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

n

The height of an arbitrary ideal is the smallest height of any prime containing the

ideal

If I has height k then I needs at least k generators otherwise the dimension

could not drop by k This is the statement of what is known as Krulls Height

Theorem Explicitly Krulls theorem states

Theorem Let S be a Noetherian ring and let I be an ideal generated

by k elements Then every minimal prime containing I has height at most k In

particular heightI k

The prop erty of a ring b eing CohenMacaulay had b een growing in imp ortance

in the s and early s Dierent groups studied it under dierent names

semiregular Macaulay even MacaulayCohen In particular Northcott and Rees

had b een studying the irreducibility of systems of parameters and relating it to the

CohenMacaulay prop erty

An ideal I is irreducible if I J K for I ( J K

Let M b e a mo dule over a lo cal ring R m The so cle of a mo dule M

denoted by so cM is the largest subspace of M whose R mo dule structure is that

of a vector space ie so cM Ann m fy M j my g

M

Proposition If R is Artinian local then is irreducible i socR is a

dimensional vector space

Proof Let V b e the so cle If dimV then simply pick two one dimen

sional subspaces of V which intersect in to show is not irreducible

For the converse the key p oint is that R is Artinian implies V R is essential

Any two ideals must intersect the so cle but if it is onedimensional they then

contain the so cle 

For example the ring R k X Y X Y has a onedimensional so cle gen

erated by XY every nonzero ideal in R contains this element and hence is

irreducible The intersection of any two nonzero ideals must contain XY On the

other hand in R k X Y X X Y Y the so cle is two dimensional generated by

the images of X and Y and XR Y R

Irreducible ideals were already present in the famous pro of of Emmy No ether

that ideals in No etherian commutative rings have a primary decomp osition No

She rst observed that the No etherian prop erty implied every ideal is a nite in

tersection of irreducible ideals For example the ideal X X Y Y ab ove is the

intersection of X Y and X Y which are b oth irreducible This irreducible

decomp osition is closely related to the ideas in the theory of Gorenstein rings

The results of Northcott and Rees are the following

CRAIG HUNEKE

Theorem NR If R m is a local Noetherian ring such that the ideal

generated by an arbitrary system of parameters is irreducible then R is Cohen

Macaulay

Theorem NR A regular local ring has the property that every ideal

generated by a system of parameters is irreducible

Combining these two theorems recovers a result rst due to Cohen regular

lo cal rings are CohenMacaulay In fact as we shall see this irreducible prop erty

characterizes Gorenstein rings From this p ersp ective the theorems of Northcott

and Rees say among other things that regular rings are Gorenstein and Goren

stein rings are CohenMacaulay

Gorenstein Rings

Grothendieck Gr develop ed duality theory for singular varieties He

constructed a rank one mo dule which on the nonsingular lo cus of an algebraic

variety agreed with the dierential forms of degree d the dimension of the variety

Let S b e a p olynomial ring and R b e a homomorphic image of S of dimension

d One can dene a dualizing mo dule or canonical module by setting

nd

R S Ext

R

S

where the dimension of S is n When R is nonsingular this mo dule is isomorphic

to the mo dule of dierential forms of degree d on R The b est duality is not

j

obtained unless the ring R is homologically trivial in the sense that Ext R S

S

for j n d Such rings are exactly the CohenMacaulay rings

The duality is p erfect on R mo dules whose pro jective dimension over the

regular ring S is smallest p ossible These are the CohenMacaulay mo dules Let

M b e such a mo dule of dimension q Then

i

Ext M

R

dq

for i d q Ext M is again CohenMacaulay and

R

dq dq

Ext Ext M M

R R

According to Bass in the ubiquity pap er Grothendieck intro duced the following

denition

Definition Let S b e a regular lo cal ring A lo cal ring R which is a

homomorphic image of S is Gorenstein if it is CohenMacaulay and its dualizing

nd

mo dule or canonical module Ext R S is free of rank where n dim S

S

and d dimR We say a p ossibly nonlo cal No etherian ring R which we assume

is the homomorphic image of a regular ring is Gorenstein if R is Gorenstein for

P

all prime ideals P

All plane curves are Gorenstein but even more we shall see that all complete

intersections are Gorenstein This denition will only b e temp orary in the sense

that the work of Bass made it clear how to extend the denition to all lo cal rings

and thereby all rings without necessarily having a canonical mo dule

The earliest printed reference I could nd giving a denition of Gorenstein rings

is in Se p The date given for his seminar is Novemb er He

HYMAN BASS AND UBIQUITY GORENSTEIN RINGS

denes an algebraic variety W to b e Gorenstein if it is CohenMacaulay and the

dualizing sheaf is lo cally free of rank He go es on to say that for plane curves

W

R the equality dimT R dimR C is equivalent to b eing Gorenstein and

Q Q Q Q

cites Se He further mentions that complete intersections are Gorenstein in this

context In Se in Serre had proved these claims and mentioned the pap ers

of Rosenlicht Samuel and Gorenstein as well as the work of Grothendieck on the

duality approach

With the denition of GrothendieckSerre it is not dicult to see that complete

intersections are Gorenstein We only need to use the free resolutions of such rings

These resolutions give a p owerful to ol not only to prove results ab out Gorenstein

rings but to actually compute with them

Proposition Let S m be a regular local ring of dimension n and let

R SI be a quotient of S of dimension d Let

k 1

k

F R F F

k k

be a minimal free S resolution of R Then R is Gorenstein i k n d and

S F

k

Proof We use a famous formula of Auslander and Buchsbaum to b egin the

pro of If S is a regular lo cal ring and M a nitely generated mo dule then

depth M p d M dim S

S

Applying this formula with R M we obtain that R is CohenMacaulay i

depthR dimR by denition i k n d

nd

R S can b e computed from the free S resolution of R We The mo dule Ext

S

simply apply Hom S to the resolution and take homology The n d homology

S

  

is the cokernel of the transp ose of from F F where Hom S

k S

k k

If R is Gorenstein this is free of rank by denition The minimality of the

resolution together with Nakayamas lemma shows that the minimal numb er of

nd

generators of Ext R S is precisely the rank of F Hence it must b e rank

k

S

Conversely supp ose that the rank of F is Since I kills R it also kills

k

nd nd

Ext R S It follows that Ext R S is isomorphic to SJ for some ideal

S S

J I However the vanishing of the other Ext groups gives that the transp osed

complex

 

