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Hyman Bass and Ubiquity: Gorenstein Rings Contemp orary Mathematics Volume 00 XXXX Hyman Bass and Ubiquity Gorenstein Rings Craig Huneke Dedicated to Hyman Bass Abstract This pap er is based on a talk given by the author in Octob er at a conference in Columbia University in celebration of Hyman Basss th birthday The pap er details some of the history of Gorenstein rings and their uses Contents Intro duction Plane Curves Commutative Algebra circa Gorenstein Rings Examples and Low Co dimension Injective Mo dules and Matlis Duality Ubiquity Homological Themes Inverse Powers and dimensional Gorenstein Rings Hilb ert Functions Invariants and Gorenstein Rings Ubiquity and Mo dule Theory Bibliography Intro duction In an article app eared in Mathematische Zeitschrift with an interesting title On the ubiquity of Gorenstein rings and fascinating content The author Mathematics Subject Classication Primary CDH Secondary B Key words and phrases Gorenstein rings The author was supp orted in part by the NSF c XXXX American Mathematical So ciety CRAIG HUNEKE was Hyman Bass The article has b ecome one of the mostread and quoted math articles in the world A journal survey done in Ga showed that it ranked third among mostquoted pap ers from core math journals For every young student in commutative algebra it is at the top of a list of pap ers to read The rst thing I did as a graduate student when I op ened the journal to this pap er was to reach for a dictionary Definition Ubiquity The state or capacity of b eing everywhere esp ecially at the same time omnipresent This article will give some of the historical background of Gorenstein rings explain a little of why they are ubiquitous and useful and give practical ways of computing them The prop erty of a ring b eing Gorenstein is fundamentally a statement of symmetry The Gorenstein of Gorenstein rings is Daniel Gorenstein the same who is famous for his role in the classication of nite simple groups A question o ccurring to everyone who studies Gorenstein rings is Why are they called Gorenstein rings His name b eing attached to this concept go es back to his thesis on plane curves written under Oscar Zariski and published in the Transactions of the American Mathematical So ciety in Go As we shall see they could p erhaps more justiably b e called Bass rings or Grothendieck rings or Serre rings or Rosenlicht rings The usual denition now used in most textb o oks go es back to the work of Bass in the ubiquity pap er Plane Curves The origins of Gorenstein rings go back to the classical study of plane curves Fix a eld k and let f X Y b e a p olynomial in the ring k X Y By a plane curve we will mean the ring R k X Y f If k is algebraically closed Hilb erts Nullstellensatz allows us to identify the maximal ideals of this ring with the solutions of f X Y which lie in k via the corresp ondence of solutions k with maximal ideals X Y In his thesis Gorenstein was interested in the prop erties of the socalled adjoint curves to an irreducible plane curve f However his main theorem had to do with the integral closure of a plane curve Definition Let R b e an integral domain with fraction eld K The integral closure of R is the set of all elements of K satisfying a monic p olynomial with co ecients in R The integral closure is a ring T containing R which is itself integrally closed An imp ortant measure of the dierence b etween R and its integral closure T is the conductor ideal denoted C CT R C fr R j r T R g It is the largest common ideal of b oth R and T Example Let R k t t k X Y X Y The element t t t is in the fraction eld of R and is clearly integral over R It follows that the integral closure of R is T k t The dimension over k of T R can b e computed by counting HYMAN BASS AND UBIQUITY GORENSTEIN RINGS p owers of t in T but not in R since the p owers of t form a k basis of b oth R and T The conductor is the largest common ideal of b oth T and R It will contain some c least p ower of t say t and then must contain all higher p owers of t since it is an ideal in T To describ e R T and C it suces to give the exp onents of the p owers of t inside each of them For T For R For C A surprising phenomenon can b e found by examining this chart the numb er of monomials in T but not in R is the same as the numb er in R but not in the conductor C In this example there are six such monomials More precisely the vector space dimension of T R is equal to that of R C Is this an accident It is no accident and this equality is the main theorem of Gorenstein lo cally on the curve In fact indep endently Apery A in Samuel Sa in and Gorenstein Go in all proved that given a prime Q of a plane curve R with integral closure T