On reflexive and I-Ulrich modules over curve singularities
aa r X i v : . [ m a t h . A C ] F e b ON REFLEXIVE AND I -ULRICH MODULES OVER CURVESINGULARITIES HAILONG DAO, SARASIJ MAITRA, AND PRASHANTH SRIDHAR
Abstract.
We study reflexive modules over one dimensional Cohen-Macaulay rings.Our key technique exploits the concept of I -Ulrich modules. Introduction
Notions of reflexivity have been studied throughout various branches of mathematics.Over a commutative ring R , recall that a module M is called reflexive if the naturalmap M → M ∗∗ is an isomorphism, where M ∗ denotes Hom R ( M, R ). When R is a field,any finite dimensional vector space is reflexive, a fundamental fact in linear algebra.Over general rings, from what we found in the literature, these modules were studiedin the works of Dieudonn´e [D + S ) and reflexive in codimensionone (over an ( S ) ring, see [BH98, Proposition 1.4.1]), understanding the one dimensional(local) case is key to understanding reflexivity in general.Assume now that ( R, m ) is a one dimensional Noetherian local Cohen-Macaulay ring.The primary examples are curve singularities or localized coordinate rings of points inprojective space. It turns out that the answers to our basic question can be quite subtle.If R is Gorenstein, then any maximal Cohen-Macaulay module is reflexive (so in ourdimension one situation, any torsionless module is reflexive). However, reflexive modulesor even ideals are poorly understood when R is not Gorenstein. Some general facts areknown, for instance if R is reduced, any second syzygy or R -dual module is reflexive.However, we found very few concrete examples in the literature: only the maximal idealand the conductor of the integral closure of R . How many can there be? Can we classifythem? When is an ideal of small colength reflexive? For instance, if R is C [[ t , t , t ]] thenany indecomposable reflexive module is either isomorphic to R or the maximal ideal, butthe reason is far from clear.Our results will give answers to the above questions in many cases. A key point in ourinvestigation is a systematic application of the concept of I -Ulrich modules, where I isany ideal of height one in R . A module M is called I -Ulrich if e I ( M ) = ℓ ( M/IM ) where e I ( M ) denotes the Hilbert Samuel multiplicity of M with respect to I and ℓ ( · ) denoteslength. This is a straight generalization of the notion of Ulrich modules, which is justthe case I = m ([Ulr84] and [BHU87]). Note also that I is I -Ulrich simply says that I isstable, a concept heavily used in Lipman’s work on Arf rings [Lip71]. Key words and phrases. reflexive modules, I -Ulrich modules, trace ideals, conductors, birational ex-tensions, Gorenstein rings, canonical ideal. Of course, the study of m -Ulrich modules and certain variants has been an active areaof research for quite some time now. The papers closest to the spirit of our work areperhaps [GOT + + ω R -Ulrich modules are reflexive, and they form a category criticalto the abundance of reflexive modules (here ω R is a canonical ideal of R ). For instance,any maximal Cohen-Macaulay module over an ω R -Ulrich finite extension of R is reflexive.Furthermore, a reflexive birational extension of R is Gorenstein if and only if its conductor I is I -Ulrich and ω R -Ulrich.We also make frequent use of birational extensions of R and trace ideals. This is heavilyinspired by some recent interesting work from Kobayashi [Kob17], Goto-Isobe-Kumashiro[GIK20], Faber [Fab19] and Herzog-Hibi-Stamate [HHS19].We now describe the structure and the main results of our paper. Sections 2 and 3 collectbasic results on reflexive modules, trace ideals and birational extensions to be used in latersections. Section 4 develops the concept of I -Ulrich modules for any ideal I of height onein R , see Definition 4.1. We give various characterizations of I -Ulrichness (Theorem 4.6).We show the closedness of the subcategory of I -Ulrich modules under various operations,prompting the existence of a lattice like structure for I -Ulrich ideals, which can be referredto as an Ulrich lattice. We establish tests for I -Ulrichness using blow-up algebras and the core of I (recall that the core of an ideal is the intersection of all minimal reductions).Finally, we show that an ω R -Ulrich M satisfies Hom R ( M, R ) ∼ = Hom R ( M, ω R ), and thatsuch a module is reflexive.The later sections deal with applications. In Section 5, under mild conditions, weare able to completely characterize extensions S of R that are “strongly reflexive” in thefollowing sense: any maximal Cohen-Macaulay S -module is reflexive over R . Interestingly,in the birational case, this classification involves the core of the canonical ideal of R . Theorem A (Theorem 5.2 and Theorem 5.5) . Suppose that R is a one-dimensionalCohen-Macaulay local ring with a canonical ideal ω R . Let S be a module finite R -algebrasuch that S is a maximal Cohen-Macaulay module over R . The following are equivalent(for the last two, assume that S is a birational extension and the residue field of R isinfinite):(1) Any maximal Cohen-Macaulay S -module is R -reflexive.(2) ω S is R -reflexive.(3) Hom R ( S, R ) ∼ = Hom R ( S, ω R ) .(4) ω R S ∼ = S .(5) S is ω R -Ulrich as an R -module.(6) S is R -reflexive and the conductor of S to R lies inside ( x ) : ω R for some principalreduction x of ω R .(7) S is R -reflexive and the conductor of S to R lies inside core( ω R ) : R ω R . The theorem above extends a result of Kobayashi [Kob17, Theorem 2.14]. Also, for S satisfying one of the conditions of Theorem A, any contracted ideal IS ∩ R is reflex-ive (Proposition 3.11). Such a statement generalizes a result by Corso-Huneke-Katz-Vasconcelos that if R is a domain and the integral closure R is finite over R , then anyintegrally closed ideal is reflexive [CHKV05, Proposition 2.14].Our next Section 6 deals with various “finiteness results”, where we study when certainsubcategories or subsets of CM( R ) are finite or finite up to isomorphism. One main result N REFLEXIVE AND I -ULRICH MODULES OVER CURVE SINGULARITIES 3 roughly says that if the conductor of R has small colength, then there are only finitelymany reflexive ideals that contain a regular element, up to isomorphism. Theorem B (Theorem 6.8) . Let R be a one-dimensional Cohen-Macaulay local ring withinfinite residue field and conductor ideal c . Assume that either:(1) ℓ ( R/ c ) ≤ , or(2) ℓ ( R/ c ) = 4 and R has minimal multiplicity.Then the category of regular reflexive ideals of R is of finite type. We also characterize rings with up to three trace ideals (Proposition 6.3). We ob-serve that if S = End R ( m ) has finite representation type, then R has only finitely manyindecomposable reflexive modules up to isomorphism (Proposition 6.12). In particular,seminormal singularities have “finite reflexive type” (Corollary 6.14).In Section 7 we give some further applications on almost Gorenstein rings. We showthat in such a ring, all powers of trace ideals are reflexive (Proposition 7.3). We alsocharacterize reflexive birational extensions of R which are Gorenstein: Theorem C (Theorem 7.10) . Suppose that R is a one-dimensional Cohen-Macaulay localring with a canonical ideal ω R . Let S be a finite birational extension of R which is reflexiveas an R -module. Let I = c R ( S ) be the conductor of S in R . The following are equivalent:(1) S is Gorenstein.(2) I is I -Ulrich and ω R -Ulrich. That is I ∼ = I ∼ = Iω R . We end the paper with a number of examples and open questions.2.
Preliminaries I: general Noetherian rings
Throughout this article, assume that all rings are commutative with unity and areNoetherian, and that all modules are finitely generated.Let R denote a Noetherian ring with total ring of fractions Q ( R ). Let R denote theintegral closure of R in Q ( R ). Let Spec R denote the set of prime ideals of R . For any R -module M , if the natural map M → M ⊗ R Q ( R ) is injective, then M is called torsion-free . It is called a torsion module if M ⊗ R Q ( R ) = 0. The dual of M , denoted M ∗ , is themodule Hom R ( M, R ); the bidual then is M ∗∗ . The bilinear map M × M ∗ → R, ( x, f ) f ( x ) , induces a natural homomorphism h : M → M ∗∗ . We say that M is torsionless if h isinjective, and M is reflexive if h is bijective.For R -submodules M, N of Q ( R ), we denote M : R N = { a ∈ R | aN ⊆ M } M : N = { a ∈ Q ( R ) | aN ⊆ M } . We will need the notion of trace ideals. We first recall the definition.
Definition 2.1.
The trace ideal of an R -module M , denoted tr R ( M ) or simply tr( M )when the underlying ring is clear, is the image of the map τ M : M ∗ ⊗ R M → R definedby τ M ( φ ⊗ x ) = φ ( x ) for all φ ∈ M ∗ and x ∈ M . HAILONG DAO, SARASIJ MAITRA, AND PRASHANTH SRIDHAR
Say that an ideal I is a trace ideal if I = tr( M ) for some module M . Since tr(tr( M )) =tr( M ), I is a trace ideal if and only if I = tr( I ). It is clear from the definition, that if M ∼ = N , then tr( M ) = tr( N ).There are various expositions on trace ideals scattered through the literature, see forexample [HHS19],[Lin17],[KT19a],[GIK20],[Fab19], etc. For the purposes of this paper,we shall mainly need the following properties of trace ideals. Proposition 2.2. [KT19a, Proposition 2.4]
Let M be an R -submodule of Q ( R ) containinga nonzero divisor of R . Then the following statements hold.(1) tr( M ) = ( R : M ) M .(2) The equality M = tr( M ) holds if and only if M : M = R : M in Q ( R ) . Recall that a finitely generated R -submodule I of Q ( R ) is called a fractional ideal andit is regular if it is isomorphic to an R -ideal of grade one. Remark 2.3.