F F

k

nd

is actually acyclic and hence a free S resolution of Ext R S Therefore

S

nd nd

Ext Ext R S S R

S S

by dualizing the complex back and now the same reasoning shows that J kills R

nd

ie J I Thus Ext R S R and R is Gorenstein 

S

Observe that the end of the pro of proves that the resolution of a Gorenstein

quotient R of S is essentially selfdual if you ip the free resolution you again get

a free resolution of R In particular the ranks of the free mo dules are symmetric

around the middle of the resolution

CRAIG HUNEKE

Remark Prop osition is esp ecially easy to apply when the dimension

of R is In that case the depth of R is which forces the pro jective dimension to

b e n so that the rst condition of is automatically satised one only needs

to check that the rank of the last free mo dule in the minimal free resolution of R

is exactly to conclude that R is Gorenstein

L

R is graded with R k a Remark The Graded Case If R

n

n

eld and is No etherian we can write R SI with S k X X and I is

n

a homogeneous ideal of S where the X are given weights In this case the free

i

resolution of SI over S can b e taken to b e graded If we twist the free mo dules

so that the maps have degree then the resolution has the form

X X

b b

1i ni

S SI S i S i

i i

For example b is the numb er of minimal generators of I in degree i

i

Let M b e the unique homogeneous maximal ideal of S generated by the X

i

Then R is Gorenstein i R is Gorenstein This follows from the characterization

M

ab ove One uses that the free resolution of R over S can b e taken to b e graded

and hence the lo cus of primes P where R is CohenMacaulay is dened by the a

P

homogeneous ideal and the lo cus where the canonical mo dule is free is also homo

geneous as the canonical mo dule is homogeneous This means that if there exists a

prime P in R such that R is not Gorenstein then P must contain either the ho

P

mogeneous ideal dening the non CohenMacaulay lo cus or the homogeneous ideal

dening the lo cus where the canonical mo dule is not free In either case these lo ci

are nonempty and will also b e contained in M forcing R not to b e Gorenstein

M

To prove that complete intersections are Gorenstein one only needs to know

ab out the Koszul complex Let x x b e a sequence of elements in a ring R

n

The tensor pro duct of the complexes

x

i

R R

is called the Koszul complex of x x This complex is a complex of nitely

n

generated free R mo dules of length n If the x form a regular sequence then this

i

complex provides a free resolution of R x x

n

Corollary Let S be a regular local ring and let I be an ideal of height

k generated by k elements Then R SI is Gorenstein

Proof Since S is CohenMacaulay I is generated by a regular sequence and

the Koszul complex provides a free S resolution of R The length of the resolution

is k and the last mo dule in this resolution is S Applying Prop osition gives

that R is Gorenstein 

The following sequence of implications is learned by all commutative alge

braists

regular complete intersection Gorenstein CohenMacaulay

If M is a graded S mo dule then the twist M t is the same mo dule but with the dierent

grading given by M t M

n

tn

There is another p erhaps more fundamental way to think of the Koszul complex as an

exterior algebra See BH

HYMAN BASS AND UBIQUITY GORENSTEIN RINGS

In general none of these implications is reversible The last implication is reversible

if the ring is a UFD and a homomorphic image of a Gorenstein ring as we shall

discuss later

The language and results of free resolutions give a useful to ol to analyze Goren

stein rings esp ecially in low co dimension

Examples and Low Co dimension

Let S b e a regular lo cal ring If an ideal I has height one and SI is Gorenstein

then in particular I is unmixed Since S is a UFD it follows that I f is

principal and SI is a complete intersection

Let I b e a height two ideal dening a Gorenstein quotient ie R SI is

Gorenstein of dimension n The resolution for R must lo ok like

l

S S S SI R

But counting ranks shows that l which means that I is generated by

elements and is a complete intersection This recovers a result of Serre Se

A natural question is whether all Gorenstein rings are complete intersections

The answer is a resounding NO but it remains an extremely imp ortant question

how to tell if a given Gorenstein ring is a complete intersection this question plays

a small but imp ortant role in the work of Wiles for example

In co dimension three the resolution of a Gorenstein R lo oks like

l l

S S S S SI R

The question is whether l is forced The answer is no But surprisingly the

minimal numb er of generators must b e o dd

Example Let S k X Y Z and let

I X Y X Z Y Z X Y X Z

The nilradical of I is the maximal ideal m X Y Z which has height three But

I requires ve generators and so is not a complete intersection On the other hand

I is Gorenstein One can compute the resolution of SI over S and see that the

last Betti numb er is The resolution lo oks like

S S S S SI

The graded resolution is

S S S S SI

This example is a sp ecial case of a famous theorem of Buchsbaum and Eisenbud

BE To explain their statement we rst recall what Pfaans are Let A b e a skew

symmetric matrix of size n by n Then the determinant of A is the square of

an element called the Pfaan of A the sign is determined by convention If A is

skew symmetric and of size n the ideal of Pfaans of order i i n of A is

the ideal generated by the Pfaans of the submatrices of A obtained by cho osing

i rows and the same i columns The theorem of Buchsbaum and Eisenbud states

CRAIG HUNEKE

Theorem Let S be a regular local ring and let I be an ideal in S of height

Set R SI Then R is Gorenstein i I is generated by the norder Pfaans

of a skewsymmetric n by n alternating matrix A In this case a minimal

free resolution of R over S has the form

A

n n

S S R S S

Unfortunately there is no structure theorem for height four ideals dening

Gorenstein rings although there has b een a great deal of work on this topic See

the references in VaVi and KM

Thinking back to the rst example of monomial plane curves it is natural to

n n

1

k

is Gorenstein t ask when the ring k t

Definition A semigroup T n n is said to b e symmetric if

k

there is a value c T such that m T i c m T

n n

1

k

is Gorenstein t Theorem HeK The monomial curve R k t

i T n n is symmetric

k

It follows that every semigroup T generated by two relatively prime integers a

and b is symmetric since this is a plane curve hence Gorenstein For such examples

