Canonical Degrees of Cohen-Macaulay Rings and Modules: a Survey
J.P. Brennan, L. Ghezzi, J. Hong, L. Hutson, W.V. Vasconcelos
aa r X i v : . [ m a t h . A C ] J un CANONICAL DEGREES OF COHEN-MACAULAY RINGS AND MODULES: ASURVEY
J. P. BRENNAN, L. GHEZZI, J. HONG, L. HUTSON AND W. V. VASCONCELOS
Contents
1. Introduction 12. Setting up and calculating canonical degrees 23. The basic canonical degree 44. Extremal values of the canonical degree 65. The bi-canonical degree 76. Canonical index 97. Roots of canonical ideals 138. Minimal values of bi-canonical degrees 179. Change of rings 1910. Monomial subrings 2211. Rees algebras 2412. Canonical degrees of A = m : m Abstract.
The aim of this survey is to discuss invariants of Cohen-Macaulay local rings thatadmit a canonical module. Attached to each such ring R with a canonical ideal C , there areintegers–the type of R , the reduction number of C –that provide valuable metrics to expressthe deviation of R from being a Gorenstein ring. We enlarge this list with other integers–theroots of R and several canonical degrees. The latter are multiplicity based functions of theRees algebra of C . We give a uniform presentation of three degrees arising from commonroots. Finally we experiment with ways to extend one of these degrees to rings where C is notnecessarily an ideal. Key Words and Phrases:
Anti-canonical degree, bi-canonical degree, canonical degree, Cohen-Macaulay type, analytic spread, roots, reduction number. Introduction
Let ( R , m ) be a Cohen-Macaulay local ring of dimension d that has a canonical ideal C .Our central viewpoint is to look at the properties of C as a way to refine our understandingof R . In [18] several metrics are treated aimed at measuring the deviation from R beingGorenstein, that is when C ≃ R . Here we explore another pathway but still with the sameoverall goal. Unlike [18] the approach here is arguably more suited for computation in classesof algebras such as Rees algebras and monomial subrings. First however we outline the general underpin of these developments. The organizing principle to set up a canonical degree is torecast numerically criteria for a Cohen-Macaulay ring to be Gorenstein.We shall now describe how this paper is organized. For a Cohen-Macaulay local ring ( R , m )of dimension d with a canonical ideal C , we are going to attach a non-negative integer c ( R )whose value reflects divisorial properties of C and provide for a stratification of the class ofCohen-Macaulay rings. We have noted two such functions in the current literature ([25], [18])and here we will build a third degree.In Section 2 we recall from the literature the needed blocks to put together the degrees.Section 3 quickly assembles three degrees and begins the comparison of its properties. Theseassemblages turn out to provide effective symbolic calculation [we used Macaulay2 ([23]) inour experiments] but turn out useful for theoretical calculations in special classes of rings. Thenew degree is labelled the bi-canonical degree of R and is given bybideg( R ) = deg( C ∗∗ / C ) = X height p =1 bideg( R p ) deg( R / p ) = X height p =1 [ λ ( R p / C p ) − λ ( R p / C ∗∗ p )] deg( R / p ) . This is a well-defined finite sum independent of the chosen canonical ideal C . It leads immedi-ately to comparisons to two other degrees, the canonical degree of [18], cdeg( R ) = deg( C / ( s ))for a minimal reduction ( s ) of C [in dimension one and suitably assembled as above to alldimensions], and the residue of R of [25] tdeg( R ) = deg( R / trace( C )), where trace( C ) is thetrace ideal of C . Arising naturally is a comparison conjecture, that cdeg( R ) ≥ bideg( R ). Weengage in a brief discussion on how to recognize that a codimension one ideal I is actually acanonical ideal. We finally recall the notion of the rootset of R ([18]), perhaps one of leastunderstood sets attached to C and raise questions on how it affects the values of the degrees.We begin in Section 4 a study of algebras according to the values of one of the c ( R ). If c ( R ) = 0, for all canonical degrees, R is Gorenstein in codimension one. It is natural to askwhich rings correspond to small values of c ( R ). In dimension one, cdeg( R ) ≥ r ( R ) − R ) ≥
1, where equality corresponding to the almost Gorenstein rings of ([3, 20, 22])and nearly Gorenstein rings of [25], respectively.We begin in Sections 5, 6, 7, 8, 9, 10, 11, 12 calculations of cdeg( R ) and bideg( R ) forvarious classes of algebras. Unlike the case of cdeg( R ), already for hyperplane sections thebehaviour of bideg( R ) is more challenging. Interestingly, for monomial rings k [ t a , t b , t c ] thetechnical difficulties are reversed. In two cases, augmented rings and [tensor] products, veryexplicit formulas are derived. More challenging is the case of Rees algebras when we are oftenlimited to deciding the vanishing of degrees. The most comprehensive results resolve around m : m .The ring A = m : m has a special role in the literature of low dimensional rings. For instance,if R is a Buchsbaum ring of dimension ≥ A is its S -ification ([17,Theorem 4.2]).Sections 13, 14, 15 discuss various possible generalizations and open questions.2. Setting up and calculating canonical degrees
In this section we describe the canonical degrees known to the authors and extend them tomore general structures.
ANONICAL DEGREES OF COHEN-MACAULAY RINGS AND MODULES: A SURVEY Divisorial basics of C . We are going to make use of the basic facts expressed in the codimensionone localizations of R . Theorem 2.1.
Let ( R , m ) be a local ring of dimension one let Q be its total ring of fractions.Assume that R has a canonical module C . (1) R has a canonical ideal if and only if the total ring of fractions of b R is Gorenstein. [1, 5, 26] . (2) The m -primary ideal I is a canonical ideal if and only if I : R m = I : Q m = ( I, s ) for some s ∈ R . [26, Theorem 3.3] . (3) R is Gorenstein if and only if C is a reflexive module. [26, Corollary 7.29] . (4) If R is an integral domain with finite integral closure then I ∗∗ is integral over I . [10,Proposition 2.14] . We often assume harmlessly that R has an infinite residue field and has arbitrary Krulldimension. For a finitely generated R -module M , the notation deg( M ) = e ( m , M ) refers tothe multiplicity defined by the m -adic topology. The Cohen-Macaulay type of R is denoted by r ( R ). Remark 2.2.
Among the ways we can set up the comparison of C to a principal ideal we havethe following.(1) In dimension one, select an element c of C and define cdeg( R ) = deg( C / ( c )). The choiceshould yield the same value for all C . In [18] ( c ) is chosen as a minimal reduction of C when then cdeg( R ) = e ( C , R ) − deg( R / C ).(2) The choice is more straightforward in one case: Set bideg( R ) = deg( C ∗∗ / C ), where C ∗∗ isthe bidual of C .(3) A standard metric is simply tdeg( R ) = deg( R /τ ( C )), where τ ( C ) is the trace ideal of C :( f ( x ) , f ∈ C ∗ , x ∈ C ). It is often used to define the Gorenstein locus of R (see [25] for adiscussion). For a method to calculate the trace of a module see [25, Proposition 3.1], [38,Remark 3.3].(4) These are distinct [in dimension d >
1] numbers, which are independent of the choice of C , that share a common property:(i) R is Gorenstein if and only if one of cdeg( R ), bideg( R ) or tdeg( R ) vanishes (in whichcase all three vanish).(ii) If R is not Gorenstein, cdeg( R ) ≥ r ( R ) − ≥
1, tdeg( R ) ≥
1, bideg( R ) ≥
1, and theminimal values are attained. In dimension one bideg = tdeg, see Proposition 2.3.(5) The cases when the minimal values are reached have the following designations:(i) cdeg( R ) was introduced in [18] and called the canonical degree of R : cdeg( R ) = r ( R ) − R is an almost Gorenstein ring ([20, 22]).(ii) tdeg( R ) was introduced in [25] and called it the residue of R : res( R ). It can alsobe called the trace degree of R . Another possible terminology is to call it the anti-canonical degree of R : tdeg( R ) = 1 if and only if R is a nearly Gorenstein ring .(iii) bideg( R ) was introduced in [19] and called the bi-canonical degree of R : bideg( R ) = 1if and only if R is a so-called Goto ring .There are some relationships among these invariants.
J. P. BRENNAN, L. GHEZZI, J. HONG, L. HUTSON AND W. V. VASCONCELOS
Proposition 2.3. [J. Herzog, personal communication]
Let R be Cohen-Macaulay local ringof dimension with a canonical ideal C . Then bideg( R ) = λ ( R / trace( C )) . Proof.
We will show that λ ( C ∗∗ / C ) = λ ( R / trace( C )) . From trace( C ) = C · C ∗ , we haveHom(trace( C ) , C ) = Hom( C · C ∗ , C ) = Hom( C ∗ , Hom( C , C )) = C ∗∗ . Now dualize the exact sequence0 → trace( C ) −→ R −→ R / trace( C ) → , into C to obtain,0 = Hom( R / trace( C ) , C ) → C = Hom( R , C ) → C ∗∗ = Hom(trace( C ) , C ) → Ext ( R / trace( C ) , C ) → , which shows that C ∗∗ / C and Ext ( R / trace( C ) , C ) are isomorphic. Since by local dualityExt ( · , C ) is self-dualizing on modules of finite support, bideg( R ) = λ ( R / trace( C )) . (cid:3) A similar argument applies to a slightly different class of ideals. We say that an ideal I is closed if Hom( I, I ) = R in the terminology of [4]. Proposition 2.4.
Let R be a local ring of dimension one and finite integral closure, and let I be a closed ideal. If I is reflexive then I is principal.Proof. We argue by contradiction. From trace( I ) = L = I · I ∗ , we haveHom(trace( I ) , R ) = Hom( I · I ∗ , R ) = Hom( I ⊗ I ∗ , R ) = Hom( I, Hom( I ∗ , R )) = Hom( I, I ∗∗ ) = R , and thus L ∗∗ = R x . If x is a unit L ∗ = R and therefore L has height greater than one, which isnot possible. Thus we have L = M x , where M ⊂ m . Since L ∗∗ = R x is integral over L = M x ([10, Proposition 2.14]), we have for some positive integer n R x n = M x R x n − , and thus R = M , which is impossible. (cid:3) The basic canonical degree
Let ( R , m ) be a Cohen-Macaulay local ring. Suppose that R has a canonical ideal C . In thissetting we introduce a numerical degree for R and study its properties. The starting point ofour discussion is the following elementary observation. We denote the length function by λ . Proposition 3.1.
Let ( R , m ) be a -dimensional Cohen-Macaulay local ring with a canonicalideal C . Then the integer cdeg( R ) = e ( C ) − λ ( R / C ) is independent of the canonical ideal C .Proof. If x is an indeterminate over R , in calculating these differences we may pass from R to R ( x ) = R [ x ] m R [ x ] , in particular we may assume that the ring has an infinite residue field.Let C and D be two canonical ideals. Suppose ( a ) is a minimal reduction of C . Since D ≃ C ([6, Theorem 3.3.4]), D = q C for some fraction q . If C n +1 = ( a ) C n by multiplying it by q n +1 ,we get D n +1 = ( qa ) D n , where ( qa ) ⊂ D . Thus ( qa ) is a reduction of D and C / ( a ) ≃ D / ( qa ).Taking their co-lengths we have λ ( R / ( a )) − λ ( R / C ) = λ ( R / ( qa )) − λ ( R / D ) . ANONICAL DEGREES OF COHEN-MACAULAY RINGS AND MODULES: A SURVEY Since λ ( R / ( a )) = e ( C ) and λ ( R / ( qa )) = e ( D ), we havee ( C ) − λ ( R / C ) = e ( D ) − λ ( R / D ) . (cid:3) We can define cdeg( R ) in full generality as follows. Theorem 3.2.
Let ( R , m ) be a Cohen-Macaulay local ring of dimension d ≥ that has acanonical ideal C . Then cdeg( R ) = X height p =1 cdeg( R p ) deg( R / p ) = X height p =1 [e ( C p ) − λ (( R / C ) p )] deg( R / p ) is a well-defined finite sum independent of the chosen canonical ideal C . In particular, if C isequimultiple with a minimal reduction ( a ) , then cdeg( R ) = deg( C / ( a )) = e ( m , C / ( a )) . Proof.
By Proposition 3.1, the integer cdeg( R p ) does not depend on the choice of a canonicalideal of R . Also cdeg( R ) is a finite sum since, if p / ∈ Min( C ), then C p = R p so that R p isGorenstein. Thus cdeg( R p ) = 0. The last assertion follows from the associativity formula:cdeg( R ) = X height p =1 λ (( C / ( a )) p ) deg( R / p ) = deg( C / ( a )) . (cid:3) Definition 3.3.
Let ( R , m ) be a Cohen-Macaulay local ring of dimension d ≥ canonical degree of R is the integercdeg( R ) = X height p =1 cdeg( R p ) deg( R / p ) . Corollary 3.4. cdeg( R ) ≥ and vanishes if and only if R is Gorenstein in codimension . Corollary 3.5.