Let R be any ring with total ring of fractions Q ( R ). For any two regularfractional ideals I , I , we have I : I ∼ = Hom R ( I , I ) where the isomorphism is as R -modules, see for example [HK71, Lemma 2.1].By abuse of notation, we will identify these two R -modules and use them interchange-ably in the remainder of the paper. Remark 2.4.
By Remark 2.3, we can identify I ∗ := Hom R ( I, R ) with R : I whenever I contains a non zero divisor. This is also denoted as I − . Moreover for any non zerodivisor x in I , we have xI ∗ = x : R I . Hence I ∗ ∼ = x : R I as R -modules.Next, we discuss some general statements about reflexive modules that will be needed.Recall that an R -module M is called totally reflexive if M is reflexive and Ext iR ( M, R ) =Ext iR ( M ∗ , R ) = 0 for all i > R . This condition is satisfied when R is generically Gorenstein or the module is locallyfree on the minimal primes. Lemma 2.5.
Let R be a Noetherian ring satisfying condition ( S ) . Consider modules M, N that are locally totally reflexive on the minimal primes of R .(1) Assume there is a short exact sequence → M → N → C → . If N is reflexiveand C is torsionless then M is reflexive.(2) If N is reflexive, then Hom R ( M, N ) is reflexive.Proof. For (1), see [Mas98, Proposition 8] and the remark following it.For (2), we first prove the case N = R . Let f : M ∗ → M ∗∗∗ be the natural map. Let g : M ∗∗∗ → M ∗ be the dual of the natural map M → M ∗∗ . Then g ◦ f = id, so f splits,and we get M ∗∗∗ = M ∗ ⊕ M . But for any minimal prime P , ( M ) P = 0, so M haspositive grade. As M ∗∗∗ embeds in a free module and R is ( S ), M = 0.Now, start with a short exact sequence 0 → C → F → N ∗ → F is free.Dualizing, we get 0 → N → F ∗ → D → D is torsionless (by [Mas98, Proposition8], a submodule of a torsionless module is torsionless). Take Hom R ( M, − ) to get the exactsequence 0 → Hom R ( M, N ) → Hom R ( M, F ∗ ) → K ′ → K ′ is a sub-module ofHom R ( M, D ). We can apply part (1) and the previous paragraph to get that Hom R ( M, N )is reflexive provided we show that K ′ is torsionless.Since, D is torsionless, D embeds into D ∗∗ and hence into a free module say G . Thus,Hom R ( M, D ) embeds into Hom R ( M, G ). Finally, note that M ∗ is torsionless as it is a N REFLEXIVE AND I -ULRICH MODULES OVER CURVE SINGULARITIES 5 submodule of a free module and thus, by applying [Mas98, Proposition 8] twice, we getthat K ′ is torsionless. (cid:4) The relationship between reflexive modules and birational extensions is also naturallyof interest. We say that an extension f : R → S is birational if S ⊂ Q ( R ). Equivalently Q ( R ) = Q ( S ). Proposition 2.6. [KT19a, Proposition 2.4]
Let M ⊆ Q ( R ) be a regular fractional idealof R . Then M is a reflexive R -module if and only if there is an equality M = R : ( R : M ) in Q ( R ) . The following lemma was stated in more generality in a recent work of S. Goto, R.Isobe, and S. Kumashiro.
Lemma 2.7. [GIK20, Lemma 2.6(1), Proposition 2.9] . Let f : R → S be a finite birationalextension. Then the conductor of S to R , denoted c R ( S ) := R : S , is a reflexive traceideal of R . Thus, we get a bijective correspondence between reflexive trace ideals of R and reflexive birational extensions of R via the map α : I End R ( I ) and its inverse β : S c R ( S ) .Proof. Let S be a reflexive birational extension. Then c R ( S ) ∼ = S ∗ is reflexive byLemma 2.5. Next, note that tr( S ) = S ∗ S = c R ( S ) S = c R ( S ). So c R ( S ) is a reflexivetrace ideal. If I is a reflexive trace ideal, then Lemma 2.5(2) tells us that End R ( I ) isindeed reflexive.Finally, we have β ( α ( I )) = I and α ( β ( S )) = S by Proposition 2.2, Proposition 2.6 andthe above paragraph. (cid:4) So these birational extensions provide important sources for generating reflexive ideals.We have the following criteria for a reflexive module over R to be a module over a finitebirational extension S . This was stated in [Fab19, Theorem 3.5] for reduced one dimen-sional local rings, but the result holds for more general rings, and we restate it here witha self-contained proof: Theorem 2.8.
Let R be a Noetherian ring and M be a finite R -module. Let S be a finitebirational extension of R . Consider the following statements.(1) M is a module over S .(2) tr( M ) ⊆ c R ( S ) where c R ( S ) = R : S .Then (1) implies (2) . The converse is true if M is a reflexive R -module.Proof. Let M be a module over S . Then there exists a S -linear (hence R -linear) surjection S n → M , so tr( M ) ⊆ tr( S ) = c R ( S ).Conversely assume tr( M ) ⊆ c R ( S ) and M reflexive. Consider f ∈ M ∗ and s ∈ S ,we have s · f ∈ M ∗ by assumption. Therefore M ∗ is an S -module. From the forwardimplication, tr( M ∗ ) ⊆ c R ( S ). So repeating the argument again, we get that M ∗∗ is an S -module. Since M is reflexive, we are done. (cid:4) Preliminary II: dimension one
Throughout the rest of this paper (unless otherwise specified), ( R, m , k ) will denote aCohen-Macaulay local ring of dimension one with maximal ideal m and residue field k .Denote by Q ( R ) the total quotient ring of R . For an m -primary ideal I and a module HAILONG DAO, SARASIJ MAITRA, AND PRASHANTH SRIDHAR M , let e I ( M ) denote the Hilbert-Samuel multiplicity of M with respect to I . In the case,when I = m , we write e ( M ). Let c := R : R denote the conductor ideal of R to R . For an R -module M , let µ ( M ) and ℓ ( M ), denote the minimal number of generators of M andthe length of M respectively, as an R -module.Let CM( R ) denote the category of maximal Cohen-Macaulay R -modules and let Ref( R )denote the category of reflexive R -modules. We say a category is of finite type if it hasonly finitely many indecomposable objects up to isomorphism. Remark 3.1.
Note that the following statements are true.(1) { free R - modules } ⊂ Ref( R ) ⊂ CM( R ) . (2) R is regular if and only if { free R - modules } = CM( R ).(3) R is Gorenstein if and only if Ref( R ) = CM( R ).We shall be interested in the behaviour of Ref( R ) in the case when R is “close to” beingregular or Gorenstein. We will come back to this in Section 6 and Section 7.The conductor and maximal ideals are natural examples of reflexive trace ideals. Corollary 3.2. If ¯ R is finite over R , c is a reflexive trace ideal. If dim R = 1 and R isCohen Macaulay but not regular, m is a reflexive trace ideal.Proof. The first statement follows from Lemma 2.7. For the second statement, sincegrade( m ) = 1, R ( m ∗ and hence m is reflexive by Proposition 2.6. Since m ⊆ tr( m ) andtr( m ) = R if and only if m is principal, m is a trace ideal. (cid:4) Support and trace.
Let CM full ( R ) = { M ∈ CM( R ) | Supp( M ) = Spec R } denote thesubcategory of CM( R ) of modules with full support. Lemma 3.3.
Suppose that M ∈ CM( R ) . Then tr( M ) is a regular ideal if and only if M P has an R P -free summand for each P ∈ Min( R ) . Thus if R is reduced then M ∈ CM full ( R ) if and only if tr( M ) is a regular ideal.Proof. As we are in dimension one, clearly tr( M ) is regular if and only if tr( M ) P = R P for any P ∈ Min( R ). As trace localizes, we have tr( M ) P = tr R P ( M P ), and the resultfollows. (cid:4) Remark 3.4.
Here we discuss why when studying CM( R ), one can reduce to the caseof CM full ( R ) and hence regular trace ideals thanks to Lemma 3.3. Let Min( R ) = { P , . . . , P n } denote the set of minimal primes of R and (0) = ∩ Q i with √ Q i = P i .For a subset X ⊂ Min( R ), let R X = R/ ∩ P i ∈ X Q i . Then CM( R ) = ∪ X ⊂ Min( R ) CM full ( R X ).Thus, understanding CM( R ) amounts to understanding CM full ( R X ) for all subsets X .It is well known that c and m are reflexive trace ideals (Corollary 3.2). In particular,we can investigate other such ideals. We set up some further notation which we will usethroughout. T( R ) := { I | I is a regular trace ideal } RT( R ) := { I | I is a regular reflexive trace ideal } Note that if R is a complete local domain, then from [Mai20, Theorem 4.4] we getthat for any ideal I ⊂ R , I ∗∗ is isomorphic to an ideal which contains the conductor c .This suggests an immediate link, relevant to our study, with the conductor ideal c . Thefollowing theorem gives a generalization to this fact. N REFLEXIVE AND I -ULRICH MODULES OVER CURVE SINGULARITIES 7 Theorem 3.5.
Let R be a one-dimensional Cohen-Macaulay local ring with conductorideal c . Any regular ideal I that contains a principal reduction is isomorphic to anotherfractional ideal J such that R : J (which is isomorphic to I ∗ ) contains the conductor c .In particular, if the residue field is infinite, any reflexive regular ideal of R is isomorphicto an ideal containing c .Proof. Assume I is a regular ideal of R with a principal reduction x . Let J = Ix = { ax , a ∈ I } . Clearly I ∼ = J . Since I n +1 = xI n for some n , we have J ⊂ I n : I n ⊂ R . But then R : J ⊃ R : R = c . The last statement follows by replacing I with I ∗ and using the factthat I ∗∗ ∼ = I . (cid:4) Corollary 3.6.