the value of c as in is c a b In general c must b e chosen one less

than the least p ower of t in the conductor This example can b e used to give many

examples of Gorenstein curves which are not complete intersections or Pfaans

The easiest such example is T The ring R k t t t t t

is Gorenstein since T is symmetric The conductor is everything from t and up

and one can take c The dening ideal when we represent R as a quotient of a

p olynomial ring in variables has height and requires generators The minimal

resolution over the p olynomial ring S has the form

S S S S S R

See Br for a discussion of symmetric semigroups For a nonGorenstein exam

ple we can take R k t t t The semigroup is not symmetric Writing R as a

quotient of a p olynomial ring in three variables the dening ideal is a prime ideal

of height requiring generators From the theorem of Serre explained earlier

one knows that this cannot b e Gorenstein since it would have to b e a complete

intersection b eing height two Alternatively one could use an amazing result due

to Kunz Ku that almost complete intersections are never Gorenstein Almost

complete intersection means minimally generated by one more element than a com

plete intersection There are nontrivial examples of ideals generated by two more

elements than their height which dene Gorenstein ideals the Pfaan ideal of

generators gives such an examples while whole classes were constructed in HU

see also HM

Where do es ubiquity t into this We need another historical thread b efore

the tap estry is complete This last thread concerns the theory of injective mo dules

Injective Mo dules and Matlis Duality

There is a smallest injective mo dule containing an R mo dule M denoted

E M or just E M It is called the injective hul l of M It is the largest R

HYMAN BASS AND UBIQUITY GORENSTEIN RINGS

essential extension of M Any mo dule b oth esesential over M and injective must

b e isomorphic to the injective hull of M

Theorem Matlis Ma Let R be a Noetherian commutative ring

Every injective module is a direct sum of indecomposable injective modules and

the nonisomorphic indecomposable injective modules are exactly up to E R P

for P prime

Let R m b e a dimensional No etherian lo cal ring There is only one inde

comp osable injective mo dule E the injective hull of the residue eld of R Matlis

proved that the length of E that is the numb er of copies of the residue eld

k R m in a ltration whose quotients are k is the same as that of R Matlis

extended these ideas and came up with what is now called Matlis duality

Theorem Let R m be a Noetherian local ring with residue eld k E

b

E k and completion R Then there is a arrow reversing correspondence from

R

_

b

nitely generated Rmodules M to Artinian R modules N given by M M and

______

N M and N N N where Hom E Furthermore M

R

Restricting this correspondence to the intersection of Artinian modules and nitely

generated modules ie to modules of nite length preserves length

The work of Matlis allows one to understand the dimensional commutative

No etherian rings which are selfinjective

Theorem Let R m be a dimensional local ring with residue eld k

and E E k The fol lowing are equivalent

R

R is injective as an R module

R E

is an irreducible ideal

The socle of R is dimensional

R is Gorenstein in the sense of GrothendieckSerre

Proof Clearly implies On the other hand if R is injective it must

b e a sum of copies of E by the theorem of Matlis Since R is indecomp osable it is

then isomorphic to E

Every dimensional lo cal ring is an essential extension of its so cle V For given

any submo dule of R that is an ideal I of R there is a least p ower n of m such

n n

that m I Then m I V I and is nonzero Since the length of R and E

agree it then is clear that R is selfinjective i its so cle is dimensional ie i

is irreducible This shows the equivalence of and

To see the equivalence of rst observe that R is the homomorphic image of

a regular lo cal ring S using the Cohen structure theorem R is CohenMacaulay

b ecause it is dimensional and of course has depth So it just needs to b e shown

n

that Ext R S is isomorphic to R where dimS n Equivalently as shown in

S

Prop osition we need to prove that the rank of the last free mo dule in a minimal

n

free S resolution of R is exactly one This rank is the dimension of Tor k R by

S

computing this Tor using the minimal free resolution of R However k has a free

S resolution by a Koszul complex since the maximal ideal of S is generated by a

regular sequence as S is regular Using the Koszul complex to compute this Tor

yields the isomorphism

n

Tor k R so cR

S

CRAIG HUNEKE

Thus R is Gorenstein i the so cle is dimensional 

A natural question from Theorem and from the second theorem of Northcott

and Rees is how to calculate the so cle of a system of parameters in a regular lo cal

ring R m For if one takes a regular lo cal ring and quotients by a system of

parameters then is irreducible by Northcott and Rees and the quotient ring is

therefore a selfinjective ring It turns out there is an excellent answer to how to

calculate so cles

Theorem Let R m be a regular local ring of dimension d and write

m x x Let f f be a system of parameters Write x x A

d d d

f f for some d by d matrix A and put detA Then the image of

d

in R f f generates the socle of this Artinian algebra

d

Another way to state this theorem is f f m f f Here

d R d

I J fr R j r J I g Theorem gives a very useful computational device

R

for computing so cles One easy example is to compute a generator for the so cle of

n n

1

d

R x x In this case the matrix A can b e taken to b e a diagonal matrix

d

Q

n n

i i

x Of course when whose ith entry along the diagonal is x and

i i

i

we lift the so cle generator back to R it is unique only up to a unit multiple plus an

element of the ideal we mo d out

In the case when R is complete regular and contains a eld the Cohen Struc

ture Theorem gives that R is isomorphic with a formal p ower series over a eld In

this case the theory of residues can b e used to give another description of the so cle

