalized features. In some aspects We start with some necessary definitions. these modules resemble torsion free modules. This concept generalizes the concept of modules over IF rings and modules 1.1 Definition: Mmod-A is said to be a right F-module over rings whose injective hulls are flat. This study deals with if it is embedded in a flat module F ≠ M mod-A. Similar F-modules in a non-commutative scenario. A characterization definition holds for a left F-module. of F-modules in terms of torsion free modules is also given. Some results regarding the dual concept of homomorphic 1.2 Definition: A ring A is said to be an IF ring if its every images of flat modules are also obtained. Some possible injective module is flat [4]. applications of the theory developed above to I (F)-flat module approximation are also discussed. 1.3 Definition [Von Neumann]: A ring A is regular if, for each a A, and some another element x A we have axa Keyswords-- 16E, 13Dxx = a. -
Pure Exactness and Absolutely Flat Modules
Advances in Theoretical and Applied Mathematics. ISSN 0973-4554 Volume 9, Number 2 (2014), pp. 123-127 © Research India Publications http://www.ripublication.com Pure Exactness and Absolutely Flat Modules Duraivel T Department of Mathematics, Pondicherry University, Puducherry, India, 605014, E-mail: [email protected] Mangayarcarassy S Department of Mathematics, Pondicherry Engineering College, Puducherry, 605014, India, E-mail: [email protected] Athimoolam R Department of Mathematics, Pondicherry Engineering College, Puducherry, 605014, India, E-mail: [email protected] Abstract We show that an R-module M is absolutely flat over R if and only if M is R- flat and IM is a pure submodule of M, for every finitely generated ideal I. We also prove that N is a pure submodule of M and M is absolutely flat over R, if and only if both N and M/N are absolutely flat over R. AMS Subject Classification: 13C11. Keywords: Absolutely Flat Modules, Pure Submodule and Pure exact sequence. 1. Introduction Throughout this article, R denotes a commutative ring with identity and all modules are unitary. For standard terminology, the references are [1] and [7]. The concept of absolutely flat ring is extended to modules as absolutely flat modules, and studied in [2]. An R module M is said to be absolutely flat over R, if for every 2 Duraivel T, Mangayarcarassy S and Athimoolam R R-module N, M ⊗R N is flat R-module. Various characterizations of absolutely flat modules are studied in [2] and [4]. In this article, we show that an R-module M is absolutely flat over R if and only if for every finitely generated ideal I, IM is a pure submodule of M and M is R- flat. -
Arxiv:2002.10139V1 [Math.AC]
SOME RESULTS ON PURE IDEALS AND TRACE IDEALS OF PROJECTIVE MODULES ABOLFAZL TARIZADEH Abstract. Let R be a commutative ring with the unit element. It is shown that an ideal I in R is pure if and only if Ann(f)+I = R for all f ∈ I. If J is the trace of a projective R-module M, we prove that J is generated by the “coordinates” of M and JM = M. These lead to a few new results and alternative proofs for some known results. 1. Introduction and Preliminaries The concept of the trace ideals of modules has been the subject of research by some mathematicians around late 50’s until late 70’s and has again been active in recent years (see, e.g. [3], [5], [7], [8], [9], [11], [18] and [19]). This paper deals with some results on the trace ideals of projective modules. We begin with a few results on pure ideals which are used in their comparison with trace ideals in the sequel. After a few preliminaries in the present section, in section 2 a new characterization of pure ideals is given (Theorem 2.1) which is followed by some corol- laries. Section 3 is devoted to the trace ideal of projective modules. Theorem 3.1 gives a characterization of the trace ideal of a projective module in terms of the ideal generated by the “coordinates” of the ele- ments of the module. This characterization enables us to deduce some new results on the trace ideal of projective modules like the statement arXiv:2002.10139v2 [math.AC] 13 Jul 2021 on the trace ideal of the tensor product of two modules for which one of them is projective (Corollary 3.6), and some alternative proofs for a few known results such as Corollary 3.5 which shows that the trace ideal of a projective module is a pure ideal. -