Let R be as in Theorem 3.5. For any regular ideal I with a principalreduction, tr( I ) ⊃ c .Proof. Let J be the fractional ideal as in Theorem 3.5. Note that R ⊂ J . Since R : J ⊃ c ,we have tr( I ) = tr( J ) = J ( R : J ) ⊃ R : J ⊃ c . (cid:4) Lemma 3.7.
Let R be as in Theorem 3.5. Suppose that I is a regular ideal and x ∈ I bea non zero divisor. Then tr( I ) = I (( x ) : R I ) : R x .Proof. Let J = Ix ∼ = I . Then tr( I ) = tr( J ) = J ( R : J ) = Ix (( x ) : R I ). (cid:4) Corollary 3.8.
Let R be as in Theorem 3.5. Suppose that I is a regular ideal and x ∈ I be a non zero divisor. Then tr( I ) ⊇ ( x ) : R I .Proof. By Lemma 3.7, tr( I ) ⊃ x (( x ) : R I ) : R x = ( x ) : R I . (cid:4) Lemma 3.9.
Let R be as in Theorem 3.5. If I is a regular ideal and I = xI for some x ∈ I then tr I = ( x ) : R I .Proof. From Corollary 3.8, ( x ) : R I ⊆ tr I . On the other hand, I tr I = I ( II − ) = I I − = xII − = x tr I , so tr I ⊂ ( x ) : R I . (cid:4) The above allows us to classify trace ideals with reduction number one.
Corollary 3.10.
Let R be as in Theorem 3.5. Let I be a regular ideal such that I = xI for some x ∈ I . Then I is a trace ideal if and only if ( x ) : R I = I . In that case I ∼ = I ∗ and hence I is reflexive.Proof. The first assertion is obvious from Lemma 3.9. For the last assertion, note that( x ) : R I ∼ = I ∗ . (cid:4) It is known that under mild assumptions, all integrally closed ideals are reflexive([CHKV05, Proposition 2.14]). The following proposition vastly generalizes this fact andalso provides a way of generating reflexive or trace ideals by contracting ideals from certainbirational extensions. For how to find such extensions see Section 5.
Proposition 3.11.
Let R be as in Theorem 3.5. Let S be a finite birational extensionsuch that CM( S ) ⊂ Ref( R ) . Let I be a regular ideal of R . Then IS ∩ R ∈ Ref( R ) . If I contains c R ( S ) , then IS ∩ R ∈ RT( R ) .Proof. Let J = IS ∩ R . As IS ∈ CM( S ), we have J ∗∗ ⊂ ( IS ) ∗∗ = IS , so J ∗∗ ⊂ IS ∩ R = J ,hence J is reflexive. If I contains c R ( S ) then c R ( S ) ⊂ J . Now, tr J = J J − ⊂ J c R ( S ) − = J S , so tr J ⊂ J S ∩ R = J . (cid:4) HAILONG DAO, SARASIJ MAITRA, AND PRASHANTH SRIDHAR
The next two results are useful for studying colength two ideals.
Proposition 3.12.
Let ( R, m ) be as in Theorem 3.5 and further assume that R hasminimal multiplicity with infinite residue field. Let I be a regular ideal of colength two.Then I is reflexive if and only if it is either integrally closed or principal.Proof. If I is integrally closed then it is reflexive by Proposition 3.11. Now assume I is neither integrally closed nor principal. Then necessarily I = m . We can then pick aregular principal reduction x for m and I . Since R has minimal multiplicity we note that m = tr( m ) = ( x ) : R m by Lemma 3.9. On the other hand ( x ) : R I ⊃ ( x ) : R m = m , soequality occurs. Using Remark 2.4, we get xI ∗ = x m ∗ and hence I ∗ = m ∗ and I ∗∗ = m .Thus I is not reflexive. (cid:4) We classify colength two ideals that are contracted from End R ( m ). Proposition 3.13.
Let ( R, m ) be as in Theorem 3.5. Let S = End R ( m ) and I be an idealof colength two. Then IS ∩ R = I if and only if ℓ ( S/IS ) > type( R ) + 1 .Proof. It is clear that IS ∩ R = I if and only if S/IS is a faithful
R/I module. As
R/I ∼ = k [ t ] /t , S/IS decomposes into a direct sum of k and R/I , so it is faithful if andonly if it is not a direct sum of k ’s, in other words the length of S/IS is strictly largerthan it’s number of generators. But µ ( S ) = ℓ ( S/ m S ) = ℓ ( S/R ) + 1 = type R + 1. (cid:4) I -Ulrich Modules Throughout this section, we assume that ( R, m , k ) is a one-dimensional Cohen-Macaulay local ring. Let I be an ideal of finite colength and x a principal reduction.This section grew out of the realization that the equality xM = IM for certain modules M appears in many situations related to our investigation. For instance, it will turnout that when I is a canonical ideal, such a module is reflexive and the finite extensionssatisfying such conditions are “strongly reflexive”, see Definition 5.3.We shall call these modules I -Ulrich, and define them slightly more generally withoutusing principal reductions. Obviously the name and definition are inspired by the verywell-studied notion of Ulrich modules, which are m -Ulrich in our sense. Note that ourdefinition is very much a straight generalization of an Ulrich module, and not as restrictiveas those studied in [GOT +
14] and [GOT + Definition 4.1.
We say that M ∈ CM( R ) is I -Ulrich if e I ( M ) = ℓ ( M/IM ). Let Ul I ( R )denote the category of I -Ulrich modules.Note that if M ∼ = N in CM( R ), then the same isomorphism takes IM to IN , so ℓ ( M/IM ) = ℓ ( N/IN ) for any ideal I and so Ulrich condition is preserved under isomor-phism. Example 4.2.
Let M ∈ CM ( R ). As ℓ ( I n M/I n +1 M ) = e I ( M ) for n ≫
0, it follows that I n M is I -Ulrich for n ≫ Definition 4.3.
Let B ( I ) denote the blow-up of I , namely the ring [ n ≥ ( I n : I n ). Let b ( I ) = c R ( B ( I )), the conductor of B ( I ) to R . Remark 4.4. If x is a principal reduction of I , then it is well-known that B ( I ) = R [ Ix ],[BP95, Theorem 1]. N REFLEXIVE AND I -ULRICH MODULES OVER CURVE SINGULARITIES 9 We shall use some standard properties of Hilbert Samuel multiplicity in the proof of thenext proposition. The reader can refer to various resources like, [Ser65], [Ser97], [BH98,mainly Corollary 4.7.11], [HS06, mainly Proposition 11.1.0, 11.2.1] for further details onmultiplicity.
Proposition 4.5.
Let R be a one-dimensional Cohen-Macaulay local ring. Suppose that x ∈ I is a (principal) reduction and M ∈ CM( R ) . The following are equivalent:(1) M is I -Ulrich.(2) IM = xM .(3) IM ⊆ xM .(4) IM ∼ = M .(5) M ∈ CM( B ( I )) (see Remark 4.8).(6) M is I n -Ulrich for all n ≥ .(7) M is I n -Ulrich for infinitely many n .(8) M is I n -Ulrich for some n ≥ .Proof. As x is a reduction of I , ℓ ( M/xM ) = e I ( M ). So (1) is equivalent to ℓ ( M/xM ) = ℓ ( M/IM ), or IM = xM . The equivalence of (2) and (3) is obvious. Clearly (2) implies(4). Assuming (4), then M ∼ = I n M for n ≫
0, so M is I -Ulrich by Example 4.2. We haveestablished the equivalence of (1) through (4).Next, (3) is equivalent to Ix M ⊆ M . In other words (3) implies that M ∈ CM( B ( I )),since R [ Ix ] = B ( I ). Since B ( I ) = B ( I n ) for any n ≥
1, we have (5) ⇒ (6). Clearly(6) ⇒ (7) ⇒ (8). Finally, assume (8). We have ℓ ( M/I n M ) = e I n ( M ) = ne I ( M ). Notethat for each i , I i M is in CM( R ) and hence, using properties of multiplicities, we get ℓ ( I i M/I i +1 M ) ≤ ℓ ( I i M/xI i M ) = e I ( I i M ) = e I ( M ) for each i . Thus, equality mustoccur for each i ; in particular, it occurs for i = 0, which shows that M is I -Ulrich. (cid:4) Without any assumption on the existence of a principal reduction, the following stillholds:
Theorem 4.6.
Let R be a one-dimensional Cohen-Macaulay local ring. Let I be a regularideal and M ∈ CM( R ) . The following are equivalent:(1) IM ∼ = M .(2) M is I -Ulrich.(3) M is I n -Ulrich for all n ≥ .(4) M is I n -Ulrich for infinitely many n .(5) M is I n -Ulrich for some n ≥ .(6) M ∈ CM( B ( I )) (see Remark 4.8).Proof. Assume (1), then M ∼ = I n M for n ≫
0, so M is I -Ulrich by Example 4.2. If (2)holds, then we may pass to a local faithfully flat extension of R possessing an infiniteresidue field and apply Proposition 4.5, followed by [Gro67, Proposition 2.5.8] to see that(1) holds. The statements (2) to (5) are unaffected by local faithfully flat extensions, sowe can enlarge the residue field and apply Proposition 4.5. As I n contains a principalreduction for n ≫ B ( I ) = B ( I n ) for all n ≥
1, Proposition 4.5 implies that (4) and(6) are equivalent. (cid:4)
Corollary 4.7.