Theorem Let R k x x be a formal power series ring over a eld

d

k of characteristic and let f f be a system of parameters in R Let J be the

d

f

i

Set detJ Then generates the Jacobian matrix whose i j entry is

x

j

socle of R f f

d

If the f happ en to b e homogeneous p olynomials then this Theorem follows

i

directly from the previous one as Eulers formula can b e used to express the f in

i

terms of the x using only units times the rows of the Jacobian matrix However

j

in the nonhomogenous case this second theorem is extremely useful

Remark In the graded case in which R k R is dimensional and a

homomorphic image of a p olynomial ring S k X X one can calculate the

n

degree of the so cle generator for R by using the free resolution of R as an S mo dule

The resolution will b e of the form

X

b

1i

S R S i S m

i

The last mo dule will b e a copy of S twisted by an integer m since R is Goren

stein The degree of the so cle of R is then exactly m n For example in

we saw that the last twist is S in the resolution of the ring R k X Y Z I

where I X Y X Z X Y X Z Y Z Hence the so cle of R sits in degree

In fact the so cle can b e taken b e any quadric not in I

HYMAN BASS AND UBIQUITY GORENSTEIN RINGS

Ubiquity

In the early s Bass had b een studying prop erties of rings with nite injec

tive dimension In particular he related irreducibility to nite injective dimension

The following is from Bass Ba

Theorem Let R be a commutative Noetherian local ring The fol lowing

conditions are equivalent

R is CohenMacaulay and every system of parameters generates an irre

ducible ideal

id R

R

id R dim R

R

One can see in retrosp ect how the theme of irreducibility injective dimension

and complete intersections were coming together Most of the work discussed ab ove

was available by the late s The AuslanderBuchsbaum work was done in

Grothendiecks work on duality in Serres work in Matliss work

on injective mo dules in and the work of Northcott and Rees on irreducible

systems of parameters was done in The work of Rosenlicht and Gorenstein

was done ab out According to Bass Serre p ointed out to him that the rings

of nite injective dimension were at least in the geometric context simply the

Gorenstein rings of Grothendieck To quote Basss pap er he remarks Gorenstein

rings it is now clear have enjoyed such a variety of manifestations as to justify

p erhaps a survey of their relevance to various situations and problems The rest

as one says is history

Bass put all this together in the ubiquity pap er in Ba It remains one

of the most read pap ers in commutative algebra

Theorem Ubiquity Let R m be a Noetherian local ring The fol lowing

are equivalent

If R is the homomorphic image of a regular local ring then R is Gorenstein

in the sense of GrothendieckSerre

id R

R

id R dim R

R

R is CohenMacaulay and some system of parameters generates an irre

ducible ideal

R is CohenMacaulay and every system of parameters generates an irre

ducible ideal

h

If R E E is a minimal injective resolution of R

P

h

E R p then for each h E

R

heightph

By Theorem of Northcott and Rees one can remove the assumption that

R b e CohenMacaulay in We can now take any of these equivalent prop erties

to b e the denition of Gorenstein in the lo cal case The usual one chosen is the

second that the ring have nite injective dimension over itself In the nonlo cal

case we say R is Gorenstein if R is Gorenstein for every prime P in R If R has

P

nite Krull dimension this is equivalent to saying R has nite injective dimension

over itself

CRAIG HUNEKE

Example Let R m b e a one dimensional Gorenstein lo cal ring The

equivalent conditions ab ove say that the injective resolution of R lo oks like

R K E

where K is the fraction eld of R this is the injective hull of R and E is the

injective hull of the residue eld k R m This gives a nice description of E as

K R If we further assume that R k t a regular lo cal ring then one can

t

identify K R with the inverse p owers of t This mo dule is an essential extension of

the residue eld k which sits in K R as the R span of t This example generalizes

to higher dimensions in the form of the socalled inverse systems of Macaulay See

section

Moreover in dimension one for No etherian reduced rings this means the ring

has no nonzero nilp otent elements with nite integral closure Bass was able to

generalize the results of Gorenstein and Rosenlicht to arbitrary Gorenstein rings

Theorem Let S be a regular local ring and let R SI be a dimensional

Gorenstein domain Assume that the integral closure of R is a nite R module this

is automatic for rings essential ly of nite type over a eld or for complete local

rings Let T be the integral closure of R and let C be the conductor Then

R C T R

n

R S where Proof Let b e the canonical mo dule of R namely Ext

S

n dimS Consider the exact sequence R T D where D T R



Applying Hom R we get the sequence

R

  

D T R Ext D R Ext T R

R R

Since T is a CohenMacaulay of dimension the last Ext vanishes As D is

  