Suppose that the canonical ideal of R is equimultiple. Then we have thefollowing. (1) cdeg( R ) ≥ r ( R ) − . (2) cdeg( R ) = 0 if and only if R is Gorenstein.Proof. Let ( a ) be a minimal reduction of the canonical ideal C . Thencdeg( R ) = e ( m , C / ( a )) ≥ ν ( C / ( a )) = r ( R ) − . If cdeg( R ) = 0 then r ( R ) = 1, which proves that R is Gorenstein. (cid:3) Now we extend the above result to a more general class of ideals when the ring has dimensionone. We recall that if R is a 1-dimensional Cohen-Macaulay local ring, I is an m -primary idealwith minimal reduction ( a ), then the reduction number of I relative to ( a ) is independent ofthe reduction ([35, Theorem 1.2]). It will be denoted simply by red( I ). Proposition 3.6.
Let ( R , m ) be a -dimensional Cohen-Macaulay local ring which is not avaluation ring. Let I be an irreducible m -primary ideal such that I ⊂ m . If ( a ) is a minimalreduction of I , then λ ( I/ ( a )) ≥ r ( R ) − . In the case of equality, red( I ) ≤ .Proof. Let L = I : m and N = ( a ) : m . Then λ ( L/I ) = r ( R /I ) = 1 and λ ( N/ ( a )) = r ( R ).Thus, we have r ( R ) ≤ λ ( L/N ) + λ ( N/ ( a )) = λ ( L/ ( a )) = λ ( L/I ) + λ ( I/ ( a )) = 1 + λ ( I/ ( a )) , which proves that λ ( I/ ( a )) ≥ r ( R ) − J. P. BRENNAN, L. GHEZZI, J. HONG, L. HUTSON AND W. V. VASCONCELOS
Suppose that λ ( I/ ( a )) = r ( R ) −
1. Then L = N . By [9, Lemma 3.6], L is integral over I . Thus, L is integral over ( a ). By [11, Theorem 2.3], red( N ) = 1. Hence L = aL . Since λ ( L/I ) = 1, by [21, Proposition 2.6], we have I = aI . (cid:3) Extremal values of the canonical degree
We examine in this section extremal values of the canonical degree. First we recall the definitionof almost Gorenstein rings ([3, 20, 22]).
Definition 4.1. ([22, Definition 3.3]) A Cohen-Macaulay local ring R with a canonical module ω is said to be an almost Gorenstein ring if there exists an exact sequence of R -modules0 → R → ω → X → ν ( X ) = e ( X ). Proposition 4.2.
Let ( R , m ) be a Cohen-Macaulay local ring with a canonical ideal C . Assumethat C is equimultiple. If cdeg( R ) = r ( R ) − , then R is an almost Gorenstein ring. Inparticular, if cdeg( R ) ≤ , then R is an almost Gorenstein ring.Proof. We may assume that R is not a Gorenstein ring. Let ( a ) be a minimal reduction of C .Consider the exact sequence of R -modules0 → R ϕ → C → X → , where ϕ (1) = a. Then ν ( X ) = r ( R ) − R ) = e ( X ). Thus, R is an almost Gorenstein ring. (cid:3) Proposition 4.3.
Let ( R , m ) be a -dimensional Cohen-Macaulay local ring with a canonicalideal C . Then cdeg( R ) = r ( R ) − if and only if R is an almost Gorenstein ring.Proof. It is enough to prove that the converse holds true. We may assume that R is not aGorenstein ring. Let ( a ) be a minimal reduction of C . Since R is almost Gorenstein, thereexists an exact sequence of R -modules0 → R ψ → C → Y → ν ( Y ) = e ( Y ) . Since dim( Y ) = 0 by [22, Lemma 3.1], we have that m Y = (0). Let b = ψ (1) ∈ C and set q = ( b ). Then mq ⊆ m C ⊆ q . Therefore, since R is not a DVR and l R ( q / mq ) = 1, we get m C = mq , whence the ideal q is a reduction of C by the Cayley-Hamilton theorem, so thatcdeg( R ) = e ( Y ) = ν ( Y ) = r ( R ) − (cid:3) Now we consider the general case when a canonical ideal is not necessarily equimultiple.
Lemma 4.4.
Let ( R , m ) be a Cohen-Macaulay local ring of dimension d ≥ with infiniteresidue field and a canonical ideal C . Let a be an element of C such that (i) for ∀ p ∈ Ass( R / C ) the element a generates a reduction of C R p , (ii) a is R -regular, and (iii) a m C . Let Z = { p ∈ Ass( C / ( a )) | C 6⊆ p } . (1) cdeg( R ) = deg( C / ( a )) − X p ∈ Z λ (( C / ( a )) p ) deg( R / p ) . (2) cdeg( R ) = deg( C / ( a )) if and only if Ass( C / ( a )) ⊆ V ( C ) .Proof. It follows from Theorem 3.2 and deg( C / ( a )) = X height p =1 λ (( C / ( a )) p ) deg( R / p ). (cid:3) Theorem 4.5.
With the same notation given in Lemma , suppose that
Ass( C / ( a )) ⊆ V ( C ) .Then cdeg( R ) = r ( R ) − if and only if R is an almost Gorenstein ring. ANONICAL DEGREES OF COHEN-MACAULAY RINGS AND MODULES: A SURVEY Proof.
It is enough to prove the converse holds true. We may assume that R is not a Gorensteinring. Choose an exact sequence 0 → R → C → Y → Y ) = ν ( Y ) = r ( R ) −
1. Since Ass( C / ( a )) ⊆ V ( C ), we have deg( Y ) ≥ deg( C / ( a )).By Theorem 3.5 and Lemma 4.4, we obtain the following. r ( R ) − ≤ cdeg( R ) = deg( C/ ( a )) ≤ deg( Y ) = r ( R ) − . (cid:3) Remark 4.6. If R is a non-Gorenstein normal domain, then its canonical ideal cannot beequimultiple. Proof.
Suppose that a canonical ideal C of R is equimultiple, i.e., C n +1 = a C n . Then we wouldhave an equation ( n + 1)[ C ] = n [ C ] in its divisor class group. This means that [ C ] = [0]. Thus, C ≃ R . Hence C cannot be equimultiple. (cid:3) The bi-canonical degree
The approach in [18] is dependent on finding minimal reductions. We pick here one thatseems particularly amenable to computation. Let C ∗ = Hom( C , R ) be the dual of C and C ∗∗ its bidual. [In general, in writing Hom R we omit the symbol for the underlying ring.] In thenatural embedding 0 → C −→ C ∗∗ −→ B → , B remains unchanged when C is replaced by another canonical module, say D = s C for aregular element s ∈ Q . B vanishes if and only if R is Gorenstein as indicated above. It iseasy to see that a similar observation can be made if d >
1. That is, B = 0 if and only if R is Gorenstein in codimension 1. B embeds into the Cohen-Macaulay module R / C that hasdimension d −
1, and thus B either is zero or its associated primes are associated primes of C ,all of which have codimension one.Like in [18], we would like to explore the length of B , bideg( R ) = λ ( B ) which we view as adegree, in dimension 1 and deg( B ) in general. We stick to d = 1 for the time being. We wouldlike some interesting examples and examine relationships to the other metrics of R . We do nothave best name for this degree, but we could also denote it by ddeg( R ) (at least the double‘d’ as a reminder of ‘double dual’).Let us formalize these observations as the following. Theorem 5.1.
Let ( R , m ) be a Cohen-Macaulay local ring of dimension d ≥ that has acanonical ideal C . Then bideg( R ) = deg( C ∗∗ / C ) = X height p =1 bideg( R p ) deg( R / p ) = X height p =1 [ λ ( R p / C p ) − λ ( R p / C ∗∗ p )] deg( R / p ) is a well-defined finite sum independent of the chosen canonical ideal C . Furthermore, bideg( R ) ≥ and vanishes if and only if R is Gorenstein in codimension .Comparison of canonical degrees. If ( c ) is a minimal reduction of C how to comparecdeg( R ) = λ ( R / ( c )) − λ ( R / C ) ⇔ λ ( R / C ) − λ ( R / C ∗∗ ) = bideg( R )The point to be raised is: which is more approachable, e ( C ) or λ ( R / C ∗∗ )? we will argue,according to the method of computation below, that the latter is more efficient which wouldbe demonstrated if the following conjecture were settled. J. P. BRENNAN, L. GHEZZI, J. HONG, L. HUTSON AND W. V. VASCONCELOS
Conjecture 5.2. [Comparison Conjecture] If dim R = 1 the following inequality holdscdeg( R ) ≥ bideg( R ) . That is from the diagram ( c ) C C ∗∗ where C ∗∗ = ( c ) : (( c ) : C ), we have λ ( C / ( c )) ≥ λ ( C ∗∗ / C ). Alternativelye ( C ) + λ ( R / C ∗∗ ) ≥ · λ ( R / C ) . This would imply, by the associativity formula, that the inequality holds in all dimensions.
Computation of duals and biduals.
Let I be a regular ideal of the Noetherian ring R . If Q isthe total ring of fractions of R then Hom( I, R ) = R : Q I. A difficulty is that computer systems such as Macaulay2 ([23]) are set to calculate quotientsof the form A : R B for two ideals A, B ⊂ R , which is done with calculations of syzygies.This applies especially in the case of the ring R = k [ x , . . . , x d ] /P where k is an appropriatefield. To benefit of the efficient quotient command of this system we formulate the problem asfollows. Proposition 5.3.
Let I ⊂ R and suppose a is a regular element of I . Then (1) I ∗ = Hom( I, R ) = a − (( a ) : R I ) . (2) I ∗∗ = Hom(Hom( I, R ) , R ) = ( a ) : R (( a ) : R I ) = annihilator of Ext R ( R /I, R ) . (3) τ ( I ) = a − I · ( a ) : R I .Proof. (1) If q ∈ Hom( I, R ) = R : Q I , then qI ⊂ R ≃ qaI ⊂ ( a ) ≃ qa ∈ ( a ) : R I ≃ q ∈ a − (( a ) : R I )(3) Follows from the calculation q ∈ I ∗∗ = [ a − (( a ) : R I )] ∗ = a [ R : Q (( a ) : R I )] = a [ a − (( a ) : R (( a ) : R I )] = ( a ) : R (( a ) : R I ) . See also [38, Remark 3.3], [25, Proposition 3.1]. (cid:3)
Computation of
Hom(
A, B ) = A : Q B . Let A and B be ideals of R . A question is how to trickthe ordinary quotient command to do these computations. There are important cases needed,such as Hom( A, A ). If B = ( b , . . . , b n ), b i regular element of R , then A : Q B = n \ i =1 A : Q b i = n \ i =1 Ab − i , so setting b = Q ni =1 b i and b b i = bb − i , we have A : Q B = b − n \ i =1 A b b i . A case of interest is when I = m : A = Hom( m , m ). If R is a Cohen-Macaulay local ring ofdimension one that is not a DVR, then A = R : m : m · Hom( m , R ) ⊂ m and therefore A = m : Q m = R : Q m = 1 /x · (( x ) : R m ) , where x is a regular element of m . ANONICAL DEGREES OF COHEN-MACAULAY RINGS AND MODULES: A SURVEY Example 5.4.
Here is an example from [18]. Let L = ( X − Y Z, Y − XZ, Z − XY ) and R = A/L . Let x, y, z be the images of
X, Y, Z in R . Then C = ( x, z ) is a canonical ideal of C = ( x , x z, xz , z ) = ( x , x z, xz ) = x C , which proves that red( C ) = 2. Note thate ( C ) = λ ( R /x R ) = λ ( A/ ( x + L )) = λ ( A/ ( x, y , yz, z )) = 3 . Recognition of canonical ideals.
These methods permit answering the following question. Givenan ideal C , is it a canonical ideal? These observations are influenced by the discussion in [15,Section 2]. For other methods, see [20]. Proposition 5.5.
Let ( R , m ) be a one-dimensional Cohen-Macaulay local ring and let Q beits total ring of fractions. Then an m -primary ideal I is a canonical ideal if I is an irreducibleideal and Hom(
I, I ) = R .Proof. Note first that if q ∈ I : Q m then qI ⊂ I , that is q ∈ Hom(
I, I ). Now we invokeTheorem 2.1(2). (cid:3)
Corollary 5.6.