Let R be as in Theorem 4.6. Let I be a regular ideal. Then R is I -Ulrichif and only if I is principal. Proof.
Theorem 4.6 implies that IR ∼ = R , so I is principal. (cid:4) Remark 4.8.
Note that if M ∈ CM( R ) is I -Ulrich, the proofs of Proposition 4.5 andTheorem 4.6 show that the action of B ( I ) on M extends the action of R on M . In otherwords, there is an action of B ( I ) on M which when restricted to R yields the originalaction of R on M . In particular, if M ⊆ Q ( R ), multiplication in Q ( R ) gives an action of B ( I ) on M .We say that an extension f : R → S is birational if S ⊂ Q ( R ). Equivalently Q ( R ) = Q ( S ). Also such an f induces a bijection on the sets of minimal primes of S and R and f P is an isomorphism at all minimal primes P of R . Let Bir( R ) denote the set of finitebirational extensions of R . Corollary 4.9.
Let R be as in Theorem 4.6. Let R ⊆ S be a finite birational extensionof rings. Then S is I -Ulrich if and only if B ( I ) ⊆ S .Proof. Follows immediately from Theorem 4.6 and Remark 4.8. (cid:4)
Corollary 4.10.
Let R be as in Theorem 4.6. Let I be a regular ideal. If ¯ R is a finitelygenerated R -module then ¯ R and the conductor c are I -Ulrich.Proof. As B ( I ) ⊆ ¯ R , ¯ R ∈ CM( B ( I )) and so by Theorem 4.6, ¯ R is I -Ulrich. Since c ∈ CM( ¯ R ) ⊆ CM( B ( I )), the conclusion follows. (cid:4) Corollary 4.11.
Let R be as in Theorem 4.6. If M ∈ CM( R ) is I -Ulrich, then tr M ⊆ b ( I ) . If M ∈ Ref( R ) , then the converse holds.Proof. This follows from Theorem 4.6 and Theorem 2.8. (cid:4)
Lemma 4.12.
Let R be as in Theorem 4.6. Let → A → B → C → be an exactsequence in CM( R ) . If B is I -Ulrich then so are A, C .Proof.
We may enlarge the residue field if necessary and assume that I has a principalreduction x . Then x is a regular element and hence induces an exact sequence0 → A/xA → B/xB → C/xC → .B is I -Ulrich if and only if I kills the middle module, but if that’s the case then I killsthe other two as well. (cid:4) Corollary 4.13.
Let R be as in Theorem 4.6. Let M ∈ Ul I ( R ) . For any f ∈ M ∗ , Im( f ) ∈ Ul I ( R ) . Corollary 4.14.
Let R be as in Theorem 4.6. If ideals J, L are in Ul I ( R ) , then J + L, J ∩ L ∈ Ul I ( R ) .Proof. The assertion follows from the short exact sequence 0 → J ∩ L → J ⊕ L → J + L → (cid:4) Lemma 4.15.
Let R be as in Theorem 4.6. If M ∈ Ul I ( R ) , then Hom R ( M, N ) ∈ Ul I ( R ) for any module N ∈ CM( R ) .Proof. As above, we can assume there is a principal reduction x of I . Note that there isan embedding Hom R ( M, N ) ⊗ R R/xR → Hom R ( M/xM, N/xN )and the latter is killed by I since M ∈ Ul I ( R ). This shows that Hom R ( M, N ) ⊗ R R/xR is killed by I and this finishes the proof. (cid:4) N REFLEXIVE AND I -ULRICH MODULES OVER CURVE SINGULARITIES 11 Proposition 4.16.
Let R be as in Theorem 4.6. If M ∈ Ul I R , then tr( M ) ∈ Ul I ( R ) .Proof. Since, tr( M ) is the sum of all images of elements in M ∗ , the proof follows imme-diately from Corollary 4.13 and Corollary 4.14. (cid:4) Corollary 4.17.
Let R be as in Theorem 4.6. The set of I -Ulrich ideals is a lattice underaddition and intersection. The largest element is b ( I ) .Proof. That this set forms a lattice follows from Corollary 4.14. For the last assertion,first note that b ( I ) is a module over B ( I ), and then apply Theorem 4.6, Proposition 4.16and Corollary 4.11. (cid:4) Remark 4.18.
Aberbach and Huneke [AH96] defined the coefficient ideal of I relativeto a principal reduction x as the largest ideal J such that xJ = IJ . It follows that thecoefficient ideal is just b ( I ).From now on we assume that I contains a principal reduction. Recall that the core of I , denoted core( I ), is defined as the intersection of all (minimal) reductions of I . Proposition 4.19.
Let R be as in Theorem 4.6. Assume that I has a principal reduction x . Consider M ∈ Ul I ( R ) . We have, tr( M ) ⊆ ( x ) : R I ⊆ tr( I ) . If the residue field of R is infinite, then: tr( M ) ⊆ core( I ) : R I ⊆ ( x ) : R I ⊆ tr( I ) . Proof.
Let J = tr( M ). Note that by Proposition 4.16, J ∈ Ul I ( R ) and so IJ = xJ ⊂ ( x )for any principal reduction x . So J ⊂ ∩ (( x ) : R I ) = core( I ) : R I . The last inclusion comesfrom Corollary 3.8. If the residue field of R is infinite, then the core is the intersection ofall principal reductions, and the second part follows. (cid:4) Corollary 4.20.
Let R be as in Theorem 4.6. Suppose that I is a regular ideal with aprincipal reduction x . Then b ( I ) = tr( b ( I )) ⊆ ( x ) : R I ⊆ tr( I ) . If the residue field of R is infinite, then: b ( I ) = tr( b ( I )) ⊆ core( I ) : R I ⊆ ( x ) : R I ⊆ tr( I ) . Proof.
Since b ( I ) = tr( B ( I )) ∈ Ul I ( R ), the conclusion follows from Proposition 4.19. (cid:4) Corollary 4.21.
Let R be as in Theorem 4.6. Assume that the residue field of R isinfinite. Let M ∈ Ref( R ) . The following are equivalent.(1) M is I -Ulrich.(2) tr( M ) ⊆ b ( I ) .(3) tr( M ) ⊆ ( x ) : R I for some principal reduction x of I .(4) tr( M ) ⊆ ( x ) : R I for any principal reduction x of I .(5) tr( M ) ⊆ core( I ) : R I .Proof. Combining Corollary 4.11 and Proposition 4.19, we see that the proof would becomplete if we show (3) implies (1). If (3) holds, then for any f ∈ M ∗ , we have that I · f ⊆ Hom R ( M, xR ) = xM ∗ . Therefore by Proposition 4.5, M ∗ is I -Ulrich. Since M ∈ Ref R , by Lemma 4.15 M is I -Ulrich. (cid:4) In light of the above results, it is natural to ask when b ( I ) = tr( I ). Note that if this isthe case, then ( x ) : R I is independent of the principal reduction. Proposition 4.22.
Let R be as in Theorem 4.6. If I is a regular reflexive trace idealsuch that xI = I for some x ∈ I , then b ( I ) = tr( I ) .Proof. As I n ∼ = I for all n >
0. we get that B ( I ) = I : I . By Lemma 2.7, b ( I ) = I =tr( I ). (cid:4) We can moreover relate b ( I ) with core I : R I for a large number of cases. Theorem 4.23.
Let R be a reduced one dimensional ring with infinite residue field k . Let I be a regular ideal with reduction number r . Assume that char( k ) = 0 or char( k ) > r .Then b ( I ) = core( I ) : R I. Proof.
By [PU05, Theorem 3.4 b] we have that,core( I ) = x n +1 : R I n for suitably large n , where x is a principal reduction of I . Thus,core( I ) : R I = x n +1 : R I n +1 ∼ = ( I n +1 ) ∗ Now for large n , ( I n +1 ) ∗ is I -Ulrich by Lemma 4.15 and hence b ( I ) = core( I ) : R I byCorollary 4.17. (cid:4) The next proposition will help in establishing some finiteness results.
Proposition 4.24.
Let R be as in Theorem 4.6. Let I, J be regular ideals. Consider thefollowing statements.(1) Ul I ( R ) = Ul J ( R ) .(2) B ( I ) = B ( J ) .(3) b ( I ) = b ( J ) .Then (1) ⇐⇒ (2) = ⇒ (3) . If R is Gorenstein, then all three are equivalent.Proof. Recall that Ul I ( R ) = CM( B ( I )) by Theorem 4.6. So if B ( I ) = B ( J ) then Ul I ( R ) =Ul J ( R ). Now assume (1). Let S = B ( I ) and T = B ( J ). Then S ∈ Ul J ( R ) = CM( T ).Thus T ⊂ T S ⊂ S . By symmetry S ⊂ T , so S = T . If S = T , then c R ( S ) = c R ( T ), so (3)follows from (2). If R is Gorenstein, then B ( I ) , B ( J ) are reflexive, so b ( I ) = b ( J ) implies B ( I ) = B ( J ) by Lemma 2.7. (cid:4) Theorem 4.25.
Let R be as in Theorem 4.6. Let c be the conductor and I be a regularideal. If c ∼ = I s for some s then Ul I ( R ) = CM( R ) . If furthermore R is complete andreduced, then Ul I ( R ) has finite type.Proof. As c is a regular ideal, R is R -finite. As c ∼ = c n for all n , B ( c ) = R . On the otherhand B ( I ) = B ( I s ) = B ( c ), proving the first claim. If R is complete and reduced, then R is a product of DVRs, so Ul I ( R ) = CM( R ) has finite type. (cid:4) Proposition 4.26.