Commutative Algebra Example Sheet 2
Commutative Algebra Michaelmas 2010 Example Sheet 2 Recall that in this course all rings are commutative 1. Suppose that S is a m.c. subset of a ring A, and M is an A-module. Show that there is a natural isomorphism AS ⊗A M =∼ MS of AS-modules. Deduce that AS is a flat A-module. 2. Suppose that {Mi|i ∈ I} is a set of modules for a ring A. Show that M = Li∈I Mi is a flat A-module if and only if each Mi is a flat A-module. Deduce that projective modules are flat. 3. Show that an A-module is flat if every finitely generated submodule is flat. 4. Show that flatness is a local properties of modules. 5. Suppose that A is a ring and φ: An → Am is an isomorphism of A-modules. Must n = m? 6. Let A be a ring. Suppose that M is an A-module and f : M → M is an A-module map. (i) Show by considering the modules ker f n that if M is Noetherian as an A-module and f is surjective then f is an isomorphism (ii) Show that if M is Artinian as an A-module and f is injective then f is an isomorphism. 7. Let S be the subset of Z consisting of powers of some integer n ≥ 2. Show that the Z-module ZS/Z is Artinian but not Noetherian. 8. Show that C[X1, X2] has a C-subalgebra that is not a Noetherian ring. 9. Suppose that A is a Noetherian ring and A[[x]] is the ring of formal power series over A. -
The Historical Development of Algebraic Geometry∗
The Historical Development of Algebraic Geometry∗ Jean Dieudonn´e March 3, 1972y 1 The lecture This is an enriched transcription of footage posted by the University of Wis- consin{Milwaukee Department of Mathematical Sciences [1]. The images are composites of screenshots from the footage manipulated with Python and the Python OpenCV library [2]. These materials were prepared by Ryan C. Schwiebert with the goal of capturing and restoring some of the material. The lecture presents the content of Dieudonn´e'sarticle of the same title [3] as well as the content of earlier lecture notes [4], but the live lecture is natu- rally embellished with different details. The text presented here is, for the most part, written as it was spoken. This is why there are some ungrammatical con- structions of spoken word, and some linguistic quirks of a French speaker. The transcriber has taken some liberties by abbreviating places where Dieudonn´e self-corrected. That is, short phrases which turned out to be false starts have been elided, and now only the subsequent word-choices that replaced them ap- pear. Unfortunately, time has taken its toll on the footage, and it appears that a brief portion at the beginning, as well as the last enumerated period (VII. Sheaves and schemes), have been lost. (Fortunately, we can still read about them in [3]!) The audio and video quality has contributed to several transcription errors. Finally, a lack of thorough knowledge of the history of algebraic geometry has also contributed some errors. Contact with suggestions for improvement of arXiv:1803.00163v1 [math.HO] 1 Mar 2018 the materials is welcome. -
TENSOR PRODUCTS Balanced Maps. Note. One Can Think of a Balanced