Let ( R , m ) be a Cohen-Macaulay local ring of dimension one. (1) The m -primary ideal I is a canonical ideal if and only if both I and xI are irreducible forany regular element x ∈ m . ( This is [15, Proposition 2.2] . )(2) m is a canonical ideal if and only if R is a discrete valuation ring.Proof. If q ∈ I : Q m , qx ⊂ I ⊂ x R , so q ∈ R . Thus q ∈ I : R m . Since I is irreducible I : R m = ( I, s ) and we can invoke again Theorem 2.1(2). Apply the previous assertion to xI .The converse is well-known.Let x be a regular element in m . Since dim R = 1 there is a positive integer n such that m n ⊂ x R but m n − x R . If n = 1, m = x R there is nothing else to prove. If n >
1, wehave that L = m n − x − satisfies L · m is an ideal of R so that either L · m = R and hence m isinvertible, or L · m ⊂ m that means L ⊂ Hom( m , m ) = R . Thus m n − x − ⊂ R so m n − ⊂ x R ,which is a contradiction. (cid:3) Canonical index
Throughout the section, let ( R , m ) be a Cohen-Macaulay local ring of dimension d ≥ C exists. We begin by showing thatthe reduction number of a canonical ideal of R is an invariant of the ring. Proposition 6.1.
Let C and D be canonical ideals of R . Then red( C ) = red( D ) .Proof. Let K be the total ring of quotients of R . Then there exists q ∈ K such that D = q C .Let r = red( C ) and J a minimal reduction of C with C r +1 = J C r . Then D r +1 = ( q C ) r +1 = q r +1 ( J C r ) = ( qJ )( q C ) r = qJ D r so that red( D ) ≤ red( C ). Similarly, red( C ) ≤ red( D ). (cid:3) Definition 6.2.
Let ( R , m ) be a Cohen-Macaulay local ring of dimension d ≥ C . The canonical index of R is the reduction number of the canonical ideal C of R and is denoted by ρ ( R ). Remark 6.3.
Suppose that R is not Gorenstein. The following are known facts.(1) If the canonical ideal of R is equimultiple, then ρ ( R ) = 1. (2) If dim R = 1 and e ( m ) = 3, then ρ ( R ) = 2.(3) If dim R = 1 and cdeg( R ) = r ( R ) −
1, then ρ ( R ) = 2. Proof. (1) Suppose that ρ ( R ) = 1. Let C be a canonical ideal of R with C = a C . Then C a − ⊂ Hom( C , C ) = R so that C = ( a ). This is a contradiction.(2) It follows from the fact that, if ( R , m ) is a 1–dimensional Cohen-Macaulay ring and I an m –primary ideal, then red( I ) ≤ e ( m ) − (cid:3) Sally module.
We examine briefly the Sally module associated to the canonical ideal C in ringsof dimension 1. Let Q = ( a ) be a minimal reduction of C and consider the exact sequence offinitely generated R [ Q T ]-modules0 → C R [ Q T ] −→ C R [ C T ] −→ S Q ( C ) → . Then the Sally module S = S Q ( C ) = L n ≥ C n +1 / C Q n of C relative to Q is Cohen-Macaulayand, by [17, Theorem 2.1], we havee ( C ) = cdeg( R ) + ρ ( R ) − X j =1 λ ( C j +1 /a C j ) = ρ ( R ) − X j =0 λ ( C j +1 /a C j ) . Remark 6.4.
Let R be a 1-dimensional Cohen-Macaulay local ring with a canonical ideal C .Then the multiplicity of the Sally module s ( S ) = e ( C ) − e ( C ) + λ ( R / C ) = e ( C ) − cdeg( R )is an invariant of the ring R , by [20, Corollary 2.8] and Proposition 3.1.The following property of Cohen-Macaulay rings of type 2 is a useful calculation that wewill use to characterize rings with minimal canonical index. Proposition 6.5.
Let R be a -dimensional Cohen-Macaulay local ring with a canonical ideal C . Let ( a ) be a minimal reduction of C . If ν ( C ) = 2 , then λ ( C /a C ) = λ ( C / ( a )) .Proof. Let C = ( a, b ) and consider the exact sequence0 → Z → R → C → , where Z = { ( r, s ) ∈ R | ra + sb = 0 } . By tensoring this exact sequence with R / C , we obtain Z/ C Z g → ( R / C ) h → C / C → . Then we haveker( h ) = Im( g ) ≃ ( Z/ C Z ) / (( Z ∩ C R ) / C Z ) ≃ Z/ ( Z ∩ C R ) ≃ ( Z/B ) / (( Z ∩ C R ) /B ) , where B = { ( − bx, ax ) | x ∈ R } .We claim that Z ∩ C R ⊂ B , i.e., δ ( C ) = ( Z ∩ C R ) /B = 0. Let ( r, s ) ∈ Z ∩ C R . Then ra + sb = 0 ⇒ sa · b = − r ∈ C and sa · a = s ∈ C . Denote the total ring of fractions of R by K . Since C is a canonical ideal, we have sa ∈ C : K C = R . Therefore ( r, s ) = (cid:16) − b · sa , a · sa (cid:17) ∈ B. ANONICAL DEGREES OF COHEN-MACAULAY RINGS AND MODULES: A SURVEY Hence ker( h ) ≃ Z/B = H ( C ) and we obtain the following exact sequence0 → H ( C ) → ( R / C ) → C / C → . Next we claim that λ (H ( C )) = λ ( R / C ). Note that H ( C ) ≃ (( a ) : b ) / ( a ) by mapping ( r, s ) + B with ra + sb = 0 to s + ( a ). Using the exact sequence0 → (( a ) : b ) / ( a ) → R / ( a ) · b → R / ( a ) → R / C → , we get λ ( R / C ) = λ ((( a ) : b ) / ( a )) = λ (H ( C )) . Now, using the exact sequence0 → H ( C ) → ( R / C ) → C / C → , we get λ ( C / C ) = 2 λ ( R / C ) − λ (H ( C )) = λ ( R / C ) . Hence, λ ( C /a C ) = λ ( C /a C ) − λ ( C / C ) = λ ( C /a C ) − λ ( R / C ) = λ ( C /a C ) − λ (( a ) /a C ) = λ ( C / ( a )) . Therefore ( r, s ) = (cid:16) − b · sa , a · sa (cid:17) ∈ B. Hence ker( h ) ≃ Z/B = H ( C ) and we obtain the following exact sequence0 → H ( C ) → ( R / C ) → C / C → . Next we claim that λ (H ( C )) = λ ( R / C ). Note that H ( C ) ≃ (( a ) : b ) / ( a ) by mapping ( r, s ) + B with ra + sb = 0 to s + ( a ). Using the exact sequence0 → (( a ) : b ) / ( a ) → R / ( a ) · b → R / ( a ) → R / C → , we get λ ( R / C ) = λ ((( a ) : b ) / ( a )) = λ (H ( C )) . Now, using the exact sequence0 → H ( C ) → ( R / C ) → C / C → , we get λ ( C / C ) = 2 λ ( R / C ) − λ (H ( C )) = λ ( R / C ) . Hence, λ ( C /a C ) = λ ( C /a C ) − λ ( C / C ) = λ ( C /a C ) − λ ( R / C ) = λ ( C /a C ) − λ (( a ) /a C ) = λ ( C / ( a )) . (cid:3) Example 6.6.
Let H = h a, b, c i be a numerical semigroup which is minimally generated bypositive integers a, b, c with gcd( a, b, c ) = 1. If the semigroup ring R = k [[ t a , t b , t c ]] is not aGorenstein ring, then r ( R ) = 2 (see [20, Section 4]). Example 6.7.
Let A = k [ X, Y, Z ], let I = ( X − Y Z, Y − XZ, Z − XY ) and R = A/I .Let x, y, z be the images of
X, Y, Z in R . By [2, Theorem 10.6.5], we see that C = ( x, z ) is acanonical ideal of R with a minimal reduction ( x ). It is easy to see that ρ ( R ) = 2, e ( C ) = 2and cdeg( R ) = 1. Example 6.8.
Let A = k [ X, Y, Z ], let I = ( X − Y Z , Y − X Z , Z − X Y ) and R = A/I .Let x, y, z be the images of
X, Y, Z in R . Then C = ( x , z ) is a canonical ideal of R with aminimal reduction ( x ). We have that ρ ( R ) = 2, e ( C ) = 16 and cdeg( R ) = 8. Lower and upper bounds for the canonical index.
Example 6.9.
Let e ≥ H = h e, { e + i } ≤ i ≤ e − , e + 1 , e + 2 i . Let k be a field and V = k [[ t ]] the formal power series ring over k . Consider the semigroup ring R = k [[ H ]] ⊆ V .(1) The conductor of H is c = 2 e + 3.(2) The canonical module is K R = h , t i and K e − R ( K e − R = V .(3) The canonical ideal of R is C = ( t c K R ) and Q = ( t c ) is a minimal reduction of C . Moreover, ρ ( R ) = red( C ) = e − R ) = λ ( C /Q ) = λ ( K R / R ) = 3.(5) In particular, cdeg( R ) ≤ ρ ( R ). Theorem 6.10.
Let ( R , m ) be a -dimensional Cohen-Macaulay local ring with a canonicalideal C . Suppose that the type of R is . Then we have the following. (1) e ( C ) ≤ ρ ( R ) cdeg( R ) . (2) ρ ( R ) = 2 if and only if e ( C ) = 2 cdeg( R ) .Proof. Let C = ( a, b ), where ( a ) is a minimal reduction of C .(1) For each j = 0 , . . . , ρ ( R ) −
1, the module C j +1 /a C j is cyclic and annihilated by L =ann( C / ( a )). Hence we obtaine ( C ) = ρ ( R ) − X j =0 λ ( C j +1 /a C j ) ≤ ρ ( R ) λ ( R /L ) = ρ ( R ) cdeg( R ) . (2) Note that ρ ( R ) = 2 if and only if e ( C ) = X j =0 λ ( C j +1 /a C j ). Since ν ( C ) = r ( R ) = 2, byProposition 6.5, λ ( C / ( a )) = λ ( C /a C ). Thus, the assertion follows from X j =0 λ ( C j +1 /a C j ) = 2 λ ( C/ ( a )) = 2 cdeg( R ) . (cid:3) Proposition 6.11.
Let ( R , m ) be a Cohen-Macaulay local ring with infinite residue field anda canonical ideal C . Suppose that R p is a Gorenstein ring for ∀ p ∈ Spec( R ) \ { m } and that C is equimultiple. (1) C n has finite local cohomology for all n > . (2) Let I ( C n ) denote the Buchsbaum invariant of C n . Then the nonnegative integer β ( R ) =sup n> I ( C n ) is independent of the choice of C . (3) ρ ( R ) ≤ deg( R ) + β ( R ) − .Proof. (1) The assertion follows from C n R p = a n R p where Q = ( a ) is a reduction of C .(2) We have C n ∼ = D n for any canonical ideal D and C n +1 = a C n for ∀ n ≫ q be a minimal reduction of m . Then, for ∀ n >
0, we have ν ( C n ) = λ ( C n / m C n ) ≤ λ ( C n / q C n ) ≤ e ( q , C n ) + I ( C n ) ≤ deg( R ) + β ( R ) . The conclusion follows from [13, Theorem 1]. (cid:3)
ANONICAL DEGREES OF COHEN-MACAULAY RINGS AND MODULES: A SURVEY Example 6.12.
Consider the following examples of 1-dimensional Cohen-Macaulay semigrouprings R with a canonical ideal C such that cdeg( R ) = r ( R ).(1) Let a ≥ H = h a, a + 3 , . . . , a − , a + 1 , a + 2 i . Let R = k [[ H ]].Then the canonical module of R is ω = (cid:10) , t, t , t , . . . , t a − (cid:11) . The ideal Q = ( t a +3 ) is aminimal reduction of C = ( t a +3 , t a +4 , t a +6 , t a +7 , . . . , t a +2 ),cdeg( R ) = a − ν ( C ) , and red( C ) = 2 . (2) Let a ≥ H = h a, a + 1 , a + 4 , . . . , a − , a + 2 , a + 3 i . Let R = k [[ H ]]. Then the canonical module of R is ω = (cid:10) , t, t , t , . . . , t a − (cid:11) . The ideal Q = ( t a +4 ) is a minimal reduction of C = ( t a +4 , t a +5 , t a +8 , t a +9 , . . . , t a +3 ).cdeg( R ) = a − ν ( C ) , and red( C ) = 3 . Roots of canonical ideals
Another phenomenon concerning canonical ideals is the following.
Definition 7.1.
Let ( R , m ) be a Cohen-Macaulay local ring of dimension d ≥ C . An ideal L is called a root of C if L n ≃ C for some n . In this case, we write τ L ( C ) = min { n | L n ≃ C} . Then the rootset of R is the set root( R ) = { τ L ( C ) | L is a root of C} .The terminology ‘roots’ of C already appears in [4]. Here is a simple example. Example 7.2. ([4, Example 3.4]) Let R = k [[ t , t , t , t ]]. Then C = ( t , t , t ) is a canonicalideal of R . Let I = ( t , t ). Then I = t C , that is, I is a square root of C .It was studied in [18] for its role in examining properties of canonical ideals. Here is oneinstance: Proposition 7.3.
