The Constructible Topology on Spaces of Valuation Domains
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Dedekind Domains
Dedekind Domains Mathematics 601 In this note we prove several facts about Dedekind domains that we will use in the course of proving the Riemann-Roch theorem. The main theorem shows that if K=F is a finite extension and A is a Dedekind domain with quotient field F , then the integral closure of A in K is also a Dedekind domain. As we will see in the proof, we need various results from ring theory and field theory. We first recall some basic definitions and facts. A Dedekind domain is an integral domain B for which every nonzero ideal can be written uniquely as a product of prime ideals. Perhaps the main theorem about Dedekind domains is that a domain B is a Dedekind domain if and only if B is Noetherian, integrally closed, and dim(B) = 1. Without fully defining dimension, to say that a ring has dimension 1 says nothing more than nonzero prime ideals are maximal. Moreover, a Noetherian ring B is a Dedekind domain if and only if BM is a discrete valuation ring for every maximal ideal M of B. In particular, a Dedekind domain that is a local ring is a discrete valuation ring, and vice-versa. We start by mentioning two examples of Dedekind domains. Example 1. The ring of integers Z is a Dedekind domain. In fact, any principal ideal domain is a Dedekind domain since a principal ideal domain is Noetherian integrally closed, and nonzero prime ideals are maximal. Alternatively, it is easy to prove that in a principal ideal domain, every nonzero ideal factors uniquely into prime ideals. -
Factorization in Integral Domains, II
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector JOURNAL OF ALGEBRA 152,78-93 (1992) Factorization in Integral Domains, II D. D. ANDERSON Department of Mathematics, The University of Iowa, Iowa City, Iowa 52242 DAVID F. ANDERSON* Department of Mathematics, The Unioersity of Tennessee, Knoxville, Tennessee 37996 AND MUHAMMAD ZAFRULLAH Department of Mathematics, Winthrop College, Rock Hill, South Carolina 29733 Communicated by Richard G. Swan Received June 8. 1990 In this paper, we study several factorization properties in an integral domain which are weaker than unique factorization. We study how these properties behave under localization and directed unions. 0 1992 Academic Press, Inc. In this paper, we continue our investigation begun in [l] of various factorization properties in an integral domain which are weaker than unique factorization. Section 1 introduces material on inert extensions and splitting multiplicative sets. In Section 2, we study how these factorization properties behave under localization. Special attention is paid to multi- plicative sets generated by primes. Section 3 consists of “Nagata-type” theorems about these properties. That is, if the localization of a domain at a multiplicative set generated by primes satisfies a certain ring-theoretic property, does the domain satisfy that property? In the fourth section, we consider Nagata-type theorems for root closure, seminormality, integral * Supported in part by a National Security Agency Grant. 78 OO21-8693/92$5.00 Copyright 0 1992 by Academic Press, Inc. All righrs of reproduction in any form reserved. FACTORIZATION IN INTEGRAL DOMAINS, II 79 closure: and complete integral closure. -
A Zariski Topology for Modules
A Zariski Topology for Modules∗ Jawad Abuhlail† Department of Mathematics and Statistics King Fahd University of Petroleum & Minerals [email protected] October 18, 2018 Abstract Given a duo module M over an associative (not necessarily commutative) ring R, a Zariski topology is defined on the spectrum Specfp(M) of fully prime R-submodules of M. We investigate, in particular, the interplay between the properties of this space and the algebraic properties of the module under consideration. 1 Introduction The interplay between the properties of a given ring and the topological properties of the Zariski topology defined on its prime spectrum has been studied intensively in the literature (e.g. [LY2006, ST2010, ZTW2006]). On the other hand, many papers considered the so called top modules, i.e. modules whose spectrum of prime submodules attains a Zariski topology (e.g. [Lu1984, Lu1999, MMS1997, MMS1998, Zah2006-a, Zha2006-b]). arXiv:1007.3149v1 [math.RA] 19 Jul 2010 On the other hand, different notions of primeness for modules were introduced and in- vestigated in the literature (e.g. [Dau1978, Wis1996, RRRF-AS2002, RRW2005, Wij2006, Abu2006, WW2009]). In this paper, we consider a notions that was not dealt with, from the topological point of view, so far. Given a duo module M over an associative ring R, we introduce and topologize the spectrum of fully prime R-submodules of M. Motivated by results on the Zariski topology on the spectrum of prime ideals of a commutative ring (e.g. [Bou1998, AM1969]), we investigate the interplay between the topological properties of the obtained space and the module under consideration. -