Let R be as in Theorem 4.6. Assume that I is a regular ideal. Let S = End R ( I ) (which is a birational extension of R ). If M is I -Ulrich, then Hom R ( M, I ) ∼ =Hom R ( M, S ) . N REFLEXIVE AND I -ULRICH MODULES OVER CURVE SINGULARITIES 13 Proof.
We have an exact sequence 0 → L → I ⊗ M → IM → L has finitelength. Take Hom R ( − , I ) we get an isomorphism Hom R ( IM, I ) ∼ = Hom R ( I ⊗ M, I ).The first is isomorphic to Hom R ( M, I ) as IM ∼ = M , and the second is isomorphic toHom R ( M, Hom R ( I, I )) = Hom R ( M, S ) by Hom-tensor adjointness. (cid:4)
Corollary 4.27.
Let R be as in Theorem 4.6 and further assume that R has a canonicalideal ω R . The following are equivalent:(1) M ∈ Ul ω R ( R ) (2) Hom R ( M, R ) ∼ = Hom R ( M, ω R ) .Proof. (1) implies (2) by Proposition 4.26. Conversely, note that R ∼ = End R ( ω R ).Hence, using Hom-Tensor adjointness, statement (2) is the same as Hom R ( ω R M, ω R ) ∼ =Hom R ( M, ω R ). Hence dualizing with respect to ω R and using Theorem 4.6 finishes theproof. (cid:4) Corollary 4.28.
Let R be as in Theorem 4.6. Assume that R has a canonical ideal ω R and M ∈ Ul ω R ( R ) . Then M is reflexive.Proof. Corollary 4.27 implies that M ∗ ∼ = M ∨ , where M ∗ = Hom R ( M, R ) and M ∨ =Hom R ( M, ω R ). By Lemma 4.15, M ∗ is still in Ul ω R ( R ), so we have M ∗∗ ∼ = M ∗∨ ∼ = M ∨∨ ∼ = M , as desired. (cid:4) Corollary 4.29.
Let R be as in Theorem 4.6. Suppose that R has a canonical ideal ω R .Then for large enough n , the ideal I = ω nR is reflexive and satisfies I ∗ ∼ = I ∨ . We end this section by looking into the question when the maximal ideal m is I -Ulrich.This will be applied when we discuss almost Gorenstein rings in the last section. Proposition 4.30.
Let ( R, m , k ) be as in Theorem 4.6. Suppose there is an exact sequence → R → I → k ⊕ s → . Then m is I -Ulrich.Proof. We can assume that R is not regular, for if R is regular the conclusion followseasily. The assumption is equivalent to I m ⊂ ( a ) for some a ∈ I . We need to show that I m = a m . As a m ⊂ I m ⊂ ( a ) and ℓ (( a ) /a m ) = 1, it is enough to show that I m is notequal to ( a ). Suppose that I m = ( a ). Then E = I m : I m = ( a ) : ( a ) = R . As m : m ⊂ E ,it follows that m : m = R . But m : m ∼ = m ∗ , and as m is reflexive, it follows that m ∼ = R ,which is impossible if R is not regular. (cid:4) Applications: Strongly reflexive ring extensions
Throughout this section, we assume that ( R, m , k ) is a one-dimensional Cohen-Macaulay local ring. Suppose that R has a canonical ideal ω R . In this section we areinterested in the following question. Let S be a finite extension of R . When is any CM S -module R -reflexive? Of course if S = R , this is equivalent to R being Gorenstein, or ω R ∼ = R . It turns out that there is a pleasant generalization to any finite extension S thatis in CM( R ): CM( S ) ⊂ Ref( R ) if and only if ω R S ∼ = S , in other words S is ω R -Ulrich.We start with a useful lemma that will be used repeatedly in the proof of our maintheorem. Lemma 5.1.
Let ( R, m , k ) be a one-dimensional Cohen-Macaulay local ring. Let S be amodule finite R -algebra such that S ∈ CM( R ) . Let M ∈ CM( S ) . (1) The map F : M Hom R ( M, R ) is an S -linear, contravariant functor from CM( S ) to CM( S ) .(2) If M ∈ Ref( R ) , then M ∼ = F ( F ( M )) in CM( S ) .Proof. Note that Hom R ( M, R ) is naturally an S -module via the action( s · f )( m ) := f ( sm ) , s ∈ S, m ∈ M, f ∈ Hom R ( M, R )and that this extends the action of R . Thus, the conclusion of part (1) follows. For part(2), notice that the canonical R -linear map M → M ∗∗ is S -linear with respect to theaction above. (cid:4) Theorem 5.2.
Let ( R, m , k ) be a one-dimensional Cohen-Macaulay local ring. Assumethat R has a canonical ideal ω R . Let S be a module finite R -algebra such that S ∈ CM( R ) .The following are equivalent:(1) CM( S ) ⊂ Ref( R ) .(2) ω S ∈ Ref( R ) .(3) ω S ∼ = S ∗ .(4) ω R S ∼ = S .(5) S is ω R -Ulrich.(6) S is reflexive and tr( S ) ⊂ b ( ω R ) .Proof. Clearly (1) implies (2). For the converse, assume that ω S ∈ Ref( R ). Take M ∈ CM( S ). Then M ∨ = Hom R ( M, ω R ) ∈ CM( S ). Take a free S -cover of M ∨ and apply ∨ ,we obtain an exact sequence 0 → M → ( S n ) ∨ → N → R ). Thus M ∈ Ref( R )by Lemma 2.5.That (3) implies (2) follows by Lemma 2.5. For the converse, assume that ω S ∈ Ref( R ).Take a free S -cover of ω ∗ S and apply ∗ . From Lemma 5.1 we get an exact sequence inCM( S ): 0 → ω S → ( S n ) ∗ → N →
0. (Note that N is in CM( S ) as it is a submoduleof a torsionfree R -module and also has an S -module structure.) This has to split inCM( S ) (since Ext S ( N, ω S ) = 0), so that ω S is a direct summand of ( S n ) ∗ in CM( S ).Since S ∈ Ref( R ) by (2) implies (1), ω ∗ S is a direct summand of S n in CM( S ) usingLemma 5.1(2). Thus ω ∗ S is S -projective. Since S is semi-local, ω ∗ S is S -free of rank one.Now applying Lemma 5.1(1) one gets that ω ∗∗ S is isomorphic to S ∗ as S -modules, so as R -modules as well and hence (3) follows.The equivalence of (3), (4) and (5) follows from Theorem 4.6 and Corollary 4.27. Theequivalence of (5) and (6) follows from Corollary 4.21. (cid:4) Definition 5.3.
We shall call an extension of R satisfying the equivalent conditions ofTheorem 5.2 a strongly reflexive extension. Remark 5.4.
The notions of reflexive extensions and totally reflexive extensions, overnot necessarily commutative rings, have been defined and studied by X. Chen in [Che13,Definition 2.3, 3.3]. They are related to but not the same as ours. For instance, a reflexiveextension in Chen’s notion would require S ∈ Ref( R ) and Hom R ( S, R ) ∼ = S , and wouldimply Ref( S ) ⊂ Ref( R ).Strongly reflexive birational extensions satisfy even more interesting characterizations. N REFLEXIVE AND I -ULRICH MODULES OVER CURVE SINGULARITIES 15 Theorem 5.5.
Let ( R, m , k ) be a one-dimensional Cohen-Macaulay local ring. Let S ∈ Bir( R ) . Assume R has a canonical ideal ω R admitting a principal reduction. Let a be anarbitrary principal reduction of ω R and set K := a − ω R (see Remark 5.6). The followingare equivalent:(1) CM( S ) ⊂ Ref( R ) .(2) ω S ∈ Ref( R ) .(3) ω S ∼ = S ∗ .(4) S is ω R -Ulrich.(5) S = KS .(6) K ⊆ S .(7) S ∈ Ref( R ) and c R ( S ) = tr( S ) ⊆ R : K = ( a ) : ω R .(8) S ∈ Ref( R ) and c R ( S ) = tr( S ) ⊆ core( ω R ) : R ω R (assuming the residue field isinfinite).Proof. We already know that (1) through (4) are equivalent from Theorem 5.2. Theequivalence of (4) and (5) follows from Proposition 4.5 and that of (4) and (6) fromRemark 4.4 and Corollary 4.9. Since a is an arbitrary principal reduction of ω R we seethat (7) holds if and only if (8) holds, as the core is the intersection of all principalreductions if the residue field is infinite.The equivalence of (4), (7), (8) follows from Corollary 4.21. (cid:4) Remark 5.6.
The condition that R has a canonical ideal with principal reduction issatisfied for instance when ˆ R is generically Gorenstein with infinite residue field, see[GMP13]. Corollary 5.7.
Let R be as in Theorem 5.2. Assume that R has a canonical ideal ω R .Let Q ( R ) ֒ → A be an extension of the total quotient ring of R . Assume that the integralclosure of R in A , say ¯ R A , is a finite R -module. Then ¯ R A ∈ Ref( R ) .Proof. From Corollary 4.10, ¯ R ∈ Ul ω R ( R ). Since ¯ R A ∈ CM( ¯ R ), by Theorem 5.2 ¯ R A ∈ Ref( R ). (cid:4) Corollary 5.8.
Let R → S be a finite extension of rings such that R is a genericallyGorenstein ( S ) ring of arbitrary dimension and S is ( S ) . If the extension R → S isstrongly reflexive in codimension one, then any finite ( S ) S -module M is R -reflexive.Proof. Since R satisfies ( S ) and M is a ( S ) R -module, M is R -reflexive if and only ifthis is true in codimension one. Since R is generically Gorenstein and S is ( S ), we mayapply Theorem 5.2 to see M is R -reflexive in codimension one. (cid:4) Corollary 5.9.