Let R be a Cohen-Macaulay local ring of dimension one. Let L be a rootof C . If L is an irreducible ideal then R is a Gorenstein ring.Proof. Note that if q ∈ Q satisfies qL ⊂ L then qL n ⊂ L n . This implies that q C ⊂ C and so q ∈ R . By Proposition 5.5, L ≃ C . We now make use of a technique of [18]. From C ≃ C n wehave C ≃ C n − · C ≃ C n − · C n = C n − . By iteration we get that
C ≃ C m for arbitrarily large values of m , and for all of them we have C m : C m = R .We may assume that the residue field of R is infinite and obtain a minimal reduction ( c ) for C ,that is an equality C r +1 = c C r for all r ≥ s for some s . This gives ( C r ) = c r C r , and therefore( C r c − r ) C r = C r . Since C r : C r = R the equality gives that C r ⊂ ( c r ) ⊂ C r and therefore C r = ( c ) r . Thus C is invertible, as desired. (cid:3) Question 7.4.
Suppose there is an ideal L such that L = C . Sometimes they imply thatcdeg( R ) is even; can it be the same with bideg( R )? Question 7.5. If L n ≃ C and L is reflexive must R be Gorenstein? Exercise 7.6.
Let ( R , m ) be a Cohen-Macaulay local ring of dimension d ≥
2. Let C be aCohen-Macaulay ideal of codimension 1. A goal is to derive a criterion for C to be a canonicalideal. Let x be regular mod C . If C /x C is a canonical ideal of R /x R then C is canonical idealof R . We make use of two facts: (1) C is a canonical module iff Ext j R ( R / m , C ) = R / m for j = d and 0 otherwise ([26, Theorem6.1]).(2) The change of rings equation asserts Ext j R ( R / m , C ) ≃ Ext j − R / ( x ) ( R / m , C /x C ), j ≤ d .More generally, how do we tell when a Cohen-Macaulay ideal I is irreducible?The set root( R ) is clearly independent of the chosen canonical ideal. To make clear thecharacter of this set we appeal to the following property. We call an ideal of R closed ifHom( L, L ) = R . Proposition 7.7.
Let ( R , m ) be a -dimensional Cohen-Macaulay local ring with a canonicalideal C . Let L be a root of C . If Hom( L n , L n ) ≃ R for infinitely many values of n then allpowers of L are closed. In this case L is invertible and R is Gorenstein.Proof. A property of roots is that they are closed ideals. More generally, it is clear that ifHom( L m , L m ) = R then Hom( L n , L n ) = R for n < m , which shows the first assertion.We may assume that R has an infinite residue field. Let s = red( L ). Then L s +1 = xL s , fora minimal reduction ( x ) and thus L s = x s L s , which gives that x − s L s ⊂ Hom( L s , L s ) ≃ R . It follows that L s ⊂ ( x s ) ⊂ L s , and thus L s = ( x s ). Now taking t such that L t ≃ C shows that C is principal. (cid:3) Corollary 7.8. If L m ≃ C ≃ L n , for m = n , then C is principal.Proof. Suppose m > n . Then
C ≃ L m = L n L m − n ≃ L m L m − n = L m − n . Iterating, L is a root of C of arbitrarily high order. (cid:3) Corollary 7.9. ([4, Proposition 3.8])
Let ( R , m ) be a -dimensional Cohen-Macaulay localring with a canonical ideal C . If R is not Gorenstein then no proper power of C is a canonicalideal. Proposition 7.10.
Let ( R , m ) be a Cohen-Macaulay local ring of dimension d ≥ with infiniteresidue field and with an equimultiple canonical ideal C . Let L be a root of C . Then τ L ( C ) ≤ min { r ( R ) − , red( L ) } .Proof. Suppose n = τ L ( C ) ≥ r ( R ). Then ν ( L n ) = ν ( C ) = r ( R ) < n + 1 = (cid:18) n + 11 (cid:19) . By [13, Theorem 1], there exists a reduction ( a ) of L such that L n = aL n − . Thus, L n − ≃ C ,which contradicts the minimality of τ L ( C ). (cid:3) Remark 7.11.
We have the following.(1) If r ( R ) = 2, then the isomorphism class of C is the only root.(2) The upper bound in Proposition 7.10 is sharp. For example, let a ≥ R = k [[ { t i } a ≤ i ≤ a − ]] ⊆ k [[ t ]]. Then the canonicalmodule of R is ω = a − X i =0 R t i = ( R + R t ) a − . ANONICAL DEGREES OF COHEN-MACAULAY RINGS AND MODULES: A SURVEY Thus R has a canonical ideal C = t a ( a − ω . Let L = ( t a , t a +1 ). Then C = L a − . Hence wehave τ L ( C ) = a − r ( R ) − Theorem 7.12.
Let ( R , m ) be a -dimensional Cohen-Macaulay local ring with a canonicalideal. If R is not Gorenstein, then root( R ) is a finite set of cardinality less than r ( R ) .Applications of roots. Proposition 7.13.
Let ( R , m ) be a -dimensional Cohen-Macaulay local ring with a canonicalideal C . Let f be the supremum of the reduction numbers of the m -primary ideals. Supposethat L n ≃ C . If p divides n , then ρ ( R ) ≤ ( f + p − /p .Proof. Since n = pm , by replacing L m by I , we may assume that I p = C . Let r = red( I ) with I r +1 = xI r . Then r = ps + q for some q such that − p + 1 ≤ q ≤
0. Since ps = r − q ≥ r , wehave C s +1 = I ps + p = x p I ps = x p C s . Thus, ρ ( R ) = red( C ) ≤ s = ( r − q ) /p ≤ ( r + p − /p ≤ ( f + p − /p . (cid:3) A computation of roots of the canonical ideal.
Let 0 < a < a < · · · < a q be integers such thatgcd( a , a , . . . , a q ) = 1. Let H = h a , a , . . . , a q i be the numerical semigroup generated by a ′ i s .Let V = k [[ t ]] be the formal power series ring over a field k and set R = k [[ t a , t a , . . . , t a q ]].We denote by m the maximal ideal of R and by e = a the multiplicity of R . Let v be thediscrete valuation of V . In what follows, let R ⊆ L ⊆ V be a finitely generated R -submoduleof V such that ν ( L ) >
1. We set ℓ = ν ( L ) −
1. Then we have the following.
Lemma 7.14.
With notation as above, m L .Proof. Choose 0 = g ∈ m so that gV ( R . Then Q = g R is a minimal reduction of the m -primary ideal I = gL of R , so that g m I . Hence 1 / ∈ m L . (cid:3) Lemma 7.15.
With notation as above, there exist elements f , f , . . . , f ℓ ∈ L such that (1) L = R + P ℓi =1 R f i , (2) 0 < v ( f ) < v ( f ) < . . . < v ( f ℓ ) , and (3) v ( f i ) H for all ≤ i ≤ ℓ .Proof. Let L = R + P ℓi =1 R f i with f i ∈ L . Let 1 ≤ i ≤ ℓ and assume that m = v ( f i ) ∈ H .We write f i = P ∞ j = m c j t j with c j ∈ k . Then c s = 0 for some s > m such that s H ,because f i R . Choose such integer s as small as possible and set h = f i − P s − j = m c j t j . Then P s − j = m c j t j ∈ R and R + R f i = R + R h . Consequently, as v ( h ) = s > m = v ( f i ), replacing f i with h , we may assume that v ( f i ) H for all 1 ≤ i ≤ ℓ . Let 1 ≤ i < j ≤ ℓ and assume that v ( f i ) = v ( f j ) = m . Then, since f j = cf i + h for some 0 = c ∈ k and h ∈ L such that v ( h ) > m ,replacing f j with h , we may assume that v ( f j ) > v ( f i ). Therefore we can choose a minimalsystem of generators of L satisfying conditions (2) and (3). (cid:3) Proposition 7.16.
With notation as above, let , f , f , . . . , f ℓ ∈ L and , g , g , . . . , g ℓ ∈ L be systems of generators of L and assume that both of them satisfy conditions (2) and (3) in Lemma 7.15 . Suppose that v ( f ℓ ) < e = a . Then v ( f i ) = v ( g i ) for all ≤ i ≤ ℓ . Proof.
We set m i = v ( f i ) and n i = v ( g i ) for each 1 ≤ i ≤ ℓ . Let us write f = α + α g + . . . + α ℓ g ℓ g = β + β f + . . . + β ℓ f ℓ with α i , β i ∈ R . Then v ( β f + . . . + β ℓ f ℓ ) ≥ v ( f ) = m >
0, whence β ∈ m because v ( g ) = n >
0. We similarly have that α ∈ m . Therefore n = v ( g ) ≥ m , since v ( β ) ≥ e > m ℓ ≥ m and v ( β f + . . . + β ℓ f ℓ ) ≥ m . Suppose that n > m . Then v ( α g + . . . + α ℓ g ℓ ) ≥ n > m and v ( α ) ≥ e > m , whence v ( f ) > m , a contradiction. Thus m = n .Now let 1 ≤ i < ℓ and assume that m j = n j for all 1 ≤ j ≤ i . We want to show m i +1 = n i +1 .Let us write f i +1 = γ + γ g + . . . + γ ℓ g ℓ g i +1 = δ + δ f + . . . + δ ℓ f ℓ with γ i , δ i ∈ R .First we claim that γ j , δ j ∈ m for all 0 ≤ j ≤ i . As above, we get γ , δ ∈ m . Let 0 ≤ k < i andassume that γ j , δ j ∈ m for all 0 ≤ j ≤ k . We show γ k +1 , δ k +1 ∈ m . Suppose that δ k +1 m .Then as v ( δ + δ f + . . . + δ k f k ) ≥ e > m k +1 , we get v ( δ + δ f + . . . + δ k f k + δ k +1 f k +1 ) = m k +1 ,so that n i +1 = v ( g i +1 ) = v ( δ + δ f + . . . + δ ℓ f ℓ ) = m k +1 , since v ( f h ) = m h > m k +1 if h > k +1.This is impossible, since n i +1 > n i = m i ≥ m k +1 . Thus δ k +1 ∈ m . We similarly get γ k +1 ∈ m ,and the claim is proved.Consequently, since v ( δ + δ f + . . . + δ i f i ) ≥ e > m i +1 and v ( f h ) ≥ m i +1 if h ≥ i + 1, wehave n i +1 = v ( g i +1 ) = v ( δ + δ f + . . . + δ ℓ f ℓ ) ≥ m i +1 . Assume that n i +1 > m i +1 . Thensince v ( γ + γ g + . . . + γ i g i ) ≥ e > m i +1 and v ( g h ) ≥ n i +1 > m i +1 if h ≥ i + 1, we have m i +1 = v ( f i +1 ) = v ( γ + γ g + . . . + γ ℓ g ℓ ) > m i +1 . This is a contradiction. Hence m i +1 = n i +1 ,as desired. (cid:3) Theorem 7.17.
With notation as above, assume that r ( R ) = 3 and write the canonical module ω = (cid:10) , t a , t b (cid:11) with < a < b . Suppose that a > b and b < e . Let L be an ideal of R and let n ≥ be an integer. If L n ∼ = ω , then n = 1 .Proof. By Proposition 7.10, we have n ≤
2. Suppose that n = 2. Let f ∈ L such that f V = LV and set M = f − L . Then R ⊆ M ⊆ V and M = f − L ∼ = ω . By Lemma 7.15,we can write M = h , f , f , . . . , f σ i , where 0 < v ( f ) < v ( f ) < · · · < v ( f σ ) and v ( f i ) H forevery 1 ≤ i ≤ σ . Then M = h , { f i } ≤ i ≤ σ , { f i f j } ≤ i ≤ j ≤ σ i . Let n i = v ( f i ) for each 1 ≤ i ≤ σ .Suppose that σ >
1. Then we claim that f
6∈ h , { f i } ≤ i ≤ σ , { f i f j } ≤ i ≤ j ≤ σ i . In order to provethe claim, we assume the contrary and write f = α + σ X i =2 α i f i + X ≤ i ≤ j ≤ σ α ij f i f j with α, α i , α ij ∈ R . Then since n < n i for i ≥ n ≤ n i < n i + n j for 1 ≤ i ≤ j ≤ σ , wehave n = v ( α ), which is impossible, because n H but v ( α ) ∈ H .Since ν ( M ) = 3, by the claim we have M = h , f , f i f j i for some 1 ≤ i ≤ j ≤ σ . In fact,the other possibility is M = h , f , f i i with i ≥
2. However, when this is the case, we get M = M so that red ( f ) L ≤
1. Thus, red ( f ) L ≤
1. Therefore, since L ∼ = ω , by Remark 6.3(1), R is a Gorenstein ring, which is a contradiction. ANONICAL DEGREES OF COHEN-MACAULAY RINGS AND MODULES: A SURVEY We now choose 0 = θ ∈ Q( V ), where Q( V ) is the quotient field of V , so that ω : M = R θ .Then ω = θM and hence θ is a unit of V . We compare ω = D , t a , t b E = h θ, θf , θf i f j i and notice that 0 < a < b < e and 0 < n < n i + n j . Then by Proposition 7.16, we get that n = a and n i + n j = b , whence b ≥ a . This is a contradiction. (cid:3) Let us give examples satisfying the conditions stated in Theorem 7.17.