C the isomorphism given by sending R and T torsion D Moreover R

a homomorphism to its evaluation at a xed nonzero element It follows that

R C Ext D R and it only remains to prove that the length of Ext D R is

R R

the same as the length of D This follows since R is Gorenstein We can use the

injective resolution of R as in Example to nd that

Hom D E Ext D R

R

R

By Matlis duality the length of this last mo dule is the same as the length of D 

Remark In fact the converse to Prop osition is also true

Homological Themes

One of the remarks Bass makes in the ubiquity pap er is the following It seems

conceivable that say for A lo cal there exist nitely generated M with nite

injective dimension only if A is CohenMacaulay The converse is true for if B A

mo dulo a system of parameters and M is a nitely generated nonzero B injective

mo dule then inj dim M

A

This b ecame known as Basss conjecture one of a celebrated group of homo

logical conjectures The search for solutions to these problems was a driving force

b ehind commutative algebra in the late s and s and continues even to day

HYMAN BASS AND UBIQUITY GORENSTEIN RINGS

However Basss conjecture has now b een solved p ositively by Peskine and Szpiro

PS in the geometric case by Ho chster Ho for all lo cal rings containing a eld

and by Paul Rob erts Rob in mixed characteristic

The following is due to Peskine and Szpiro PS and based on

their work on the conjecture of Bass

Theorem Let R m be a local ring The fol lowing are equivalent

R is Gorenstein

an ideal I such that id R I

R

an mprimary ideal I such that pd R I and I is irreducible

R

Other homological themes were intro duced in the ubiquity pap er which are now

part of the standard landscap e A mo dule M of a lo cal ring R is said to b e a k th

syzygy if there is an exact sequence

M F F

k

where the F are nitely generated free R mo dules It is imp ortant to know intrin

i

sically when a given mo dule is a k th syzygy For Gorenstein lo cal rings there is a

go o d answer due to Bass Ba Theorem

Theorem Let R be a Gorenstein local ring and k A nitely gener

i 

ated R module M is a k th syzygy i M is reexive and Ext M R for al l

R

i k



The idea of the pro of is to take a resolution of the dual mo dule M



Hom M R and then dualize it using that M M since M is reexive A

R

mo dern version of this result was given by Auslander and Bridger AB see for

example EG Theorem

Other imp ortant homological ob jects intro duced by Bass in ubiquity were the

i i

Bass numb ers p M dim Ext k p M where k p R pR are the

p p p

k p

R

p

residue elds of the lo calizations at prime ideals of R These numb ers play a very

imp ortant role in understanding the injective resolution of M and are the sub ject

of much work For example another equivalent condition for a lo cal No etherian

ring R m to b e Gorenstein is that R b e CohenMacaulay and the dth Bass num

d

b er m R This was proved by Bass Vasconcelos conjectured that one

could delete the hyp othesis that R b e CohenMacaulay This was proved by Paul

Rob erts in Rob Another line of work inspired by the work of Bass is that

of Avramov and Foxby who have develop ed and studied the concept of homomor

phisms b etween rings b eing Gorenstein For example see AvF and the references

there

Inverse Powers and dimensional Gorenstein Rings

The study of Gorenstein rings can b e approached by rst trying to understand

the dimensional Gorenstein rings These are all the dimensional Artinian rings

which are injective as mo dules over themselves Equivalently they are exactly the

dimensional Artinian rings with a dimensional so cle Given any lo cal Gorenstein

ring of arbitrary dimension one can always mo d out the ideal generated by a system

of parameters and obtain a dimensional Gorenstein ring In the dimensional

case one has the formidable results of Matlis to help

CRAIG HUNEKE

Example Let R k X X or a p ower series ring k X X An

n n

injective hull of the residue eld k R X X is given by the inverse powers

n

E k X X

n

with the action b eing determined by

a a a

a a

1 1 i

n n

X X X X X X

i

n n

i

if a If a then the pro duct is Notice the copy of k inside E is exactly

i i

the span of

Example To obtain the injective hull of the residue eld of a graded

quotient

k X X I

n

one simply takes Hom SI E This follows easily from the fact that this mo dule

S

is injective over SI using Homtensor adjointness and is also essential over k

since it can b e identied with a submo dule of E which is already essential over k

This is naturally identied with the elements of E killed by I One sees that the

graded structure of E is simply that of R upside down

R is dimensional Gorenstein lo cal i R E k In this case R Hom R E

S

which means that the annihinlator of R in the inverse p owers is cyclic The con

verse also holds The mprimary ideals I such that R SI are Gorenstein are

EXACTLY the annihilators of single elements in the inverse p owers See Prop osi

tion b elow

Example Let F X Y Z The set of elements in S killed by

F is the ideal

I X Y Y Z X Y X Z Y Z

This is the height Gorenstein ideal that we considered in Example

It is not dicult to prove that if F is a quadratic form in X Y Z then

the rank of F as a quadratic form is i the corresp onding ideal has generators

while if the rank is at most the corresp onding ideal is a complete intersection

generated by p olynomials See Ei p exer

In characteristic there is an imp ortant form of inverse p owers using the ring

of dierential op erators Let k b e a eld of characteristic and let T k y y

n

b e a p olynomial ring over k A p olynomial dierential op erator with constant

co ecients is an op erator on T of the form

X

i i

1 n

a D

i i

1 r

y y

n

i i

1 r

k We think of D as an element in a new p olynomial ring S with a

i i

1 r

Since the partials commute with k x x acting on T by letting x act as

n i

y

i

each other this makes sense Fix f T and let I I b e the ideal of all elements

f

D S such that D f Then I is an ideal in S primary to x x such that

n

SI is dimensional Gorenstein Furthermore all such dimensional Gorenstein

quotients of S arise in this manner The p oint is that we can identify T with the