and conductor C dim T R dim R C k Q Q k Q Q where T R and C are the lo calizations at the prime Q In other words we Q Q Q invert all elements outside of Q R is then a lo cal ring with unique maximal ideal Q Q It turns out that al l dimensional lo cal Gorenstein domains have this fun Q damental equality in fact it essentially characterizes such dimensional domains Plane curves are a simple example of Gorenstein rings the simplest examples are p olynomial rings or formal p ower series rings More general than plane curves are complete intersections Let S a regular ring eg the p olynomial ring k X X or a regular lo cal ring and let R SI n The co dimension of R or I is dened by co dimR dim S dim R n dimR R is said to b e a complete intersection if I can b e generated by co dim R elements Plane curves are a simple example since they are dened by one equation All complete intersections are CohenMacaulay and the dening ideal I will b e generated by a regular sequence An imp ortant example of a complete intersection is Z where is a nitely generated ab elian group In Rosenlicht Ro proved the equality dim T R dim R C k Q Q k Q Q for lo calizations of complete intersection curves ie for onedimensional domains R of the form R k X X F F n n with integral closure T and conductor C Furthermore Rosenlicht proved that a lo cal ring R of an algebraic curve had the prop erty that dim T R dim R C i the mo dule of regular dierentials k k is a free R mo dule It is interesting that b oth Gorenstein and Rosenlicht were students of Zariski As for plane curves complete intersections are also sp ecial examples of Gorenstein rings To explain the background and denitions we need to review the state of commutative algebra around CRAIG HUNEKE Commutative Algebra circa The most basic commutative rings are the p olynomials rings over a eld or the integers and their quotient rings Of particular imp ortance are the rings whose cor resp onding varieties are nonsingular To understand the notion of nonsingularity in terms of the ring structure the notion of regular lo cal rings was develop ed DefinitionTheorem A regular local ring S n is a No etherian lo cal ring which satises one of the following equivalent prop erties The maximal ideal n is generated by dimS elements Every nitely generated S mo dule M has a nite resolution by nitely gen erated free S mo dules ie there is an exact sequence k 1 k F F F M k k where the F are nitely generated free S mo dules i The residue eld Sn has a nite free resolution as in These equivalences are due to AuslanderBuchsbaum AuBu and Serre Se and marked a watershed in commutative algebra One immediate use of this theo rem was to prove that the lo calization of a regular lo cal ring is still regular Regular lo cal rings corresp ond to the lo calizations of ane varieties at nonsingular p oints The existence of a nite free resolution for nitely generated mo dules is of crucial imp ortance For example in Auslander and Buchsbaum AuBu used the existence of such resolutions to prove their celebrated result that regular lo cal rings are UFDs A resolution as in is said to b e minimal if F nF In this case it is i i i unique up to an isomorphism The length of a minimal resolution is the pro jective dimension of M denoted p d M S We say a No etherian ring is regular if it is lo cally at each of its primes Every p olynomial ring or p ower series ring over a eld is regular Other typ es of rings were b ecoming increasingly interesting and imp ortant To explain these some further denitions are helpful A lo cal ring R m is a No etherian commutative ring with a unique maximal ideal m An mprimary ideal I is any ideal such that one of the following equivalent conditions hold R I is dimension R I is Artinian The nilradical of I is m SuppR I fmg A regular sequence x x on an R mo dule M is any set of elements such t that x x M M and such that x is not a zerodivisor on M x is not t a zerodivisor on the mo dule M x M and in general x is not a zerodivisor on i M x x M i The depth of a nitely generated mo dule over a lo cal ring R is the maximal length of a regular sequence on the mo dule M is said to b e CohenMacaulay if depthM dim M This is the largest p ossible value for the depth HYMAN BASS AND UBIQUITY GORENSTEIN RINGS A sop system of parameters for a lo cal ring R of dimension d is any d elements x x which generate an mprimary ideal d A lo cal ring is CohenMacaulay i every equivalently one system of param eters forms a regular sequence A lo cal ring R is regular i the maximal ideal is generated by a sop The height of a prime ideal P in a commutative ring R is the supremum of integers n such that there is a chain of distinct prime ideals P P P P
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