Example 7.18.
Let e ≥ H = h e + i | ≤ i ≤ e − i = e − , e − i . Then ω = (cid:10) , t , t (cid:11) and r ( R ) = 3. More generally, let a, b, e ∈ Z such that 0 < a < b , b < a ,and e ≥ a + b + 2. We consider the numerical semigroup H = h e + i | ≤ i ≤ e − i = e − b − , e − a − i . Then ω = (cid:10) , t a , t b (cid:11) and r ( R ) = 3. These rings R contain no ideals L such that L n ∼ = ω forsome integer n ≥
2. 8.
Minimal values of bi-canonical degrees
Now we begin to examine the significance of the values of bideg( R ). We focus on rings ofdimension 1. Almost Gorenstein rings.
First we recall the definition of almost Gorenstein rings ([3, 20, 22]).
Definition 8.1. ([22, Definition 3.3]) A Cohen-Macaulay local ring R with a canonical module ω is said to be an almost Gorenstein ring if there exists an exact sequence of R -modules0 → R → ω → X → ν ( X ) = e ( X ). In particular if R has dimension one X =( R / m ) r − , r = r ( R ). Theorem 8.2.
Let ( R , m ) be a Cohen-Macaulay local ring of dimension with a canonicalideal C . If R is almost Gorenstein then bideg( R ) = 1 .Proof. In dimension 1, from 0 → ( c ) −→ C −→ X → ,X is a vector space k r − , r = r ( R ). To determine C ∗∗ apply Hom( · , ( c )) to the above exactsequence to get Hom( C , ( c )) = m . On the other hand, by Proposition 5.3, C ∗∗ = Hom( m , ( c )) = L , the socle of R / ( c ) [which is generated by r elements], properly containing C that is C ∗∗ = L ,the socle of C . Therefore bideg( R ) = λ ( L/C ) = 1. (cid:3)
The example below shows that the converse does not holds true.
Example 8.3.
Consider the monomial ring (called to our attention by Shiro Goto) R = Q [ t , t , t ] , m = ( x, y, z ). We have a presentation R = Q [ x, y, z ] /P , with P = ( y − xz, x − yz , z − x y ). Let us examine some properties of R . For simplicity we denote the images of x, y, z in R by the same symbols. An explanation for these calculations can be found in theproof of Theorem 10.2.(1) Let C = ( x, y ). A calculation with Macaulay2 shows that if D = C : m , then λ ( D / C ) = 1.Therefore C is a canonical ideal by Corollary 5.6.(2) ( c ) : C 6 = m so R is not almost Gorenstein. However C ∗∗ = ( c ) : [( c ) : C ] satisfies [byanother Macaulay2 calculation] λ ( C ∗∗ / C ) = 1, so C ∗∗ = L . This shows that bideg( R ) = 1. (3) This example shows that bideg( R ) = 1 holds [for dimension one] in a larger class ringsthan almost Gorenstein rings.(4) Find red( C ) and red( D ) for this example. Need the minimal reductions. Goto rings.
We now examine the significance of a minimal value for bideg( R ). Suppose R isnot a Gorenstein ring. Definition 8.4.
A Cohen-Macaulay local ring R of dimension d is a Goto ring if it has acanonical ideal and bideg( R ) = 1. Questions 8.5. (1) What other terminology should be used? Nearly Gorenstein ring hasalready been used in [25]. We will examine its relationship to Goto rings.(2) Almost Gorenstein rings of dimension one have red( G ) = 2. What about these rings?(3) What are the properties of the Cohen-Macaulay module X defined by0 → ( c ) −→ C −→ X → , nearly Ulrich or pre-Ulrich bundles ?(4) λ ( C ∗∗ / C ) = 1 implies that C ∗∗ / C = L/ C = C : m /C ≃ R / m . (5) If L = ( c ) : m , the socle of ( c ), is equal to L , then C / ( c ) is a vector space, so R is almostGorenstein and conversely.(6) In all cases L = C L , m L = m C , so C / m C ֒ → L/ m L ([9, Theorem 3.7]). Therefore if R is not almost Gorenstein, c cannot be a minimal generator of L and thus L = ( C , α ), ν ( L ) = r + 1, with α ∈ L , or L = ( L , β ), with β ∈ C .(7) This says that L = ( x , . . . , x r , β ) , x i ∈ ( c ) : m , C = ( c, x , . . . , x r − , β ) , c ∈ m L C ∗ = c − [( c ) : C ] C ∗∗ = ( c ) : (( c )) : C ) = L C ∗ = C ∗∗∗ = c − [( c ) : L ]( c ) : C = ( c ) : L = ( c ) : β. Let us attempt to get bideg( R ) for R almost Gorenstein but of dimension ≥
2. Note thatcdeg( R ) = r −
1. Assume d = r = 2: can we complete the calculation? Proposition 8.6.
Let ( R , m ) be a Cohen-Macaulay local ring with a canonical ideal C . (1) From the → ( c ) −→ C −→ X → applying Hom( · , ( c )) , we get → Hom( C , ( c )) −→ R −→ Ext( X, ( c )) −→ Ext( C , ( c )) → . (2) The image of
Hom( C , ( c )) in R is a proper ideal N , and so R /N is a submodule of Ext( X, ( c )) which is annihilated by ann( X ) ( which contains ( c )) . We note ( see [29, p.155]) that Ext( X, ( c )) ≃ Ext R ( X, R ) = Hom R / ( c )) ( X, R / ( c )) = (( c ) : N ) / ( c ) , if r = 2 . ANONICAL DEGREES OF COHEN-MACAULAY RINGS AND MODULES: A SURVEY (3) Suppose C is equimultiple and bideg( R ) = 1 . Then R is Gorenstein at all primes ofcodimension one with one exception, call it p . This means that C ∗∗ / C is p -primary and deg( C ∗∗ / C ) = bideg( R p ) · deg( R / p ) = 1 . Remark 8.7.
What sort of modules are C / ( c ) and C ∗∗ / C ? The first we know is Cohen-Macaulay, how about the second?(1) C ∗∗ = C : p : both are divisorial ideals that agree in codimension one.(2) If dim R = 2, C ∗∗ / C is Cohen-Macaulay of dimension one and multiplicity one so whatsort of module is it? It is an R / p -module of rank one and R / p is a discrete valuationdomain so C ∗∗ / C ≃ R / p .(3) In all dimensions, C ∗∗ / C has only p for associated prime and is a torsion-free R / p -moduleof rank one. Thus it is isomorphic to an ideal of R / p . C ∗∗ / C has also the condition S ofSerre from the exact sequence0 → C ∗∗ / C −→ R / C −→ R / C ∗∗ → , R / C ∗∗ has the condition S . If R is complete and contains a field by [32] R / p is a regularlocal ring and therefore C ∗∗ / C ≃ R / p .(4) Some of these properties are stable under many changes of the rings. Will check for generichyperplane section soon. Questions 8.8. (1) How to pass from bideg( A ) of a graded algebra A to bideg( B ) of one ofits Veronese subalgebras?(2) If S is a finite injective extension of R , is bideg( S ) ≤ [ S : R ] · bideg( R )?9. Change of rings
Let ϕ : R → S be a homomorphism of Cohen-Macaulay rings. We examine a few cases of therelationship between cdeg( R ) and cdeg( S ) induced by ϕ . We skip polynomial, power seriesand completion since flatness makes it straightforward. Finite extensions. If R → S is a finite injective homomorphism of Cohen-Macaulay rings and C is a canonical ideal of R , then D = Hom( S , C ) is a canonical module for S , according to [6,Theorem 3.3.7]. Recall how S acts on D : If f ∈ D and a, b ∈ S , then a · f ( b ) = f ( ab ). Augmented rings.
A case in point is that of the so-called augmented extensions. Let ( R , m )be an 1-dimensional Cohen-Macaulay local ring with a canonical ideal. Assume that R is nota valuation domain. Suppose A is the augmented ring R ⋉ m . That is, A = R ⊕ m ǫ , ǫ = 0.[Just to keep the components apart in computations we use ǫ as a place holder.]Let ( c ) be a minimal reduction of the canonical ideal C . We may assume that C ⊂ m byreplacing C by c C if necessary. Then a canonical module D of A is D = Hom R ( A , C ). Let usidentify D to an ideal of A . For this we follow [18].Let R be a commutative ring with total quotient ring Q and let F denote the set of R -submodules of Q . Let M, K ∈ F . Let M ∨ = Hom R ( M, K ) and let A = R ⋉ M denote the idealization of M over R . Then the R -module M ∨ ⊕ K becomes an A -moduleunder the action ( a, m ) ◦ ( f, x ) = ( af, f ( m ) + ax ) , where ( a, m ) ∈ A and ( f, x ) ∈ M ∨ × K . We notice that the canonical homomorphism ϕ : Hom R ( A , K ) → M ∨ × K such that ϕ ( f ) = ( f ◦ λ, f (1)) is an A -isomorphism, where τ : M → A , τ ( m ) = (0 , m ), andthat K : M ∈ F and ( K : M ) × K ⊆ Q ⋉ Q is an A -submodule of Q ⋉ Q , the idealization of Q over itself. When Q · M = Q , identifying Hom R ( M, K ) = K : M , we therefore have a naturalisomorphism Hom R ( A , K ) ∼ = ( K : M ) × K of A -modules.Using this observation, setting K = C , M = m , we get the following D = Hom R ( R ⊕ m ǫ, C ) = Hom R ( m , C ) ⊕ C ǫ = L + C ǫ because C : Q m = C : R m ⊂ C : Q C = R . Denote L = Hom R ( m , C ). Then L ≃ C : m ⊂ m sothat D is an ideal of A .Let us determine D ∗∗ . The total ring of fractions of A is Q ⋉ Qǫ . If ( a, bǫ ) ∈ Q ⋉ Qǫ is inHom( D , A ) = Hom(( C ⊕ Lǫ ) , ( R ⊕ m ǫ )) , then a C ⊂ R , aL ⊂ m and a C 6 = R as R is not Gorenstein. Thus a ∈ C ∗ . On the other hand, b C ⊂ m . Thus b ∈ C ∗ and conversely. Thus D ∗ = C ∗ ⊕ C ∗ . Suppose ( a, bǫ ) ∈ D ∗∗ , D ∗∗ = Hom(( C ∗ ⊕ C ∗ ǫ ) , ( R ⊕ m ǫ )) . Then a C ∗ ⊂ R and C ∗∗ = R . In turn bǫ C ∗ ⊂ m ǫ and so b ∈ C ∗∗ , and conversely. Thus D ∗∗ = C ∗∗ ⊕ C ∗∗ ǫ. Let us summarize this calculation. This gives D ∗∗ / D = C ∗∗ / C ⊕ C ∗∗ ǫ/Lǫ. Proposition 9.1.
Let ( R , m ) be a Cohen-Macaulay local ring with a canonical ideal and let A = R ⋉ m . Then bideg( A ) = 2 bideg( R ) − . In particular if R is a Goto ring then A is also a Goto ring. By comparison, according to [18, Theorem 6.7], cdeg( A ) = 2 cdeg( R ) + 2, so that applyingto Example 5.4 we get cdeg( A ) = bideg( A ) + 3 . Products.
Let k be a field, and let A , A be two finitely generated Cohen-Macaulay k -algebras.Let us look at the canonical degrees of the product A = A ⊗ k A .As a rule, if B i , C i are A i -modules, we use the natural isomorphismHom( B ⊗ B , C ⊗ C ) = Hom A ⊗ k A ( B ⊗ k B , C ⊗ k C ) , which works out to be Hom A ( B , C ) ⊗ k Hom A ( B , C ) . If A i , i = 1 ,
2, are localizations [of fin gen k -algebras] then B is an appropriate localization.[If m i are the maximal ideals of A i , pick primes M i in B i and a prime B over both m i .] If B i ANONICAL DEGREES OF COHEN-MACAULAY RINGS AND MODULES: A SURVEY are finite A i -modules and F i are minimal resolutions over A i [or over S i , a localization in thenext item], then F ⊗ k F is a resolution whose entries lie in appropriate primes.If S i → A i , i = 1 ,
2, are presentations of A i , S = S ⊗ S → A ⊗ A = A gives a presentationof A and from it we gather the invariants [all ⊗ over k ]. Proposition 9.2.