TENSOR PRODUCTS Balanced Maps. Note. One can think of a balanced map β : L × M → G as a multiplication taking its values in G. If instead of β(`; m) we write simply `m (a notation which is often undesirable) the rules simply state that this multiplication should be distributive and associative with respect to the scalar multiplication on the modules. I. e. (`1 + `2)m = `1m + `2m `(m1 + m2)=`m1 + `m2; and (`r)m = `(rm): There are a number of obvious examples of balanced maps. (1) The ring multiplication itself is a balanced map R × R → R. (2) The scalar multiplication on a left R-module is a balanced map R × M → M . (3) If K , M ,andN are left R-modules and E =EndRMthen for δ ∈ EndR M and ' ∈ HomR(M; P) the product ϕδ is defined in the obvious way. This gives HomR(M; P) a natural structure as a right E-module. Analogously, HomR(K; M)isaleftE-moduleinanobviousway.ThereisanE-balanced map β :HomR(M; P) × HomR(K; M) → HomR(K; M)givenby( ;')7→ '. (4) If M and P are left R-modules and E =EndRM, then, as above, HomR(M; P) is a right E-module. Furthermore, M is a left E-module in an obvious way. We then get an E-balanced map HomR(M; P) × M → P by ('; m) 7→'(m). (5) A (distributive) multiplication on an abelian group G is a Z-balanced map G × G → G. It is interesting to note that in examples (1) through (4), the target space for the balancedmapisinfactanR-module, rather than merely an abelian group. -
INTERSECTION NUMBERS, TRANSFERS, and GROUP ACTIONS by Daniel H. Gottlieb and Murad Ozaydin Purdue University University of Oklah
INTERSECTION NUMBERS, TRANSFERS, AND GROUP ACTIONS by Daniel H. Gottlieb and Murad Ozaydin Purdue University University of Oklahoma Abstract We introduce a new definition of intersection number by means of umkehr maps. This definition agrees with an obvious definition up to sign. It allows us to extend the concept parametrically to fibre bundles. For fibre bundles we can then define intersection number transfers. Fibre bundles arise naturally in equivariant situations, so the trace of the action divides intersection numbers. We apply this to equivariant projective varieties and other examples. Also we study actions of non-connected groups and their traces. AMS subject classification 1991: Primary: 55R12, 57S99, 55M25 Secondary: 14L30 Keywords: Poincare Duality, trace of action, Sylow subgroups, Euler class, projective variety. 1 1. Introduction When two manifolds with complementary dimensions inside a third manifold intersect transversally, their intersection number is defined. This classical topological invariant has played an important role in topology and its applications. In this paper we introduce a new definition of intersection number. The definition naturally extends to the case of parametrized manifolds over a base B; that is to fibre bundles over B . Then we can define transfers for the fibre bundles related to the intersection number. This is our main objective , and it is done in Theorem 6. The method of producing transfers depends upon the Key Lemma which we prove here. Special cases of this lemma played similar roles in the establishment of the Euler–Poincare transfer, the Lefschetz number transfer, and the Nakaoka transfer. Two other features of our definition are: 1) We replace the idea of submanifolds with the idea of maps from manifolds, and in effect we are defining the intersection numbers of the maps; 2) We are not restricted to only two maps, we can define the intersection number for several maps, or submanifolds. -
Positivity in Algebraic Geometry I
Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics 48 Positivity in Algebraic Geometry I Classical Setting: Line Bundles and Linear Series Bearbeitet von R.K. Lazarsfeld 1. Auflage 2004. Buch. xviii, 387 S. Hardcover ISBN 978 3 540 22533 1 Format (B x L): 15,5 x 23,5 cm Gewicht: 1650 g Weitere Fachgebiete > Mathematik > Geometrie > Elementare Geometrie: Allgemeines Zu Inhaltsverzeichnis schnell und portofrei erhältlich bei Die Online-Fachbuchhandlung beck-shop.de ist spezialisiert auf Fachbücher, insbesondere Recht, Steuern und Wirtschaft. Im Sortiment finden Sie alle Medien (Bücher, Zeitschriften, CDs, eBooks, etc.) aller Verlage. Ergänzt wird das