injective hull of the residue eld of S See Ei exer This p oint of view has

b een very imp ortant for work on the Hilb ert scheme of dimensional Gorenstein

HYMAN BASS AND UBIQUITY GORENSTEIN RINGS

quotients of xed dimension See for example the pap er of Iarrobino Ia and its

references

There is a related way to construct all dimensional Gorenstein lo cal rings

containing a eld Start with a p ower series ring S k X X Cho ose

n

an arbitrary system of parameters f f in S and an arbitrary element g

n

f f Set I f f g Then R SI is a dimensional Gorenstein

n n S

ring Conversely all dimensional Gorenstein rings are of this form The pro of

follows at once from the following prop osition

Proposition Let R be a dimensional Gorenstein ring and let J R

Then R J is Gorenstein i J R g for some nonzero element g R Precisely

g R J

R

Proof Since R is Gorenstein it is isomorphic to its own injective hull It

follows that the injective hull of R J is Hom R J R J and R J is

R R

R J i J g R and g J  Gorenstein i J

R R R

For example to construct the Gorenstein quotient k X Y Z I where I

X Y Y Z X Y X Z Y Z we can do the following First take the complete

intersection quotient T k X Y Z X Y Z Set g X Y X Z Y Z

Then I X Y Z g and is therefore Gorenstein ALL Gorenstein quotients

R K such that X Y Z K arise in this fashion K X Y Z f for

some f k X Y Z and f is unique up to an element of X Y Z and a unit

multiple

Understanding dimensional Gorenstein rings is extremely imp ortant if one

uses Gorenstein rings as a to ol For a great many problems one can reduce to a

dimensional Gorenstein ring A natural question in this regard is the following

let R m b e a No etherian lo cal ring When do es there exist a descending sequence

of mprimary ideals I such that R I are Gorenstein for all n equivalently I are

n n n

n

irreducible and such that I m Melvin Ho chster completely answered this

n

question He called such rings approximately Gorenstein

Theorem Ho A local Noetherian ring R m is approxi

mately Gorenstein if and only if its madic completion is approximately Gorenstein

An excel lent local ring R so for example a complete local Noetherian ring with

dimR is approximately Gorenstein if and only if the fol lowing two conditions

hold

depthR

If P AssR and dimR P then R P R P is not embeddable in

R

Hilb ert Functions

The Hilbert function of a graded ring over a eld is the function H n

dim R The generating function for the Hilb ert function is called the Hilb ert

k n

series ie the series

X

n

F R t dim R t

k n

n

CRAIG HUNEKE

If R is generated by oneforms and is No etherian we can always write

l

h h t h t

l

F R t

d

t

where d dimR the Krull dimension of R The vector of integers h h h is

l

called the hvector of R For example if R is dimensional then h H i

i

Let R k R We can write R SI where S k X X and I is a

n

homogeneous ideal of S In this case the free resolution of SI over S can b e taken

to b e graded as in Section ab ove The resolution has the form

X X

b b

1i ni

S SI S i S i

i i

Fixing a degree n allows one to compute the dimension of SI as the alter

n

P

j b

j i

Each term in this sum is easily computatable as S i nating sum

n

ji

the dimension of the forms of a certain degree in the p olynomial ring

To do this calculation it is more convenient to write

F S c S c

i i n i

i

Then the Hilb ert series for SI is exactly

P

t

c

i c

1i n i

i

t t

i

n

t

The free resolution of a Gorenstein ring is essentially symmetric The dual of the

nd

resolution of a CohenMacaulay ring always gives a free resolution of Ext R S

S

the canonical mo dule Here S is a regular lo cal ring of dimension n and R is

CohenMacaulay of dimension d When R is further assumed to b e Gorenstein

the canonical mo dule is isomorphic to R and the ipp ed resolution is isomorphic

to the orginial resolution This basic duality should make the following theorem no

surprise

Theorem A dimensional graded Gorenstein ring R R R R

t

with k R a eld and R has symmetric Hilbert function ie H i

t

H t i for al l i t

One sees this by considering the pairing R R R k Since R is

i ti t

essential over the so cle it is not dicult to show that this pairing is p erfect and

identies R with the dual of R

i ti

Example Let us compute the Hilb ert function of the Example The

ideal is dened by ve quadrics in three variables and every cubic is in the dening

ideal Hence the Hilb ert function is

Example Let f f b e a homogeneous system of parameters in S

n

k X X of degrees m m resp ectively generating an ideal I The Hilb ert

n n

function can b e computed from the graded free resolution of SI and this is just

the Koszul complex The various graded free mo dules in this resolution dep end

only up on the degrees of the f and so the Hilb ert function is the same as that of

i

m

m

1

n

X X This is easily computed as ab ove The so cle is generated in degree

n

P

m

i i

HYMAN BASS AND UBIQUITY GORENSTEIN RINGS

This symmetry and the fact the so cle is onedimensional is b ehind the classical

applications of the Gorenstein prop erty outlined in EGH That article details nine

versions of the socalled CayleyBacharach Theorem b eginning with the famous

theorem of Pappus proved in the fourth century AD and ending with a common

generalization of all of them that p olynomial rings are Gorenstein It is worth

rep eating one of these avatars here and hop efully ho oking the reader to lo ok at

EGH The following theorem was proved by Chasles in

Theorem Let X X P be cubic plane curves meeting in exactly nine

points If X P is any cubic containing eight of these points then it contains

the ninth as wel l

Proof First note that X and X meet in the maximum numb er of p oints by

Bezouts theorem We interpret the theorem algebraically Let S k X Y Z b e

the homogeneous co ordinate ring of the pro jective plane X and X corresp ond

to the vanishing lo ci of two homogeneous cubics F F The ideal I F F

Q Q where Q is dened by two linear forms These corresp ond to the nine

i

p oints in which X and X meet Assume that another cubic G is contained in

Q Q We want to prove that G Q ie that G I

We pro ceed to cut down to Artinian quotients by killing a general linear form

Set T S where is a general linear form Of course T is simply a p olynomial

ring in two variables We write f f g for the images of F F G in this new ring

R T f f is a complete intersection whose Hilb ert function is

The so cle of T f f sits in degree If G I then g f f and g sits in

degree Since R is Gorenstein g R must contain the so cle As g sits in degree

three the Hilb ert function of the ideal g R is and hence R g R has

dimension seven The Hilb ert function of R g R must b e However by

assumption F F G is contained in the eight linear ideals Q Q When we

cut by a general linear form it follows that the dimension must b e at least This