Lecture 1 Course Introduction, Zariski Topology
18.725 Algebraic Geometry I Lecture 1 Lecture 1: Course Introduction, Zariski topology Some teasers So what is algebraic geometry? In short, geometry of sets given by algebraic equations. Some examples of questions along this line: 1. In 1874, H. Schubert in his book Calculus of enumerative geometry proposed the question that given 4 generic lines in the 3-space, how many lines can intersect all 4 of them. The answer is 2. The proof is as follows. Move the lines to a configuration of the form of two pairs, each consists of two intersecting lines. Then there are two lines, one of them passing the two intersection points, the other being the intersection of the two planes defined by each pair. Now we need to show somehow that the answer stays the same if we are truly in a generic position. This is answered by intersection theory, a big topic in AG. 2. We can generalize this statement. Consider 4 generic polynomials over C in 3 variables of degrees d1; d2; d3; d4, how many lines intersect the zero sets of each polynomial? The answer is 2d1d2d3d4. This is given in general by \Schubert calculus." 4 3. Take C , and 2 generic quadratic polynomials of degree two, how many lines are on the common zero set? The answer is 16. 4. For a generic cubic polynomial in 3 variables, how many lines are on the zero set? There are exactly 27 of them. (This is related to the exceptional Lie group E6.) Another major development of AG in the 20th century was on counting the numbers of solutions for n 2 3 polynomial equations over Fq where q = p . -
Arxiv:1601.07660V1 [Math.AC] 28 Jan 2016 2].Tesemigroup the [26])
INTEGRAL DOMAINS WITH BOOLEAN t-CLASS SEMIGROUP S. KABBAJ AND A. MIMOUNI Abstract. The t-class semigroup of an integral domain is the semigroup of the isomorphy classes of the t-ideals with the operation induced by t- multiplication. This paper investigates integral domains with Boolean t-class semigroup with an emphasis on the GCD and stability conditions. The main results establish t-analogues for well-known results on Pr¨ufer domains and B´ezout domains of finite character. 1. Introduction All rings considered in this paper are integral domains (i.e., commutative with identity and without zero-divisors). The class semigroup of a domain R, denoted S(R), is the semigroup of nonzero fractional ideals modulo its subsemigroup of nonzero principal ideals [11, 41]. The t-class semigroup of R, denoted St(R), is the semigroup of fractional t-ideals modulo its subsemigroup of nonzero principal ideals, that is, the semigroup of the isomorphy classes of the t-ideals of R with the operation induced by ideal t-multiplication. Notice that St(R) is the t-analogue of S(R), as the class group Cl(R) is the t-analogue of the Picard group Pic(R). The following set-theoretic inclusions always hold: Pic(R) ⊆ Cl(R) ⊆ St(R) ⊆ S(R). Note that the first and third inclusions turn into equality for Pr¨ufer domains and the second does so for Krull domains. More details on these objects are provided in the next section. Divisibility properties of a domain R are often reflected in group or semigroup- theoretic properties of Cl(R) or S(R). -
Lecture 1: Overview
Lecture 1: Overview September 28, 2018 Let X be an algebraic curve over a finite field Fq, and let KX denote the field of rational functions on X. Fields of the form KX are called function fields. In number theory, there is a close analogy between function fields and number fields: that is, fields which arise as finite extensions of Q. Many arithmetic questions about number fields have analogues in the setting of function fields. These are typically much easier to answer, because they can be connected to algebraic geometry. Example 1 (The Riemann Hypothesis). Recall that the Riemann zeta function ζ(s) is a meromorphic function on C which is given, for Re(s) > 1, by the formula Y 1 X 1 ζ(s) = = ; 1 − p−s ns p n>0 where the product is taken over all prime numbers p. The celebrated Riemann hypothesis asserts that ζ(s) 1 vanishes only when s 2 {−2; −4; −6;:::g is negative even integer or when Re(s) = 2 . To every algebraic curve X over a finite field Fq, one can associate an analogue ζX of the Riemann zeta function, which is a meromorphic function on C which is given for Re(s) > 1 by Y 1 X 1 ζX (s) = −s = s 1 − jκ(x)j jODj x2X D⊆X here jκ(x)j denotes the cardinality of the residue field κ(x) at the point x, D ranges over the collection of all effective divisors in X and jODj denotes the cardinality of the ring of regular functions on D. -