Programm durch Services wie Neuerscheinungsdienst oder Zusammenstellungen von Büchern zu Sonderpreisen. Der Shop führt mehr als 8 Millionen Produkte. Introduction to Part One Linear series have long stood at the center of algebraic geometry. Systems of divisors were employed classically to study and define invariants of pro- jective varieties, and it was recognized that varieties share many properties with their hyperplane sections. The classical picture was greatly clarified by the revolutionary new ideas that entered the field starting in the 1950s. To begin with, Serre’s great paper [530], along with the work of Kodaira (e.g. [353]), brought into focus the importance of amplitude for line bundles. By the mid 1960s a very beautiful theory was in place, showing that one could recognize positivity geometrically, cohomologically, or numerically. During the same years, Zariski and others began to investigate the more complicated be- havior of linear series defined by line bundles that may not be ample. -
Commutative Algebra
Commutative Algebra Andrew Kobin Spring 2016 / 2019 Contents Contents Contents 1 Preliminaries 1 1.1 Radicals . .1 1.2 Nakayama's Lemma and Consequences . .4 1.3 Localization . .5 1.4 Transcendence Degree . 10 2 Integral Dependence 14 2.1 Integral Extensions of Rings . 14 2.2 Integrality and Field Extensions . 18 2.3 Integrality, Ideals and Localization . 21 2.4 Normalization . 28 2.5 Valuation Rings . 32 2.6 Dimension and Transcendence Degree . 33 3 Noetherian and Artinian Rings 37 3.1 Ascending and Descending Chains . 37 3.2 Composition Series . 40 3.3 Noetherian Rings . 42 3.4 Primary Decomposition . 46 3.5 Artinian Rings . 53 3.6 Associated Primes . 56 4 Discrete Valuations and Dedekind Domains 60 4.1 Discrete Valuation Rings . 60 4.2 Dedekind Domains . 64 4.3 Fractional and Invertible Ideals . 65 4.4 The Class Group . 70 4.5 Dedekind Domains in Extensions . 72 5 Completion and Filtration 76 5.1 Topological Abelian Groups and Completion . 76 5.2 Inverse Limits . 78 5.3 Topological Rings and Module Filtrations . 82 5.4 Graded Rings and Modules . 84 6 Dimension Theory 89 6.1 Hilbert Functions . 89 6.2 Local Noetherian Rings . 94 6.3 Complete Local Rings . 98 7 Singularities 106 7.1 Derived Functors . 106 7.2 Regular Sequences and the Koszul Complex . 109 7.3 Projective Dimension . 114 i Contents Contents 7.4 Depth and Cohen-Macauley Rings . 118 7.5 Gorenstein Rings . 127 8 Algebraic Geometry 133 8.1 Affine Algebraic Varieties . 133 8.2 Morphisms of Affine Varieties . 142 8.3 Sheaves of Functions . -
Flat Modules We Recall Here Some Properties of Flat Modules As
Flat modules We recall here some properties of flat modules as exposed in Bourbaki, Alg´ebreCommutative, ch. 1. The aim of these pages is to expose the proofs of some of the characterizations of flat and faithfully flat modules given in Matsumura's book Commutative Algebra. This part is mainly devoted to the exposition of a proof of the following Theorem. Let A be a commutative ring with identity and E an A-module. The following conditions are equivalent α β (a) for any exact sequence of A-modules 0 ! M 0 / M / M 00 ! 0 , the sequence 0 1E ⊗α 1E ⊗β 00 0 ! E ⊗A M / E ⊗A M / E ⊗A M ! 0 is exact; ∼ (b) for any finitely generated ideal I of A, the sequence 0 ! E ⊗A I ! E ⊗A A = E is exact; 0 b1 1 0 x1 1 t Pr . r . r (c) If bx = i=1 bixi = 0 for some b = @ . A 2 A and x = @ . A 2 E , there exist a matrix br xr s t A 2 Mr×s(A) and y 2 E such that x = Ay and bA = 0. α β We recall that, for any A-module, N, and for any exact sequence 0 ! M 0 / M / M 00 ! 0 , one gets the exact sequence 0 1N ⊗α 1N ⊗β 00 N ⊗A M / N ⊗A M / N ⊗A M ! 0 : Moreover, if αM 0 is a direct summand in M (which is equivalent to the existence of a morphism πM ! M 0 0 1N ⊗α such that πα = 1M 0 ), then the morphism N ⊗A M / N ⊗A M is injective and the image of 1N ⊗ α is a direct summand in N ⊗A M.