contradiction proves that G I 

A b eautiful result of Richard Stanley shows that the symmetry of the Hilb ert

function is not only necessary but even a sucient condition for a CohenMacaulay

graded domain to b e Gorenstein

Theorem St If R k R is a CohenMacaulay domain of dimension

d then R is Gorenstein i

d l

F R t t F R t

for some l Z

Example The converse is NOT true if the ring is not a domain For

instance I X X Y Y has Hilb ert function and satises the equation

F t t F t but is not Gorenstein

It is an op en problem p osed by Stanley to characterize which sequences of inte

gers can b e the hvector of a Gorenstein graded algebra Symmetry is a necessary

but not sucient condition Stanley gives an example of a Gorenstein dimensional

graded ring with Hilb ert function The lack of unimo dality is the

interesting p oint in this example

CRAIG HUNEKE

Stanleys theorem was at least partially motivated as far as I can tell by the

uses he makes of it One of them is to prove that invariants of tori are Gorenstein

Invariants of group actions provide a rich source of Gorenstein rings as we will

discuss in the next section

Invariants and Gorenstein rings

There are a great many sources of Gorenstein rings Complete interesections

are the most prevelant After all every nitely generated k algebra which is a

domain is birationally a complete intersection hence birationally Gorenstein The

Gorenstein prop erty b ehaves well under at maps and often b ers of such maps

are Gorenstein For example if K is a nitely generated eld extension of a eld k

and L is an arbitrary eld extension of k then K L is Gorenstein

k

A huge source of Gorenstein rings comes from invariants of groups We x

notation and denitions Let G b e a closed algebraic subgroup of the general linear

group GLV where V is a nite dimensional vector space over a ground eld k

We obtain a natural action of G on the p olynomial ring k V S and we denote

G

the ring of invariant p olynomials by R S Thus

G

S ff k V j g f f for all g Gg

We will fo cus on groups G which are linearly reductive This means that every

Gmo dule V is a direct sum of simple mo dules Equivalently every Gsubmo dule

of V has a Gstable complement Examples of linearly reductive groups include

nite groups whose order is invertible tori the classical groups in characteristic

and semisimple groups in characteristic When G is linearly reductive there is

G G

a retraction S S called the Reynolds operator which is S linear It follows

that the ring of invariants R must b e No etherian For a chain of ideals in R when

extended to S will stabilize and then applying the retraction stabilizes the chain

in R But even more is true a famous theorem of Ho chster and Rob erts HoR

says that the ring of invariants is even CohenMacaulay

Theorem Let S be a Noetherian k algebra which is regular eg a poly

nomial ring over k and let G be a linearly reductive linear algebraic group acting

G

k rational ly on S Then S is CohenMacaulay

G

A natural question is to ask when S is Gorenstein The following question

was the fo cus of several authors see Ho St St Wa and their references

Question Let k b e an algebraically closed eld and let G b e a linearly

reductive algebraic group over k acting linearly on a p olynomial ring S over k such

G

that the det is the trivial character Is S Gorenstein

Thus for example when G is not only in GLV but in SLV the ring of in

variants should b e Gorenstein This question was motivated b ecause of the known

examples it was proved for nite groups whose order is invertible by Watanab e

Wa for an algebraic torus by Stanley St or for connected semisimple groups

by a result of Murthy Mu that CohenMacaulay rings which are UFDs are Goren

stein provided they have a canonical mo dule However this question has a negative

solution Friedrich Knop in Kn gave a counterexample to Question and

has basically completed describ ed when the ring of invariants is Gorenstein

HYMAN BASS AND UBIQUITY GORENSTEIN RINGS

Murthys result quoted in the ab ove paragraph is straightforward based on

the theory of the canonical mo dule The canonical mo dule of an integrally closed

No etherian ring R is a rank reexive mo dule which therefore lives in the class

group If R is a UFD the class group is trivial and so is the canonical mo dule

But this means that it is a free R mo dule which together with R b eing Cohen

Macaulay gives Gorenstein Grith has shown in Gr Theorem that if S

is a lo cal ring which is a UFD which is nite over a regular lo cal ring R in such a

way that the extension of fraction elds is Galois then S is necessarily a complete

intersection See also AvB

Ubiquity and Mo dule Theory

In trying to recapture the material that led Bass to write his ubiquity pap er it

seems that his choice of the word ubiquity came from that fact that the Gorenstein

prop erty was arising in totally dierent contexts from many authors from North

cott and Rees work on irreducible systems of parameters and CohenMacaulay

rings from the work of Gorenstein and Rosenlicht on plane curves and complete

intersections from the work of Grothendieck and Serre on duality and from his

own work on rings of nite injective dimension They were indeed ubiquitous

But there are other ways in which they are ubiquitous Every complete lo cal

domain or nitely generated domain over a eld is birationally a complete intersec

tion hence up to integral closure every such ring is Gorenstein

Every CohenMacaulay ring R with a canonical mo dule is actually a Goren

stein ring up to nilp otents by using the idealization idea of Nagata Sp ecically

we form a new ring consisting of by matrices whose lower left hand corner is

whose diagonal is a constant element of R and whose upp er left comp onent is an

arbitrary element of Pictorially elements lo ok like

R

r y

r

where r R and y It is easy to see this ring S is commutative is an

R

ideal of S S R and What is amazing is that S is Gorenstein Up to

radical this transfers questions ab out CohenMacaulay rings to questions ab out

Gorenstein rings

Even if a lo cal ring R is not CohenMacaulay we can still closely approximate

it by a Gorenstein ring if it is the homomorphic image of a regular lo cal ring S