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. -
Number Theory
Number Theory Alexander Paulin October 25, 2010 Lecture 1 What is Number Theory Number Theory is one of the oldest and deepest Mathematical disciplines. In the broadest possible sense Number Theory is the study of the arithmetic properties of Z, the integers. Z is the canonical ring. It structure as a group under addition is very simple: it is the infinite cyclic group. The mystery of Z is its structure as a monoid under multiplication and the way these two structure coalesce. As a monoid we can reduce the study of Z to that of understanding prime numbers via the following 2000 year old theorem. Theorem. Every positive integer can be written as a product of prime numbers. Moreover this product is unique up to ordering. This is 2000 year old theorem is the Fundamental Theorem of Arithmetic. In modern language this is the statement that Z is a unique factorization domain (UFD). Another deep fact, due to Euclid, is that there are infinitely many primes. As a monoid therefore Z is fairly easy to understand - the free commutative monoid with countably infinitely many generators cross the cyclic group of order 2. The point is that in isolation addition and multiplication are easy, but together when have vast hidden depth. At this point we are faced with two potential avenues of study: analytic versus algebraic. By analytic I questions like trying to understand the distribution of the primes throughout Z. By algebraic I mean understanding the structure of Z as a monoid and as an abelian group and how they interact. -
The Family of Residue Fields of a Zero-Dimensional Commutative Ring
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Journal of Pure and Applied Algebra 82 (1992) 131-153 131 North-Holland The family of residue fields of a zero-dimensional commutative ring Robert Gilmer Department of Mathematics BlS4, Florida State University. Tallahassee, FL 32306, USA William Heinzer* Department of Mathematics, Purdue University. West Lafayette, IN 47907, USA Communicated by C.A. Weibel Received 28 December 1991 Abstract Gilmer, R. and W. Heinzer, The family of residue fields of a zero-dimensional commutative ring, Journal of Pure and Applied Algebra 82 (1992) 131-153. Given a zero-dimensional commutative ring R, we investigate the structure of the family 9(R) of residue fields of R. We show that if a family 9 of fields contains a finite subset {F,, , F,,} such that every field in 9 contains an isomorphic copy of at least one of the F,, then there exists a zero-dimensional reduced ring R such that 3 = 9(R). If every residue field of R is a finite field, or is a finite-dimensional vector space over a fixed field K, we prove, conversely, that the family 9(R) has. to within isomorphism. finitely many minimal elements. 1. Introduction All rings considered in this paper are assumed to be commutative and to contain a unity element. If R is a subring of a ring S, we assume that the unity of S is contained in R, and hence is the unity of R. If R is a ring and if {Ma}a,, is the family of maximal ideals of R, we denote by 9(R) the family {RIM,:a E A} of residue fields of R and by 9"(R) a set of isomorphism-class representatives of S(R).In connection with work on the class of hereditarily zero-dimensional rings in [S], we encountered the problem of determining what families of fields can be realized in the form S(R) for some zero-dimensional ring R. -
Integral Closures of Ideals and Rings Irena Swanson