Let R be a generically Gorenstein ( S ) ring of arbitrary dimension. Let Q ( R ) ֒ → A be an extension of the total quotient ring of R . Assume that the integralclosure of R in A , say ¯ R A , is a finite R -module. Then ¯ R A ∈ Ref( R ) .Proof. Since R → ¯ R is strongly reflexive in codimension one, by Corollary 5.8 ¯ R A ∈ Ref( R ). (cid:4) Applications: Finiteness results
Throughout this section, we assume that ( R, m , k ) is a one-dimensional Cohen-Macaulay local ring. Here we study when certain subsets of interesting ideals and modulesare “finite”. We say that a subset S of mod( R ) is of finite type if any element of S is isomorphic to a direct sum of modules from a finite set in mod( R ). Note that since wesometimes consider sets that are not subcategories which are closed under isomorphism,this notion is a bit broader than the usual notion of “finite representation type”. Rep-resentation finiteness of subcategories of CM( R ) have been studied heavily, and manybeautiful connections to the singularities of R have been discovered over the years. Ourstudy suggests that the same promise could hold for reflexive modules.Consider the following classes of ideals of R : I ( R ) := { I | I is an integrally closed regular ideal } I c ( R ) := { I | I is an integrally closed regular ideal and c ⊆ I } Ref ( R ) := { I | I is a reflexive regular ideal } . T( R ) := { I | I is a regular trace ideal } We shall look at the finiteness of these classes of ideals and the interaction betweenthem. Note that from Proposition 3.11, we have that I ⊆ Ref ( R ) and that I c ⊆ RT( R ) := Ref ( R ) ∩ T( R ).6.1. Finiteness of T( R ) . We begin by answering the following question raised by E.Faber in [Fab19, Question 3.7].
Question 6.1.
Let R be a one-dimensional complete local or graded ring. Are the follow-ing equivalent?(1) CM( R ) is of finite type.(2) There are only finitely many possibilities for tr( M ) , where M ∈ CM( R ) . The answer to this question is negative. Consider the following example.
Example 6.2.
Let R = k [[ t e , . . . , t e − ]] where R = k [[ t ]], k infinite and e ≥
4. Then theset of trace ideals is finite but CM( R ) is infinite. Proof.
Here c = m . By Corollary 3.6, there are exactly two trace ideals, R and m . Since R is an m -Ulrich R -module, µ R ( R ) = e ( R ). However, since e ( R ) = e ≥
4, CM( R ) isinfinite by [LW12, Theorem 4.2]. (cid:4) Note here that finitely many trace ideals in a ring can raise some natural classificationquestions. Of course, a single trace ideal characterizes a DVR. The following propositionprovides a strong motivation to classify such rings.
Proposition 6.3.
Let ( R, m , k ) be a complete local one-dimensional domain containingan infinite field k , so that R = k [[ t ]] . Let e ( R ) = e and let v be the valuation defining R .(1) R ) = 2 if and only if R = k [[ t e , . . . , t e − ]] .(2) The following are equivalent(a) R ) = 3 (b) R = k [[ αt e , t c , t c +1 , . . . , c t e , . . . , t c + e − ]] where α is a unit of k [[ t ]] and e + 2 ≤ c ≤ e .(c) ℓ ( R/ c ) = 2 Proof.
Note that every integrally closed ideal in R is of the form I f := { r ∈ R | v ( r ) ≥ f } where f ∈ N . Then m = I e and let c = I c where c is chosen maximally, that is c = I c +1 .For (1), first assume R ) = 2. Here m and R are the only trace ideals and so c = m . Hence, e = c and choose t e + P i β i t i ∈ m , β i ∈ k , so that it is part of a minimal N REFLEXIVE AND I -ULRICH MODULES OVER CURVE SINGULARITIES 17 generating set for m . Since c = m , we have that t e + j + t j P i β i t i ∈ m for all j ≥
1. Since R is complete, we have that t e + j ∈ m for all j ≥ R = k [[ t e , . . . , t e − ]]. Theother direction is clear using Corollary 3.6.For (2), first assume R ) = 3. By Proposition 3.11, we get that there are nointegrally closed ideals strictly between c and m . In other words there does not exist r ∈ R such that v ( r ) = f for all e < f < c . Since R is complete, c = ( t c , t c +1 , . . . , t c + e − ).Thus we can choose a principal reduction x = t e + c − e − P i =1 k i t e + i ∈ R of m where k i ∈ k .Consider the ideal I := ( x ) + c . We claim that I = m . Take any element r ∈ m . If v ( r ) > e , then r ∈ c ⊆ I . If v ( r ) = e , to show r ∈ I , after multiplication by a suitableelement of k we may assume that r = t e + c − e − P i =1 b i t e + i where b i ∈ k . If v ( r − x ) = 0, thennecessarily e < v ( r − x ) < c , which is impossible. Therefore r = x and I = m . Finally wehave 2 e ≥ c since there does not exist any element in R with valuation strictly between e and c . Since c = m , we have that c ≥ e + 2.To show (b) implies (c), assume R has the specified form. Then since c =( t c , t c +1 , . . . , t c + e − ), we have that m / c is a cyclic R -module. Moreover since c ≤ e , m ⊆ c and m / c is a k -vector space. Thus ℓ ( m / c ) = 1, that is ℓ ( R/ c ) = 2.(c) implies (a) is clear from Corollary 3.6. (cid:4) Finiteness of I and I c .Proposition 6.4. Let R be a one-dimensional Cohen-Macaulay local ring. Suppose R isa finite R -module. Then I c is a finite set. Moreover if R is a complete local domain, I is of finite type.Proof. Let MaxSpec( R ) = { n , ..., n s } . Since c is a regular ideal of R , choose an irre-dundant primary decomposition in R , c = ∩ si =1 n ( r i ) i where I ( n ) denotes the n th symbolicpower of an ideal I . Since any J ∈ I c is the contraction to R of an ideal of R containing c , J = ∩ si =1 ( n ( s i ) i ∩ R ) where 1 ≤ s i ≤ r i for each i . Thus I c is a finite set.Now assume R is a complete local domain, so that R is a DVR; thus the elements of I are totally ordered by inclusion. From the first part of this proposition, it suffices toconsider I ∈ I , I ⊆ c . Now I = IR ∩ R , but IR ⊆ c R = c . Therefore I is also an idealof R and there exists 0 = a ∈ R such that aR = I . Therefore I ∼ = R as R -modules and I has finite type. (cid:4) Finiteness of
Ref ( R ) . We first note that Ref( R ) (in fact Ref ( R )) is of infinitetype if R is not finitely generated over R . Lemma 6.5.
Let R be a one-dimensional Cohen-Macaulay local ring. Let I be a regularideal. Then B ( I ) is a finite R -module.Proof. From Example 4.2, I n is I -Ulrich for sufficiently large n . Thus I n +1 ≃ I n for such n by Theorem 4.6. Therefore End R ( I n ) stabilizes and B ( I ) is a finite R -module. (cid:4) Lemma 6.6.
Let R be a one-dimensional Cohen-Macaulay local ring. Assume Ref ( R ) has finite type and that R admits a canonical ideal ω R . Then R is a finite R -module. Inparticular, R is reduced. Proof.
It suffices to show that R is a finite R -module. Suppose on the contrary that it isnot. By Lemma 6.5, R is not a finite B ( ω R )-module. Thus, we can find an infinite chainof rings S i inside R , B ( ω R ) ( S ( · · · ( S i ( . . . such that each S i is a finite R -module.From Corollary 4.9, the S i are ω R -Ulrich and hence by Theorem 5.2, they are R -reflexive.Consider S i ( S j , and let if possible S i ≃ S j as R -modules. Then they are isomorphic as S i -modules as well. By Theorem 2.8, S i = tr S i ( S j ) ⊆ c S i ( S j ). So S i = S j , a contradiction.Therefore the S i ’s are indecomposable and mutually non-isomorphic and hence, Ref ( R )is not of finite type. (cid:4) We prove next that Ref ( R ) is of finite type when the conductor has small colength.Before stating Theorem 6.8, we summarize the cases that we will always reduce to inthe proof. Lemma 6.7 (Reduction Lemma) . Let ( R, m , k ) be a one-dimensional Cohen-Macaulaylocal ring. Assume R is finite over R and let c be the conductor ideal. Further assumethat k is infinite. For any ideal I , consider the following conditions:(1) c ⊂ I ,(2) c ( x : R I ( x : R I ( m where x is a principal reduction of I .Let Ref ′ ( R ) := { I ∈ Ref ( R ) | I satisfies (1) and (2) above } . Then Ref ( R ) is of finitetype if and only if Ref ′ ( R ) is of finite type.Proof. If Ref ( R ) is of finite type then certainly Ref ′ ( R ) is of finite type. Converselyassume that Ref ′ ( R ) is of finite type. Let I ∈ Ref ( R ) and I not principal. FromTheorem 3.5 we may assume that c ⊆ I . By Corollary 4.10, we have c ⊆ x : R I ⊆ x : R I ⊆ m If c = x : R I , then by Remark 2.4, ( I ) ∗ ∼ = c and hence c ∼ = ¯ I . But both are traceideals, and hence c = I = I . Similarly if x : R I = m , by Remark 2.4 we have I ∼ = m ∗ .Finally, if x : R I = x : R I , we get I ∗ = ( I ) ∗ by Remark 2.4. Thus I = I and hence I ∈ I c . Combining the above observations and finally using Proposition 6.4, we havethat Ref ( R ) is of finite type. (cid:4) Theorem 6.8.