eg in the geometric case or the complete case Write R SI and cho ose

a maximal regular sequence f f I where g heightI Then the ring

g

T Sf f is Gorenstein b eing a complete intersection T maps onto R and

g

dimT dimR

Regular rings are the most basic rings in the study of commutative rings How

ever Gorenstein rings are the next most basic and as the examples ab ove demon

strate one can approximate arbitrary lo cal commutative rings quite closely by

Gorenstein rings Moreover the mo dule theory over Gorenstein rings is as close to

that of regular rings as one might hop e

A partial converse to Murthys result was given by Ulrich U His construction uses the

theory of liaison or linkage in which Gorenstein ideals play an extremely imp ortant role See for

example PS or HU

CRAIG HUNEKE

Recall that nitely generated mo dules over a regular lo cal ring are of nite

pro jective dimension This characterizes regular lo cal rings so we cannot hop e

to achieve this over nonregular rings However there are two approximation

theorems for nitely generated mo dules over Gorenstein lo cal rings which show

that up to CohenMacaulay mo dules we can still approximate such mo dules by

mo dules of nite pro jective dimension The rst theorem go es back to Auslander

and Bridger AB

Theorem Let R m be a Gorenstein local ring and let M be a nitely

generated R module of dimension equal to the dimension of R Then there exists

an exact sequence

C M F Q

where F is nitely generated and free C is a CohenMacaulay module of maximal

dimension and Q is a module of nite projective dimension

A second and similar theorem in spirit is due to Auslander and Buchweitz

ABu

Theorem Let R m be a Gorenstein local ring and let M be a nitely

generated R module Then there is an exact sequence

Q C M

such that C is a nitely generated CohenMacaulay module of maximal dimension

and Q has nite projective dimension

To put this in context any CohenMacaulay mo dule of maximal dimension over

a regular lo cal ring is free so b oth of these results recover that over a regular lo cal

ring nitely generated mo dules have nite pro jective dimension The p oint is that

over a Gorenstein ring the study of mo dules often reduces to the study of mo dules

of nite pro jective dimension and CohenMacaulay mo dules of maximal dimension

This is certainly the b est one can hop e for

The mo dules C in the are called CohenMacaulay approximations of M

and are a topic of much current interest Theorem can b e thought of as a

generalization of an argument known as Serres trick which he used in the study of

pro jective mo dules Let M b e a nitely generated mo dule over a ring R and cho ose

generators for Ext M R If there are n such generators then one can use the

R

n

Yoneda denition of Ext to create an exact sequence R N M where

n

Ext N R The p oint is after dualizing the dual of R maps onto Ext M R

R R

It turns out one can continue this pro cess and at the next stage obtain a short exact

sequence P N M where now Ext N R Ext N R and

R R

P has pro jective dimension at most Continuing until the dimension d of R we

eventually obtain a sequence Q C M where Q has nite pro jective

d

dimension and Ext C R Ext C R Ext C R If R is assumed

R R R

to b e lo cal and Gorenstein the duality forces C to b e CohenMacaulay which gives

Theorem

Gorenstein rings are now part of the basic landscap e of mathematics A search

for the word Gorenstein in MathSci Net reveals around entries There are

many osho ots which were not mentioned in this article The ubiquity pap er and

all of its manifestations are indeed ubiquitous

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HYMAN BASS AND UBIQUITY GORENSTEIN RINGS

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EG G Evans and P Grith Syzygies London Math So c Lecture Notes vol Cam

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Ga E Gareld Journal Citation Studies Pure and Applied Mathematics Journals What

they cite and vice versa Current Contents

Go D Gorenstein An arithmetic theory of adjoint plane curves Trans Amer Math So c

G P Grith Normal extensions of regular local rings J Algebra

Gr A Grothendieck Theoremes de dualite pour les faiceaux algebriques coherents Seminaire

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HeK J Herzog and E Kunz Die Wertehalbgrupper eines lokalen Rings der Dimension

Ber Heidelb erger Akad Wiss

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vol SpringerVerlag

HM J Herzog and M Miller Gorenstein ideals of deviation two Comm Algebra

Ho M Ho chster Topics in the homological theory of modules over commutative rings

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Cyclic purity versus purity in excel lent Noetherian rings Trans Amer Math Ho

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HoR Ho chster M and JL Rob erts Rings of invariants of reductive groups acting on regular

rings are CohenMacaulay Advances in Math

HU C Huneke and B Ulrich The structure of linkage Annals Math

Divisor class groups and deformations American J Math HU

Ia A Iarrobino Associated graded algebra of a Gorenstein Artin algebra Memoirs Amer

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Kn F Knop Der kanonische Modul eines Invariantenrings J Alg

Ku E Kunz Almost complete intersections are not Gorenstein rings J Alg

KM A Kustin and M Miller Structure theory for a class of grade four Gorenstein ideals

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Ma E Matlis Injective modules over noetherian rings Pacic J Math

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No E No ether Idealtheorie in Ringbereichen Math Annalen

NR DG Northcott and D Rees Principal systems Quart J Math

PS C Peskine and L Szpiro Dimension projective nie et cohomologie locale IHES

Liaison des varietes algebriques Invent Math PS

Rob P Rob erts Le theoreme dintersection C R Acad Sc Paris Ser I

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Ro M Rosenlicht Equivalence relations on algebraic curves Annals Math

Sa P Samuel Singularites des varietes algebriques Bull So c Math de France

Se JP Serre Sur les modules projectifs Seminaire Dubreil

Se Sur la dimension homologique des anneaux et des modules noetheriens Pro c

Int Symp TokyoNikko Science Council of Japan

Algebraic Groups and Class Fields Graduate Texts in Mathematics Se

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St R Stanley Hilbert functions of graded algebras Advances Math

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U B Ulrich Gorenstein rings as specializations of unique factorization domains J Alg

VaVi WV Vasconcelos and R Villarreal On Gorenstein ideals of codimension four Pro c

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Wa K Watanab e Certain invariant subrings are Gorenstein I II Osaka J Math

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