Integral closures of ideals and rings Irena Swanson ICTP, Trieste School on Local Rings and Local Study of Algebraic Varieties 31 May–4 June 2010 I assume some background from Atiyah–MacDonald [2] (especially the parts on Noetherian rings, primary decomposition of ideals, ring spectra, Hilbert’s Basis Theorem, completions). In the first lecture I will present the basics of integral closure with very few proofs; the proofs can be found either in Atiyah–MacDonald [2] or in Huneke–Swanson [13]. Much of the rest of the material can be found in Huneke–Swanson [13], but the lectures contain also more recent material. Table of contents: Section 1: Integral closure of rings and ideals 1 Section 2: Integral closure of rings 8 Section 3: Valuation rings, Krull rings, and Rees valuations 13 Section 4: Rees algebras and integral closure 19 Section 5: Computation of integral closure 24 Bibliography 28 1 Integral closure of rings and ideals (How it arises, monomial ideals and algebras) Integral closure of a ring in an overring is a generalization of the notion of the algebraic closure of a field in an overfield: Definition 1.1 Let R be a ring and S an R-algebra containing R. An element x S is ∈ said to be integral over R if there exists an integer n and elements r1,...,rn in R such that n n 1 x + r1x − + + rn 1x + rn =0. ··· − This equation is called an equation of integral dependence of x over R (of degree n). The set of all elements of S that are integral over R is called the integral closure of R in S. -
Weakly Integrally Closed Domains and Forbidden Patterns
Weakly Integrally Closed Domains and Forbidden Patterns by Mary E. Hopkins A Dissertation Submitted to the Faculty of The Charles E. Schmidt College of Science in Partial Ful…llment of the Requirements for the Degree of Doctor of Philosophy Florida Atlantic University Boca Raton, Florida May 2009 Acknowledgements I am deeply grateful to my advisor, Dr. Fred Richman, for his patience and gentle guidance, and for helping me to understand algebra on a deeper level. I would also like to thank my family and Dr. Lee Klingler for supporting me personally through this challenging process. Many thanks are given to Dr. Timothy Ford and Dr. Jorge Viola-Prioli for their insightful remarks which proved to be very helpful in writing my dissertation. Lastly, I will be forever grateful to Dr. James Brewer for taking me under his wing, opening my eyes to the beautiful world of algebra, and treating me like a daughter. I dedicate this dissertation to my grandmother and Dr. Brewer. iii Abstract Author: Mary E. Hopkins Title: Weakly Integrally Closed Domains and Forbidden Patterns Institution: Florida Atlantic University Dissertation advisor: Dr. Fred Richman Degree: Doctor of Philosophy Year: 2009 An integral domain D is weakly integrally closed if whenever there is an element x in the quotient …eld of D and a nonzero …nitely generated ideal J of D such that xJ J 2, then x is in D. We de…ne weakly integrally closed numerical monoids similarly. If a monoid algebra is weakly integrally closed, then so is the monoid. A pattern F of …nitely many 0’s and 1’s is forbidden if whenever the characteristic binary string of a numerical monoid M contains F , then M is not weakly integrally closed. -
Arxiv:1509.06425V4 [Math.AG] 9 Dec 2017 Nipratrl Ntegoercraiaino Conformal of Gen Realization a Geometric the (And in Fact Role This Important Trivial
TRIVIALITY PROPERTIES OF PRINCIPAL BUNDLES ON SINGULAR CURVES PRAKASH BELKALE AND NAJMUDDIN FAKHRUDDIN Abstract. We show that principal bundles for a semisimple group on an arbitrary affine curve over an algebraically closed field are trivial, provided the order of π1 of the group is invertible in the ground field, or if the curve has semi-normal singularities. Several consequences and extensions of this result (and method) are given. As an application, we realize conformal blocks bundles on moduli stacks of stable curves as push forwards of line bundles on (relative) moduli stacks of principal bundles on the universal curve. 1. Introduction It is a consequence of a theorem of Harder [Har67, Satz 3.3] that generically trivial principal G-bundles on a smooth affine curve C over an arbitrary field k are trivial if G is a semisimple and simply connected algebraic group. When k is algebraically closed and G reductive, generic triviality, conjectured by Serre, was proved by Steinberg[Ste65] and Borel–Springer [BS68]. It follows that principal bundles for simply connected semisimple groups over smooth affine curves over algebraically closed fields are trivial. This fact (and a generalization to families of bundles [DS95]) plays an important role in the geometric realization of conformal blocks for smooth curves as global sections of line bundles on moduli-stacks of principal bundles on the curves (see the review [Sor96] and the references therein). An earlier result of Serre [Ser58, Th´eor`eme 1] (also see [Ati57, Theorem 2]) implies that this triviality property is true if G “ SLprq, and C is a possibly singular affine curve over an arbitrary field k.