Let ( R, m , k ) be a one-dimensional Cohen-Macaulay local ring. Let R befinite over R and let c be the conductor ideal. Further assume that k is infinite. Considerthe following.(1) ℓ ( R/ c ) ≤ (2) ℓ ( R/ c ) = 4 and R has minimal multiplicity.Then in all the above cases, Ref ( R ) is of finite type.Proof. (1) follows immediately from Lemma 6.7.Suppose (2) holds. Note that by Lemma 6.7, we can assume that I ∈ Ref ′ ( R ) and ℓ ( R/x : R I ) = 2 where x is a principal reduction of I .Since J = x : R I is reflexive and R has minimal multiplicity, by Proposition 3.12 weget that J is integrally closed. Thus, I ∼ = J ∗ where J ∈ I c and the proof is now completeby Proposition 6.4. (cid:4) Corollary 6.9.
Let R be a complete one-dimensional local domain containing an infinitefield such that R ) = 3 . Then Ref ( R ) is of finite type.Proof. This follows from Proposition 6.3 and Theorem 6.8. (cid:4)
N REFLEXIVE AND I -ULRICH MODULES OVER CURVE SINGULARITIES 19 Remark 6.10.
Theorem 6.8 is true if we only assume that | Min( ˆ R ) | ≤ | k | . To see this,first note that since R is one dimensional and CM, ¯ R is a finite R -module if and onlyif R is analytically unramified ( see for example [LW12, Theorem 4.6]). In this case,¯ˆ R = ¯ R ⊗ R ˆ R , so c ˆ R ⊆ c ˆ R . Therefore l ( ˆ R/ c ˆ R ) ≤ l ( ˆ R/ c ˆ R ) = l ( R/ c ). Since ˆ R is reduced, thenumber of maximal ideals in its integral closure is equal to its number of minimal primes.By [FO18, Corollary 3.3], every ideal of ˆ R admits a principal reduction. Then from theargument in Theorem 6.8, Ref ( ˆ R ) has finite type. By [Gro67, Proposition 2.5.8], Ref ( R )has finite type. Remark 6.11.
Since c ∈ I c , by Theorem 3.5 and [GHK03, Proposition 2.9] if R/ c isGorenstein, then Ref ( R ) is of finite type. Moreover in this case, by [CHV98, Theorem3.7], we have that µ ( c ) = µ ( m ), so that R necessarily has minimal multiplicity here. Inparticular finiteness of Ref ( R ) in the cases ℓ ( R/ c ) ≤ R necessarily has minimal multiplicity.6.4. Finiteness of
Ref( R ) . In this subsection, we give a criterion for finiteness of Ref( R )and derive that over seminormal singularities, the category of reflexive modules is of finitetype. Proposition 6.12.
Let ( R, m ) be a one-dimensional Cohen-Macaulay local ring. Let S = End R ( m ) . If CM( S ) is of finite type, then Ref( R ) is of finite type.Proof. Let M be an indecomposable, non-free reflexive module over R . Then tr( M ) ⊂ m = c R ( S ), so M ∈ CM( S ) by Theorem 2.8. Finally, note that if two S -modules areisomorphic, then they are also isomorphic as R -modules. (cid:4) Corollary 6.13.
Assume that ( R, m ) is complete, reduced, one-dimensional and the con-ductor c of R is equal to m . Then Ref( R ) is of finite type.Proof. By assumption S = End R ( m ) = ¯ R , which is a product of DVRs and thereforeCM( S ) is of finite type. Thus, Proposition 6.12 applies. (cid:4) As a consequence of the above, we can study finiteness of Ref( R ) for ‘seminormal’ rings.R. Swan [Swa80] defined a seminormal ring as a reduced ring R such that whenever b, c ∈ R satisfy b = c , there is an a ∈ R with a = b, a = c . For a detailed exposition on variousresults related to seminormality (including generalizations to the above definition), werefer the reader to [Vit11]. Seminormality has also been studied in the context of studying F -singularities in characteristic p >
0. For instance, F -injective rings constitute a classof examples for seminormal rings [DM19, Corollary 3.6]. Corollary 6.14.
Suppose that ( R, m ) is a seminormal complete reduced local ring ofdimension one. Then Ref( R ) is of finite type.Proof. By [Vit11, Proposition 2.10(1)] (with A = R, B = R ), we get that c = m , soCorollary 6.13 applies. (cid:4) Further applications, examples and questions
We begin this section by discussing notions of being ‘close to Gorenstein’ as promisedin Remark 3.1. Throughout this section, ( R, m , k ) denotes a one-dimensional CohenMacaulay local ring. Definition 7.1. R is called almost Gorenstein if a : R ω R ⊇ m for some principalreduction a of ω R . R is called nearly Gorenstein if tr( ω R ) ⊇ m .These classes of rings have attracted a lot of attention lately, the reader can refer to[HHS19], [BF97], [GMP13], [GTT15], [DKT20] amongst other sources.In our language: Proposition 7.2.
Assume that ( R, m ) is a one-dimensional Cohen-Macaulay local ringwhich has a canonical ideal ω R with some principal reduction a . R is almost Gorensteinif and only if m is ω R -Ulrich.Proof. This follows from Proposition 4.30. (cid:4)
It is clear from Corollary 3.8 that in this situation we get tr( ω R ) ⊇ m . This provides aproof for a well-known fact that almost Gorenstein rings are nearly Gorenstein.One would expect that for rings close to Gorenstein, it would be easier to find reflexivemodules. We now give supporting evidence for that statement. Proposition 7.3.
Let ( R, m ) be almost Gorenstein and let I be a regular ideal with S =End R ( I ) .(1) If S is reflexive and strictly larger than R , then CM( S ) ⊆ Ref( R ) . Thus IM ∈ Ref( R ) for any M ∈ CM( R ) . In particular all powers of I are reflexive.(2) If I is a proper trace ideal, then IM is reflexive for any M ∈ CM( R ) . In particular I and all of its powers are reflexive.Proof. As S is reflexive and c R ( S ) ⊂ m ⊆ R : ω R a by hypothesis, we are done by Theo-rem 5.5. Since IM ∈ CM( S ), the proof of (1) is complete. For part (2), just note that S = I ∗ is reflexive and not equal to R . (cid:4) In particular, if R is almost Gorenstein, m n is reflexive for all n . Remark 7.4.
Suppose R is almost Gorenstein but not Gorenstein, and take I = ω R . I isnot reflexive. Note that here End R ( I ) is reflexive but does not contain R properly. Thusthe conditions on I in Proposition 7.3 are needed.The following example shows that in general, m can fail to be reflexive. Example 7.5.
Let R = k [[ t , t , t ]]. Then m = ( t , t , t , t , t , . . . ). Thus ( m ) ∗ =(1 , t , t , t , . . . ) and t ∈ ( m ) ∗∗ . But t / ∈ m .With a bit more work one can even find an example where none of m n , n ≥ Example 7.6.
Let R = k [[ t , t , t , t , t ]]. Then m = ( t , t , t , . . . ). Thus, ( m ) ∗ = t − ( t , t , t , t , t , t ) = t − c . Thus, t ∈ ( m ) ∗∗ but t m , so m is not reflexive.However, note that t is a minimal reduction of m and t m = m . Thus, none of thehigher powers of m can be reflexive.Next, we classify when Ref( R ) is of finite type for almost Gorenstein rings. Proposition 7.7.
Suppose that ( R, m ) is almost Gorenstein. Let S = End R ( m ) . Then Ref( R ) is of finite type if and only if CM( S ) is of finite type.Proof. The ‘if’ direction is Proposition 6.12. The other direction follows from Proposi-tion 7.3(1). (cid:4)
N REFLEXIVE AND I -ULRICH MODULES OVER CURVE SINGULARITIES 21 Remark 7.8.
Let S = End R ( m ). Using the notations in [Kob17], we thus get CM( S ) ⊂ Ref( R ) = Ω CM( R ), so CM( S ) = Ω CM ′ ( R ) by Theorem 2.8. It follows that Ω CM ′ ( R )has finite type if and only if CM( S ) has finite type. This recovers results by T. Kobayashi[Kob17, Corollary 1.3]. Proposition 7.9.
Let ( R, m, k ) be a one-dimensional, reduced, complete local ring con-taining Q and further assume that k is algebraically closed. Consider the following state-ments. (a) CM( R ) is of finite type. (b) Ref( R ) is of finite type. (c) Ref ( R ) is of finite type. (d) RT( R ) is finite.If R is Gorenstein, then all the four statements are equivalent. If R is an almost Goren-stein domain, then ( b ) , ( c ) and ( d ) are equivalent.Proof. Assume first that R is Gorenstein. Clearly, ( a ) = ⇒ ( b ) = ⇒ ( c ) = ⇒ ( d ).Now suppose CM( R ) is not of finite type. Then by [LW12, Theorem 4.13(ii)(a)] andLemma 2.7, we get RT( R ) is not of finite type. This completes the first part of the proof.Next assume that R is an almost Gorenstein domain. We only need to show ( d ) = ⇒ ( b ). Assume that Ref( R ) is of infinite type, and hence CM( S ) is of infinite type byProposition 7.7, where S = End R ( m ). Thus, there are infinitely many non-isomorphicfinite reflexive birational extensions of R by [LW12, Theorem 4.13(ii)(a)] and by Propo-sition 7.3(1). The proof is now complete using Lemma 2.7. (cid:4) Next, we classify birational reflexive extensions of R that are Gorenstein. Our resultwas inspired by and extends [GMP13, Theorem 5.1]. Theorem 7.10.