Let A i , i = 1 , , be as above. Then (1) A is Cohen-Macaulay (2) C = C ⊗ C , C ∗∗ = C ∗∗ ⊗ C ∗∗ , r ( A ) = r ( A ) · r ( A )(3) If ( c i ) is a minimal reduction of C i , i = 1 , , then ( c ) = ( c ) ⊗ ( c ) is a minimal reductionfor C and C / ( c ) = C / ( c ) ⊗ C ⊕ C ⊗ C / ( c ) , C ∗∗ / C = C ∗∗ / C ⊗ C ⊕ C ⊗ C ∗∗ / C , cdeg( A ) = cdeg( A ) · deg( A ) + deg( A ) · cdeg( A ) , bideg( A ) = bideg( A ) · deg( A ) + deg( A ) · bideg( A ) . (4) If A i is almost Gorenstein, that is cdeg( A i ) = r ( A i ) − , then cdeg( A ) = ( r − · deg( A ) + deg( A ) · ( r − . Suppose further, A = A , that is A is the square. Then cdeg( A ) = 2 deg( A )( r − . In this case, A is almost Gorenstein if cdeg( A ) = r − , that is A ) = r + 1 . Questions 9.3. (1) How to pass from bideg( A ) of a graded algebra A to bideg( B ) of one ofits Veronese subalgebras?(2) If S is a finite injective extension of R , is bideg( S ) ≤ [ S : R ] · bideg( R )? Hyperplane sections.
A desirable comparison is that between bideg( R ) and bideg( R / ( x ) forappropriate regular element x . [The so-called ‘Lefschetz type’ assertion.] We know that if C isa canonical module for R then C /x C is a canonical module for R / ( x ) with the same numberof generators, so type is preserved under specialization. However C /x C may not be isomorphicto an ideal of R / ( x ). Here is a case of good behavior. Suppose x is regular modulo C . Thenfor the sequence 0 → C −→ R −→ R / C → , we get the exact sequence 0 → C /x C −→ R / ( x ) −→ R / ( C , x ) → , so the canonical module C /x C embeds in R / ( x ). We set S = R / ( x ) and D = ( C , x ) / ( x ) forthe image of C in S . We need to compare bideg( R ) = deg( R / C ) − deg( R / C ∗∗ ) and bideg( S ) =deg( S / D ) − deg( S / D ∗∗ ).We don’t know how to estimate the last term. We can choose x so that deg( R / C ) =deg( S / D ), but need, for instance to show that C ∗∗ maps into D ∗∗ . Let c be as in Proposition 5.3and pick x so that c, x is a regular sequence. Set C = ( c ) : C , C = ( c ) : C , D = ( c ) : D , D = ( c ) : D . We have C D ⊂ ( c ) S and thus C S ⊂ D . This shows that D ∗∗ = ( c ) : S D ⊂ ( c ) : S C , and thus D C ⊂ ( e ) but not enough to show D ⊂ C S .A model for what we want to have is [18, Proposition 6.12] asserting that if C is equimultiplethen cdeg( R ) ≤ cdeg( R / ( x )). For bideg( R ), in consistency with Conjecture 5.2, we have Conjecture 9.4.
Under the conditions above, bideg( R ) ≥ bideg( R / ( x )). Proposition 9.5.
Let ( R , m ) be a Cohen–Macaulay local ring of dimension ≥ . If x ∈ m is aregular element of R and C is a canonical ideal of R , then C /x C is a canonical ideal of R /x R if and only if height ( x, trace( C )) ≥ .Proof. If C is a canonical module of R , C /x C is a canonical module of R /x R . We must showthat it is an ideal. For any minimal prime p of x C p is an ideal which cannot contain its trace byhypothesis by the height condition. It follows that C p is principal and thus the localization R p is a Gorenstein ring. This implies that modulo x it is also Gorenstein, and thus its canonicalmodulo is isomorphic to the ring. The converse is just a reversion of the steps. (cid:3) Monomial subrings
Let H be a finite subset of Z ≥ , { a < a < . . . < a n } . For a field k we denote by R = k [ H ]the subring of k [ t ] generated by the monomials t a i . We assume that gcd( a , . . . , a n ) = 1. Wealso use the bracket notation H = h a , a , . . . , a n i for the exponents. For reference we shalluse [14, p. 553] and [40, Section 8.7].There are several integers playing roles in deriving properties of R , with the emphasis onthose that lead to the determination of bideg( R ).(1) There is an integer s such that t n ∈ R for all n ≥ s . The smallest such s is called the conductor of H or of R , which we denote by c and t c k [ t ] is the largest ideal of k [ t ] containedin k [ H ]. c − R and reads its multiplicity, c − R ).(2) For any integer a >
0, say a = c of H , the subring A = k [ t a ] provides for a Noethernormalization of R . This permits the passage of many properties A to R , and vice-versa. R is a free A -module and taking into account the natural graded structure we can write R = m M j =1 A t α j . Note that s = P mj =1 α j .(3) How to read other invariants of R such as its canonical ideal C and red( C ) and its canonicaldegrees cdeg( R ) and bideg( R )? Monomial curves.
Let R = k [ t a , t b , t c ], a < b < c, gcd( a, b, c ) = 1. Assume R is not Goren-stein, ω = (1 , t s ).It would be helpful to have available descriptions of the canonical ideal and attached invari-ants. Some of the information is available from the Frobenius equations: x m − y m z m y m − x m z m z m − x m y m which can be expressed as the 2 × ϕ = x a z c z c y b y b x a ANONICAL DEGREES OF COHEN-MACAULAY RINGS AND MODULES: A SURVEY Calculating the canonical ideal and its bidual.
Let R = A /P be a Cohen-Macaulay integraldomain and A a Gorenstein local ring. If codim P = g , ω = Ext g A ( R , A ) is a canonical modulefor R . Since R is an integral domain, ω may be identified to an ideal of R . A canonical moduleof R is obtained by deleting one row and a column according to the following comments andmapping the remaining entries to R .(1) Let L = ( x , . . . , x g ) be a regular sequence contained in P . Then ω = Ext g A ( A /P, A ) = Hom A /L ( A /P, A /L ) = ( L : P ) /L. The simplest case to apply this formula is when P = ( L, f ) so this becomes take ω = ( L : f ) /L. (2) Suppose P = I ( ϕ ) is a prime ideal of codimension two where ϕ = a c c b b a We are going to argue that if the 2 × x , x , x = a a − b c x = a b − c c that arise from keeping the first row form a regular sequence then for f = b b − c a , wehave ( x , x ) : f = ( a , c ) . With these choices we have ( L : f ) /L ≃ ( a , c , P ) /P. Indeed the nonvanishing mapping ω = ( L : f ) /L R /P of modules of rank one over the domain R /P must be an embedding. Example 10.1.
Let R = k [ t a , t b , t c ], b − a = c − b = d : Then b = a + d, c = a + 2 d and ϕ = x yz p x q y z , Note p ( a + 2 d ) − qa = d from ( p + 1)( a + 2 d ) = qa + ( a + d ) and ( x, y )
7→ C = ( t a , t b ) = t a (1 , t d ).By Proposition 5.3 we have ( x ) : ( x, y ) = ( x, y, z p ),( x ) : ( x, y ) = ( x, y, z p ) , C ∗∗ = ( x ) : ( x, y, z p ) = ( x, z, y ) , bideg( R ) = λ ( R / C ) − λ ( R / C ∗∗ ) = λ ( R / ( x, y )) − λ ( R / ( x, y, z )) = p − . Proposition 10.2.
Let R = k [ t a , t b , t c ] be a monomial ring and denote by ϕ its Herzog matrix ϕ = x a z c z c y b y b x a . Then bideg( R ) = a · b · c . Proof.
From ϕ we take C = ( x a , z c ), where harmlessly we chose a ≤ a . Then( x a ) : ( x a , z c ) = ( x a , z c , y b ) , C ∗∗ = ( x a ) : ( x a , y b , z c ) = ( x a , y b , z c ) , bideg( R ) = λ ( R / C ) − λ ( R / C ∗∗ ) = λ ( R / ( x a , z c )) − λ ( R / ( x a , y b , z c )) . and since ( x a , z c ) = ( x a , y b y b , z c )we have bideg( R ) = a · ( b + b ) · c − a · b · c = a · b · c . (cid:3) Remark 10.3.
For the monomial algebra R = k [ t a , t b , t c ] The value of bideg( R ) is alsocalculated in [25, Proposition 7.9]. According to [20, Theorem 4.1], cdeg( R ) = a · b · c ,which supports the Comparison Conjecture 5.2.11. Rees algebras
Let R be Cohen-Macaulay local ring and I an ideal such that the Rees algebra S = R [ It ] isCohen-Macaulay. We consider a few classes of such algebras. We denote by C a canonical idealof R and set G = gr I ( R ). Rees algebras with expected canonical modules. (See [28] for details.) This means ω S = ω R (1 , t ) m ,for some m ≥ −
1. This will be the case when G = gr I ( R ) is Gorenstein ([28, Theorem 2.4,Corollary 2.5]). We actually assume I of codimension at least 2. We first consider the case C = R . Set D = (1 , t ) m , pick a is a regular element in I and its initial form a is regular on G ,finally replace D by a m D ⊂ S . Proposition 11.1.
Let R be a Gorenstein local ring, I an ideal of codimension at least twoand S its Rees algebra. If the canonical module of S has the expected form then S is Gorensteinin codimension less than codim( I ) , in particular bideg( S ) = 0 .Proof. It is a calculation in [28, p. 294] that S : (1 , t ) m = I m S . It follows that( I m , I m t ) ⊂ I m S · (1 , t ) m . Since codim( I m , ( It ) m ) = codim( I, It ) = codim( I ) + 1, the assertion follows. Note that thisimplies that ω S is free in codimension one and therefore it is reflexive by a standard argument.Finally by Proposition 5.1, bideg( S ) = 0, and therefore that cdeg( S ) = 0. (cid:3) Remark 11.2.
How about the general case, when R is not Gorenstein? Let us see whetherwe can argue these points directly.(1) a m S : I m S = ( a, at ) m : S ∩ a m S : SI m +1 ∩ a m IS : I m +1 S = ( a m I + a m I t + · · · + a m I i +1 + · · · ) : S I m = I m S. ANONICAL DEGREES OF COHEN-MACAULAY RINGS AND MODULES: A SURVEY (2) Let us calculate bideg( S ) = deg( S/ D ) − deg( S/ D ∗∗ ):deg( S/I m − S ) = i = m − X i =0 deg( I i S/I i +1 S ) = m deg( G ) . (3) According to [28], a i (1 , t ) i is a Cohen-Macaulay ideal for 0 ≤ i ≤ m . We also have that S = S/a (1 , t ) = R [ It ], where R = R /a R , I = I/a R .(4) The ideal J = a (1 , t ) is free in codimension 1: ( a, at ) m · ( a, at ) − m = (1 , t ) m · I m S is an idealof codimension 2. Therefore for all i ≥
0, deg( J i /J i +1 ) = deg( S/J ).(5) From deg(
S/a m (1 , t ) m ):deg( S/a (1 , t )) + deg( a (1 , t ) /a (1 , t ) ) + · · · + deg( a m − (1 , t ) m − /a m (1 , t ) m ) . (6) We get deg( S/ D ) = m deg( S ). Rees algebras of minimal multiplicity.
Let ( R , m ) be a Cohen-Macaulay local ring, I an m -primary ideal, and J a minimal reduction of I with I = J I . From the exact sequence0 → I R [ J t ] → R [ J ] → R [ J t ] /I R [ J t ] = gr J ( R ) ⊗ R R /I → , with the middle and right terms being Cohen-Macaulay, we have that I R [ J t ] = I R [ It ] isCohen-Macaulay. With gr I ( R ) Cohen-Macaulay and I an ideal of reduction number at most1, we have if dim R > R [ It ] is Cohen-Macaulay. A source of these ideals arises fromirreducible ideals in Gorenstein rings such as I = J : m , where J is a parameter ideal.Let S = R [ J t ] and S = R + ItS . If R is Gorenstein, then the canonical module of S is ω S = (1 , t ) d − S and by the change of rings formula, we have ω S = Hom S ( S, (1 , t ) d − S ) = (1 , t ) d − S : S. If d = 2, then the canonical module ω S is the conductor of S relative to S . In the case m S is a prime ideal of S , so the conductor could not be larger as it would have grade at least twoand then S = S : ω S = m S . If d >
2, then m (1 , t ) d − S will work.Note that S is Gorenstein in codimension 1. If P is such prime and P ∩ R = q = m ,then ( S ) q = S q , so S P = ( S ) P is Gorenstein in codimension 1. Thus we may assume P ∩ R = m , so that P = m S . For S P to be Gorenstein would mean Hom( S, S ) ≃ S at P , that is ( S : S ) P is a principal ideal of S P (see next Example). From [6, Theorem 3.3.7], D = Hom( S, S ) = m S = m S . Thus deg( S/ D ) = 2 and since D ∗∗ = D , deg( S/ D ∗∗ ) = 1. Itfollows that bideg( S ) = 1. Example 11.3.
Let R = k [ x, y ], and I = ( x , x y , y ). Then for the reduction Q = ( x , y ),we have I = QI . The Rees algebra S = R [ It ] = k [ x, y, u, v, w ] /L , where L = ( x u − xv, y w − yv, v − xyuw ) is given by the 2 × ϕ = v xwyu vx y , whose content is ( x, y, v ). Itfollows that S is not Gorenstein in codimension 1 as it would require a content of codimensionat least four. Indeed, setting A = k [ x, y, u, v, w ], a projective resolution of S over A is definedby the mapping ϕ : A → A which dualizing gives C = Ext A ( A /L, A ) = Ext A ( L, A ) = coker( ϕ ∗ ) . It follows that C is minimally generated by two elements at the localizations of A that contain( x, y, v ). Rees algebras of ideals with the expected defining relations.
Let I be an ideal of the Cohen-Macaulay local ring ( R , m ) [or a polynomial ring R = k [ t , . . . , t d ] over the field k ] with apresentation R m ϕ −→ R n −→ I = ( b , . . . , b n ) → . Assume that codim( I ) ≥ ϕ lie in m . Denote by L its ideal of relations0 → L −→ S = R [ T , . . . , T n ] ψ −→ R [ It ] → , T i b i t. L is a graded ideal of S and its component in degree 1 is generated by the m linear forms f = [ f , . . . , f m ] = [ T , . . . , T n ] · ϕ. Sylvester forms.
Let f = { f , . . . , f s } be a set of polynomials in L ⊂ S = R [ T , . . . , T n ] andlet a = { a , . . . , a s } ⊂ R . If f i ∈ ( a ) S for all i , we can write f = [ f , . . . , f s ] = [ a , . . . , a q ] · A = a · A , where A is an s × q matrix with entries in S . We call ( a ) a R -content of f . Since a L , thenthe s × s minors of A lie in L . By an abuse of terminology, we refer to such determinants as Sylvester forms , or the
Jacobian duals , of f relative to a . If a = I ( ϕ ), we write A = B ( ϕ ),and call it the Jacobian dual of ϕ . Note that if ϕ is a matrix with linear entries in the variables x , . . . , x d , then B ( ϕ ) is a matrix with linear entries in the variables T , . . . , T n . Definition 11.4.
Let I = ( b , . . . , b n ) be an ideal with a presentation as above and let = a ( a , . . . , a s ) = I ( ϕ ). The Rees algebra R [ It ] has the expected relations if L = ( T · ϕ, I s ( B ( ϕ ))) . There will be numerous restrictions to ensure that R [ It ] is a Cohen-Macaulay ring and thatit is amenable to the determination of its canonical degrees. We consider some special casesgrounded on [31, Theorem 1.3] and [36, Theorem 2.7] Theorem 11.5.
Let R = k [ x , . . . , x d ] be a polynomial ring over an infinite field, let I bea perfect R -ideal of grade with a linear presentation matrix ϕ , assume that ν ( I ) > d andthat I satisfies G d ( meaning ν ( I p ) ≤ dim R p on the punctured spectrum ) . Then ℓ ( I ) = d ; r ( I ) = ℓ ( I ) − ; R [ It ] is Cohen-Macaulay, and L = ( T · ϕ, I d ( B ( ϕ ))) . The canonical ideal of these rings is described in [36, Theorem 2.7] (for g = 2) Proposition 11.6.
Let I be an ideal of codimension two satisfying the assumptions of Theo-rem 11.5. Then bideg( R [ It ]) = 0 and similarly cdeg( R [ It ]) = 0 .Proof. Let J be a minimal reduction of I and set K = J : I . Then C = Kt R [ It ] is a canonicalideal of R [ It ] by [36, Theorem 2.7]. Let a be a regular element of K . Then( a ) : C = X L i t i ⊂ R [ It ] , and bt i ∈ L i if and only if bKI j ⊂ aI i + j . Thus for b ∈ L , b · K ⊂ ( a ) so b = r · a since codim( K ) ≥
2. Hence L = ( a ). For bt ∈ L , bK ⊂ aI , b = ra with r ∈ I : K . As rKI ⊂ I , we have L = a ( I : K ) t . For i ≥ b ∈ L i ,we have b = rat i with rK ⊂ I i . Hence L i = a ( I i : K ) t i . In general we have L i = a ( I i : K ) t i .Therefore, we have( a ) : C = a ( R + ( I : K ) t + ( I : K ) t + · · · ) , C ∗∗ = ( a ) : (( a ) : C ) = X j L j t j = R [ It ] : ( R + ( I : K ) t + ( I : K ) t + · · · ) . ANONICAL DEGREES OF COHEN-MACAULAY RINGS AND MODULES: A SURVEY For b ∈ L and i ≥ b ( I i : K ) ⊂ I i and thus b ∈ T i I i : ( I i : K ) = L . In general it isclear that KI i ⊂ L i . Note that L = R : ( R : K ) = R . It follows that C 6 = C ∗∗ and hencebideg( R [ It ]) = 0. Recall that the vanishing of either of the functions cdeg or bideg holds ifand only if the ring is Gorenstein in codimension 1. Therefore cdeg( R [ It ]) = 0. (cid:3) Canonical degrees of A = m : m Let ( R , m ) be a CM local ring of dimension 1, and set A = Hom( m , m ) = m : Q m . Assumethat R is not a DVR. Let C be its canonical ideal. (We can also discuss some of the samequestions by replacing m by a prime ideal p such that R p is not a DVR.) General properties.
We begin by collecting elementary data on A .(1) A = m : m ⊂ R : m and since m · ( R : m ) = R [as otherwise m is principal] R : m = A .(2) If R is a Cohen-Macaulay local ring of dimension one that is not a DVR, then A = R : m as m · Hom( m , R ) ⊂ m and therefore A = m : Q m = R : Q m = 1 /x · (( x ) : R m ) . Indeed, if q ∈ m : Q m and x is a regular element of m , let a = qx ∈ m . Then q = (1 /x ) a and a m = qx m = xq m ⊂ x m . Thus A = (1 /x )(( x ) : m ), which makes A amenable by calculation using software such as Macaulay2 ([23]).(3) A relevant point is to know when A is a local ring. Let us briefly consider some cases. Let L be an ideal of the local ring R and suppose L = I ⊕ J , L = I + J , I ∩ J = 0, is a non-trivial decomposition. Then I · ( J : I ) = 0 and J · ( J : I ) = 0 and thus if grade( I, J ) = 1[maximum possible by the Abhyankar–Hartshorne Lemma], then I : J = 0. It follows thatHom( L, L ) = Hom(
I, I ) × Hom(
J, J ) , and therefore Hom( L, L ) is not a local ring.(4) Suppose R is complete, or at least Henselian. If A is not a local ring, by the Krull-SchmidtTheorem A admit a non-trivial decomposition of R -algebras A = B × C . Since m = m A , we have a decomposition m = m B ⊕ m C . If we preclude such decompo-sitions then A is a local ring. Among these cases are: analytically irreducible rings, inparticular they include the localization of any monomial ring. Number of generators of A . (1) Since m is an ideal of both R and A , R : A = m . Thus A / R ≃ ( R / m ) n = k n . The exactsequence 0 → R −→ A −→ k n → , yields 0 → R / m −→ A / m = A / m A −→ k n → , gives ν ( A ) = n + 1 since R m A = m . (2) D = C : A is the canonical ideal of A , according to [6, Theorem 3.3.7]. Applying Hom( · , C )to the exact sequence above we get0 → D −→ C −→ Ext ( k n , C ) ≃ k n → . Thus C / D ≃ k n and m C ⊂ D . As0 → m C −→ C −→ k r → D / m C ≃ k r − n . Theorem 12.1.
Let ( R , m ) be a Cohen-Macaulay local ring of dimension one and type r ( R ) = r . Then A = Hom( m , m ) = m : Q m = x − · (( x ) : R ) for any regular element x ∈ m . Furthemore ν R ( A ) = r + 1 . Proof.
Since A = x − (( x ) : m ) for any regular element of m , we have A = 1 /x · (( x ) : m ), which gives ν ( A ) = λ (( x ) : m ) and since(( x ) : m ) /x R ≃ k r then r ≤ ν (( x ) : m ) ≤ r + 1 . Writing ( x ) : m = ( x, y , . . . , y r ), n = r − x is not a minimal generatorof ( x ) : m , so x ∈ m (( x ) : R m ) . In particular x ∈ m . If you preclude this with an initial choice of x ∈ m \ m , we would have r = n always. (cid:3) Corollary 12.2. If ( R , m ) is a local ring and C is the canonical ideal, resp. module, of R then D = m C is the canonical ideal, resp. module, of A .Canonical invariants of A . The driving questions are what are r ( A ), cdeg A and bideg( A ) inrelation to the invariants of R . Our main calculation is the following result: Theorem 12.3.
Suppose ( A , M ) is a local ring. Then cdeg( A ) = e − [cdeg( R ) + e ( m ) − r ] . Proof.
The equality n = r means that D = m C . In particular if ( c ) is a minimal reduction of C and ( x ) is a minimal reduction of m then ( cx ) is a minimal reduction of D . This gives thatif [ e = A /M : R / m ] is the relative degree thene ( D , A ) = e − · λ R ( A /xc A ) = e − [ λ R ( A /x A ) + λ R ( x A /xc A )]= e − · [ λ R ( R /x R ) + λ R ( R /c R )]= e − [e ( C ) + e ( m )] . On the other hand λ A ( A / D ) = e − · λ R ( A / m C ) = e − · [ λ R ( A / R ) + λ R ( R / C ) + λ R ( C / m C )]= e − · [2 r + λ R ( R / C )] . ANONICAL DEGREES OF COHEN-MACAULAY RINGS AND MODULES: A SURVEY These equalities givecdeg( A ) = e − · [e ( C ) − λ R ( R / C ) + e ( m ) − r ] = e − [cdeg( R ) + e ( m ) − r ] , as desired. (cid:3) Corollary 12.4. A is a Gorenstein ring if and only if cdeg( R ) + e ( m ) − r = 0 . In particular m C = A z , so m and C A are principal ideals of A . Remark 12.5. If m is a maximal ideal but A is semilocal with maximal ideals { M , . . . , M s } ,we can still obtain a formula for cdeg( A ) as a summation of the cdeg( A M i ), ascdeg( A ) = X i cdeg( A M i ) = [cdeg( R ) + e ( m ) − r ] · ( X i e − i ) . Example 12.6.
Let ( R , m ) be a Stanley-Reisner ring of dimension one. If R = k [ x , . . . , x n ] /L , L is generated by all the binomials x i x j , i = j . Note that m = ( x ) ⊕ · · · ⊕ ( x n ) and since for i = j the annihilator of Hom(( x i ) , ( x j )) has grade positive, the observations above show that A = Hom( m , m ) = Hom(( x ) , ( x )) × · · · × Hom(( x n ) , ( x n )) = k [ x ] × · · · × k [ x n ] . To determine cdeg( R ), we already have that e ( m ) = n so let us calculate r = r ( R ). TheHilbert series of R is easily seen to be 1 + ( n − t − t , so r = n −
1, which yieldscdeg( R ) = 2 r − e ( m ) = 2( n − − n = n − r − . Thus R is almost Gorenstein ([18, Theorem 5.2]). This also follows from [20, Theorem 5.1].Some of this argument applies in higher dimension. For instance:If R = k [ x , . . . , x n ] /L is an unmixed Stanley-Reiner ring, L = T ≤ i ≤ s P i is an irreduciblerepresentation then the natural mapping R Y ≤ i ≤ s k [ x , · · · , x n ] /P i gives an embedding of R into its integral closure. Note that each factor is a polynomial ringof the same dimension as R .Let us determine other invariants of A : Corollary 12.7.
For A as above (1) The type of A is r ( A ) = ν A ( D ) = e − ν R ( D ) = e − ν R ( m C ) = e − λ ( m C / m C ) . (2) For A to be AGL requires cdeg( A ) = r ( A ) − e − λ ( m C / m C ) − e − [cdeg( R ) + e ( m ) − r ] . Additional invariants. (1) Next we would like to calculate under the same conditionsbideg( A ) = λ A ( A / D ) − λ A ( A / D ∗∗ ) . The first term is as above λ A ( A / D ) = 1 /e · [2 r + λ ( R / C )] . (2) We need D ∗∗ or D · D ∗ . We must pick an element x ∈ D .(3) Since A = x − (( x ) : m ) and D = m C AA : m =? D ∗ = [ A : m ] : C = D ∗ · D = [[ A : m ] : C ] · C m = D ∗ = A : D = x − (( x ) : m ) : m C = x − [( x ) : m ) : m ] : C (4) Would like at least to argue thatcdeg( R ) ≥ bideg( R ) = ⇒ cdeg( A ) ≥ bideg( A ) . Example 12.8.
We return to the ring R = Q [ t , t , t ] , m = ( x, y, z ). We have a presentation R = Q [ x, y, z ] /P , with P = ( y − xz, x − yz , z − x y ). Let us examine some properties of R and A = m : m .(1) The canonical module is C = ( x, y ), and a minimal reduction ( c ) = ( x ). It gives red( C ) = 4.(2) ( c ) : C 6 = m so R is not almost Gorenstein. However C ∗∗ = ( c ) : [( c ) : C ] satisfies [byanother Macaulay2 calculation] λ ( C ∗∗ / C ) = 1, so C ∗∗ = L . This shows thatbideg( R ) = λ ( C ∗∗ /C ) = 1 and cdeg( R ) = λ ( C / ( c )) = 2 . (3) Since e ( m ) = 5, r = 2 and e = 1, we havecdeg( A ) = 2 − . (4) red( D ) = red( m C ) = 2. Note that (x) is a minimal reduction of both C and m .(5) To calculate D ∗∗ we change C to x C . We then get λ ( A / D ) = 11 and λ ( A / D ∗∗ ) = 9, andso bideg( A ) = λ ( A / D ) − λ ( A / D ∗∗ ) = 2.13. Effective computation of bi-canonical degrees
A basic question is how to calculate canonical degrees. We will propose an approach thatapplies to the bi–canonical degree and suggests an extension to rings when the canonicalmodule C is not an ideal. Definition 13.1.
Let R be a Cohen-Macaulay local ring and C one canonical module. Considerthe natural mapping 0 → E −→ C −→ C ∗∗ −→ E → , define bideg C ( R ) = deg( E ) + deg( E ) . The primary setting of our calculations is the following construction of Auslander [2].
ANONICAL DEGREES OF COHEN-MACAULAY RINGS AND MODULES: A SURVEY Definition 13.2.
Let E be a finitely generated R -module with a projective presentation F ϕ −→ F −→ E → . The
Auslander dual of E is the module D ( E ) = coker( ϕ ∗ ),0 → E ∗ −→ F ∗ ϕ ∗ −→ F ∗ −→ D ( E ) → . (1)The module D ( E ) depends on the chosen presentation but it is unique up to projectivesummands. In particular the values of the functors Ext i R ( D ( E ) , · ) and Tor R i ( D ( E ) , · ), for i ≥
1, are independent of the presentation. Its use here lies in the following result (see [2,Proposition 2.6]):
Proposition 13.3.
Let R be a Noetherian ring and E a finitely generated R -module. Thereare two exact sequences of functors: → Ext R ( D ( E ) , · ) −→ E ⊗ R · −→ Hom R ( E ∗ , · ) −→ Ext R ( D ( E ) , · ) → → Tor R ( D ( E ) , · ) −→ E ∗ ⊗ R · −→ Hom R ( E, · ) −→ Tor R ( D ( E ) , · ) → . (3) Corollary 13.4.
Let R be a Cohen-Macaulay local ring and C one of its canonical modules.If D ( C ) is the Auslander dual of C , E = Ext R ( D ( C ) , R ) , E = Ext R ( D ( C ) , R ) . Now we give a presentation of D ( C ) in an important case suitable for computation. Remark 13.5.
Recall that to calculate Ext i R ( M, N ) we invoke the command Ext i ( f, g ) where f and g are the presentations of M and N , respectively. Theorem 13.6.
Let S be a Gorenstein local ring and R = S /I a Cohen-Macaulay ring witha with minimal projective resolution → S p ϕ −→ S m −→ · · · −→ S n f −→ S −→ R = S /I → . Then C = coker( ϕ ∗ ⊗ R ) ,D ( C ) = coker( ϕ ⊗ R ) , and therefore bideg( R ) = deg(Ext R ( ϕ ⊗ R , R )) + deg(Ext R ( ϕ ⊗ R , R )) . Furthermore, C is an ideal if and only if height ( I p − ( ϕ )) ≥ p + 1 . Example 13.7.
Consider the ring R = S /I , where S is a Gorenstein local ring and I is aCohen-Macaulay ideal given by a minimal resolution0 → S n − ϕ −→ S n −→ S −→ R → . This gives 0 → S −→ S n ϕ ∗ −→ S n − −→ C → C is the canonical module of R . We get the exact complexes R n ¯ ϕ ∗ −→ R n − −→ C → , → C ∗ −→ R n − ϕ −→ R n −→ D ( C ) → . (1) In general we have that C is an ideal if and only codim ( I n − ( ϕ )) ≥
3. For instance if ϕ = (cid:20) x yz xzy zx xy (cid:21) I = I ( ϕ ) = ( x z − y z, x y − xy z, x z − xy z )but I ( ϕ ) has codimension 2 so C is not an ideal.(2) Setting Z = ker( ϕ ) we have the exact sequences0 → C ∗ −→ R −→ Hom( Z, R ) −→ Ext S ( C , R ) → → Hom( Z, R ) −→ R −→ R −→ Ext S ( Z, R ) = Ext S ( C , R ) → . For the homogeneous ring defined by ϕ = (cid:20) x yz xzy zx xy (cid:21) , we found deg( E ) = 1 and deg( E ) = 6, so bideg( R ) = 7. Notes and Questions. (1) D ( C ) is fed into the Ext calculation to determine E , E . We only did homogeneouscalculations.(2) Would like some very explicit formulas. For instance suppose I has type 2 and C is anideal 0 → S ϕ −→ S −→ I → . Then height I ( ϕ ) = 3. We have C = coker( ϕ t ) = H n − ( K ). This is to be completedand extended to all strongly Cohen-Macaulay ideals.(3) In our calculations, we asked also for dim( E ), dim( E ) besides deg( E ), deg( E ).(4) We would like to verify that if E = 0 then dim E = dim E .(5) What are the properties of bideg C ( R )?(6) If E = 0, E = C ∗∗ /C and it is either 0 or Cohen-Macaulay of dimension d − E = 0 it must have the same dimension as C [which is a MCM], that is d . We guessthat E has also dimension d . If dim E < d , localizing at one prime not in the supportget a case with E = 0 but E = 0.(8) What if C is reflexive? • Let R m −→ C ∗ →
0, then we have 0 → C ∗∗ −→ R m . If C is reflexive it will thenembed in R m . • Suppose R has dimension ≤
1. Localizing at a regular element we get C x splits off Q m since it is an injective Q -module (in fact its injective envelope). Thus C x splits offa free module. • From Hom( C x , C x ) = Q we get that C x = Q . Thus C x is an ideal of Q and therefore C is an ideal of R . • If C is reflexive, in dimension one by an earlier criterion R is Gorenstein. Corollary 13.8. E = 0 if and only if C is an ideal. ANONICAL DEGREES OF COHEN-MACAULAY RINGS AND MODULES: A SURVEY Now must examine the vanishing of E . Consider0 → H −→ C ⊗ C ∗ −→ trace( C ) = L ⊂ R We have 0 → Hom( L, C ) −→ Hom( C , R ) −→ Hom( H, C ) → . Other generalizations
Let R be a Noetherian local ring and M a finitely generated MCM R -module. We would liketo define a bicanonical degree bideg( M ) with properties similar to bideg( R ).(1) One of these is bideg( M ) = 0 if and only if M is a Gorenstein module in codimension 1.A Gorenstein module is a maximal Cohen-Macaulay module of finite injective dimension.(2) We should be careful here: m is a bidual but not always principal. This shows the needfor additional restrictions.(3) In Proposition 15.2 we give a class of rings and ideals with the required conditions.(4) Let M be a MCM module and set L = Hom( M, C ). In analogy of the case above set0 → E −→ L −→ L ∗∗ −→ E → , and define bideg( M ) = deg( E ) + deg( E ) . If C is an ideal then E = 0. Here we would like to show that if bideg( M ) = 0 then M is Gorenstein in codimension one. This should mean L is free. The usual approach is set A = Hom( L, L ) and consider the natural mapping ϕ : L ⊗ L ∗ −→ Hom(
L, L ) = A . The image τ of ϕ is the ideal of A of the endomorphisms of L that factor thru projectives.We would like to see whether a version of Proposition 2.3 works here, that is Proposition 14.1.
Let R be Cohen-Macaulay local ring of dimension with a canonical ideal C . If M is torsionless, that is a submodule of a free module, then bideg( M ) = λ ( A /τ ) , where τ is the trace of M .Proof. We will show that λ ( L ∗∗ /L ) = λ ( A /τ ) . From the exact sequence 0 → K −→ L ⊗ L ∗ −→ τ → K has finite support and thereforeHom( τ, C ) = Hom( L ⊗ L ∗ , C ) = Hom( L, Hom( L ∗ , C )) = Hom( L, Hom(Hom( L, R ) , C ))From trace( C ) = C · C ∗ , we haveHom(trace( C ) , C ) = Hom( C · C ∗ , C ) = Hom( C ∗ , Hom( C , C )) = C ∗∗ . Now dualize the exact sequence0 → trace( C ) −→ R −→ R / trace( C ) → , into C to obtain0 = Hom( R / trace( C ) , C ) → C = Hom( R , C ) → C ∗∗ = Hom(trace( C ) , C ) → Ext ( R / trace( C ) , C ) → , which shows that C ∗∗ / C and Ext ( R / trace( C ) , C ) are isomorphic. Since by local dualityExt ( · , C ) is self-dualizing on modules of finite support, bideg( R ) = λ ( R / trace( C )) . (cid:3) Precanonical Ideals
Let I be an ideal of the Cohen-Macaulay local ring R . To attach a degree to I with propertiesthat mimic bideg( R ) we would want it to satisfy three properties:(1) I is closed, that is Hom( I, I ) = R .(2) If I is principal then R is Gorenstein, at least in codimension one.(3) If I is reflexive then I is principal.This could be used to define: bideg I ( R ) = deg( I ∗∗ /I ). For a class I of such ideals, a questionis when this degree is independent of I , for non-principal ideals.(1) Of course the issue is to know what the conditions above mean. A promising case toconsider it that of semidualizing ideals/modules. These modules arose in several sources.In [37] a class of modules, termed spherical were introduced by conditions akin to canonicalmodules: the closedness Hom R ( D, D ) = R and the rigidity conditions Ext i ( D, D ) = 0, i ≥
1. See [7, 8] .We want to consider the question: If D is such ideal and it is reflexive, is it principal incodimension 1? This would be a generalization of [26, Corollary 7.29]. In [37, p. 109] apresumed proof is not clear, but see [30, Lemma 2.9].(2) Another path is to consider is deg( I/ ( c )) , where c is some distinguished element of I . Question 15.1.
Where to find semidualizing modules? How are the semidualizing modulesof R , A = m : m and B = R ⋉ m related?Here is a teaser: Proposition 15.2.
Let R be a local ring of dimension one and finite integral closure, and let I be a closed ideal. If I is reflexive then I is principal.Proof. From trace( I ) = L = I · I ∗ , we haveHom(trace( I ) , R ) = Hom( I · I ∗ , R ) = Hom( I ⊗ I ∗ , R ) = Hom( I, Hom( I ∗ , R )) = Hom( I, I ) = R , and thus L ∗∗ = R x . If x is a unit L ∗ = R and therefore has height greater than one, which isnot possible. Thus we have L = M x , where M ⊂ m . Since L ∗∗ = R x is integral over L = M x ([10, Proposition 2.14]), we have for some positive integer n R x n = M x R x n − , and thus R = M , which is a contradiction. (cid:3) Better see [30, Lemma 2.9].
ANONICAL DEGREES OF COHEN-MACAULAY RINGS AND MODULES: A SURVEY Almost semidualizing ideals.
Theorem 15.3.
Let ( R , m ) be a one-dimensional Cohen-Macaulay local ring that has a canon-ical ideal and S is an m -primary ideal such that (1) Hom( S, S ) = R , and (2) Ext ( S, S ) = 0 .If c ∈ S is such that S/ ( c ) ≃ ( R / m ) n , for some n , then S is a canonical ideal.Proof. Consider the exact sequence of natural maps0 → ( c ) −→ S −→ V → . Suppose V = k n . Applying Hom( · , S ) we get0 → R −→ c − S −→ Ext ( V, S ) = Ext ( k, S ) n = Hom( k, S/cS ) n −→ Ext ( S, S ) = 0 . Thus Hom( k, S/cS ) = k . Since R /S ≃ ( c ) /cS embeds in S/cS , the socle of R /S is R / m , so S is an irreducible ideal.The assertion would hold for xS for any regular element of R . We now invoke Corolary 5.6. (cid:3) Comments: (1) The ideals with these two conditions include the semidualizing ones. Theorem 15.3 assertsthat the almost Gorenstein rings of Goto ([20, 22]) can only occur when S is a canonicalideal.(2) What other interesting modules occur as V = S/ ( s )? We are going to assume that S isCohen-Macaulay of codimension one and s is a regular element, in particular V is a Cohen-Macaulay module. We will also assume that S is closed. It always leads to V = Ext ( V, S ).(3) Maybe closed plus Ext ( S, S ) = 0 could be called strongly closed, or perhaps 1-closed...then those ∞ -closed would be the semidualizing ideals.(4) Suppose S is 2-closed and V = T n with T = R / m . References [1] Y. Aoyama, Some basic results on canonical modules,
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