Suppose that R is a one-dimensional Cohen-Macaulay local ring with acanonical ideal ω R . Let S ∈ Ref( R ) be a birational extension of R . Let I = c R ( S ) . Thefollowing are equivalent:(1) S is Gorenstein.(2) I is I -Ulrich and ω R -Ulrich. That is I ∼ = I ∼ = Iω R .Proof. Note that I is a trace ideal by Lemma 2.7. Suppose (1) holds. Then S = End R ( I ) =End S ( I ) so I ∼ = S by [Kob17, Lemma 2.9]. Thus I = aS for some a ∈ I , necessarilya regular element, and since S = S , we have aI = I . By Corollary 3.8, we have I = tr( I ) ⊇ ( a ) : R I ⊇ I , so I = ( a ) : R I . Thus I ∼ = I ∗ . However, since S is Gorenstein,we have S ∼ = S ∨ , so S ∗ ∼ = S ∨ . By Corollary 4.27, S , and hence I is ω R -Ulrich.Assume (2). We can assume that R has infinite residue field and thus I has a reduction a . Thus I = aI , and the same argument in the preceding paragraph shows that I ∼ = I ∗ ∼ = S . So S is ω R -Ulrich, which (by Corollary 4.27) implies S ∨ ∼ = S ∗ ∼ = S , thus S isGorenstein. (cid:4) Corollary 7.11. [GMP13, Theorem 5.1]
Suppose that ( R, m ) is a one-dimensional Cohen-Macaulay local ring with a canonical ideal ω R . Let S = End R ( m ) . The following areequivalent:(1) S is Gorenstein.(2) R has minimal multiplicity and is almost Gorenstein. Proof.
Note that minimal multiplicity and almost Gorenstein are just m being m -Ulrichand ω R -Ulrich, respectively. (cid:4) Example 7.12. (trace ideals are not always reflexive) Let R = k [[ t , t , t ]]. Here c = m .Let I = ( t , t ). Then I ∈ T( R ) but I ∗∗ ∼ = ( t , t , t ) and hence I RT( R ). Proof. c = m is clear. A straight forward computation gives J := ( t : ( t , t )) =( t , t , t ), so that tr( I ) = II ∗ = I ( t ) − J = ( t , t ). Therefore I ∈ T( R ).However, (( t ) : R J ) = m . So I is not reflexive. (cid:4) Example 7.13.
Let R = k [[ t , t , t ]], which is a complete intersection domain of multi-plicity 4. The conductor c is m , with colength 4. The set RT( R ) is infinite and classifiedby [GIK20, Example 3.4(i)]. We note a few features that illustrate our results:(1) It shows that the category of reflexive (regular) ideals is not of finite type, soour Theorem 6.8 is sharp, as the conductor has colength 4 but R does not haveminimal multiplicity.(2) The finite set of integrally closed ideals are { ( t n ) n ≥ = m , ( t n ) n ≥ , ( t n ) n ≥ , ( t n ) n ≥ = c } .(3) The rest is the infinite family { I a = ( t − at , t ) , a ∈ k } . We have t = ( t − at ) t + at t which shows that t ∈ I a and so I a = m . So none of the I a areintegrally closed. However let S = End R ( m ) = k [[ t , t , t , t ]]. Then ℓ ( S/I a ) = 3and ℓ ( R/I a ) = 2, so I a S ∩ R = I a by Proposition 3.13. In other words all traceideals in R are contracted from S . Example 7.14.
Let R = k [[ t , t , t , t ]]. Then clearly c = ( t , t , t , t ) and thereforeT( R ) = { c , m , R } . Thus we are in the situation of Corollary 6.9.Next note that End R ( m ) = k [[ t , t ]]. By [LW12, Theorem 4.18], CM(End( m )) is offinite type. Hence Ref( R ) is of finite type by Proposition 6.12. Example 7.15 ( Reflexivity is not preserved under going modulo a non-zerodivisor in general ) . Let M ∈ Ul( R ) ∩ Ref( R ). Let l be a principal reduction of m .Then M/lM is a finite dimensional k -vector space. Since, R/l is Artinian, k is reflexive ifand only if R is Gorenstein (recall that Hom R ( k, R ) is the non-zero socle if R is Artinian).So, if R is not Gorenstein, then M/lM Ref(
R/l ).We end the paper with a number of open questions. The following finiteness questionsare very natural:
Question 7.16.
Let R be a complete Cohen-Macaulay local ring of dimension one.(1) Can we classify when R has finitely many trace ideals?(2) Can we classify when Ref ( R ) is of finite type?(3) Can we classify when Ref( R ) is of finite type? Since the trace of a reflexive ideal is a natural place to look for reflexive trace ideals,we have the following question.
Question 7.17.
Let R be a Cohen-Macaulay local ring of dimension one. If an ideal I ⊆ R is reflexive, is tr( I ) reflexive? As evidenced by Theorem 6.8 and Example 7.13, the first obstruction seems to be incolength two.
N REFLEXIVE AND I -ULRICH MODULES OVER CURVE SINGULARITIES 23 Question 7.18.
Let R be a Cohen-Macaulay local ring of dimension one. When is anideal of colength two a trace ideal? When is it reflexive? We conclude with the following question about rings of positive characteristic.
Question 7.19.
Let R be a Cohen-Macaulay local ring of dimension one. Suppose R hascharacteristic p > . When does R /q belong to Ref( R ) for q = p i large enough? The answer is always positive for one-dimensional complete local domains with alge-braically closed residue field where for large q , R /q becomes a module over R and thus R -reflexive. Acknowledgments
This project is a part of the Kansas Extension Seminar ( http://people.ku.edu/~hdao/CAS.html ).We thank the Department of Mathematics at the University of Kansas for partial sup-port. The first author was partly supported by Simons Collaboration Grant FND0077558.We thank Craig Huneke, Daniel Katz, Takumi Murayama and Ryo Takahashi for helpfulcomments on the topics of the paper. We are also grateful to Toshinori Kobayashi forpointing out Proposition 7.9.
References [AH96]
Ian Aberbach and Craig Huneke . A theorem of Brian¸con-Skoda type for regular localrings containing a field . Proceedings of the American Mathematical Society , 124(3):707–713,1996.[Bas60]
Hyman Bass . Finitistic dimension and a homological generalization of semi-primary rings . Trans. Amer. Math. Soc. , 95:466–488, 1960. ISSN 0002-9947. doi:10.2307/1993568.[BF97]
Valentina Barucci and Ralf Fr¨oberg . One-dimensional almost gorenstein rings . Jour-nal of Algebra , 188(2):418–442, 1997.[BH98]
Winfried Bruns and H. J¨urgen Herzog . Cohen-Macaulay Rings . Cambridge Stud-ies in Advanced Mathematics. Cambridge University Press, 2 edition, 1998. doi:10.1017/CBO9780511608681.[BHU87]
Joseph P Brennan, J¨urgen Herzog and Bernd Ulrich . Maximally generated cohen-macaulay modules . Mathematica Scandinavica , 61:181–203, 1987.[Bou65]
N Bourbaki . Diviseurs, in “Alg`ebre Commutative” . Chap. VII. Hermann, Paris , 1965.[BP95]
Valentina Barucci and Kerstin Pettersson . On the biggest maximally generated idealas the conductor in the blowing up ring . manuscripta mathematica , 88(1):457–466, 1995.[Che13] Xiao-Wu Chen . Totally reflexive extensions and modules . J. Algebra , 379:322–332, 2013.ISSN 0021-8693. doi:10.1016/j.jalgebra.2013.01.014.[CHKV05]
Alberto Corso, Craig Huneke, Daniel Katz and Wolmer V Vasconcelos . Integralclosure of ideals and annihilators of homology . Commutative algebra , 244:33–48, 2005.[CHV98]
Alberto Corso, Craig Huneke and Wolmer V Vasconcelos . On the integral closureof ideals . manuscripta mathematica , 95(1):331–347, 1998.[D + Jean Dieudonn´e et al.
Remarks on quasi-frobenius rings . Illinois Journal of Mathematics ,2(3):346–354, 1958.[DKT20]
Hailong Dao, Toshinori Kobayashi and Ryo Takahashi . Trace of canonical mod-ules, annihilator of ext, and classes of rings close to being gorenstein . arXiv preprintarXiv:2005.02263 , 2020.[DM19] Rankeya Datta and Takumi Murayama . Permanence properties of f -injectivity . arXivpreprint arXiv:1906.11399 , 2019.[Fab19] Eleonore Faber . Trace ideals, Normalization Chains, and Endomorphism Rings . arXivpreprint arXiv:1901.04766 , 2019. [FO18] Louiza Fouli and Bruce Olberding . Generators of reductions of ideals in a local noe-therian ring with finite residue field . Proceedings of the American Mathematical Society ,146(12):5051–5063, 2018.[GHK03]
Shiro Goto, Futoshi Hayasaka and Satoe Kasuga . Towards a theory of Gorenstein m -primary integrally closed ideals . In Commutative algebra, singularities and computer algebra(Sinaia, 2002) , volume 115 of
NATO Sci. Ser. II Math. Phys. Chem. , pages 159–177. KluwerAcad. Publ., Dordrecht, 2003.[GIK20]
Shiro Goto, Ryotaro Isobe and Shinya Kumashiro . Correspondence between TraceIdeals and birational extensions with application to the analysis of the Gorenstein property ofrings . J. Pure Appl. Algebra , 224(2):747–767, 2020. ISSN 0022-4049. doi:10.1016/j.jpaa.2019.06.008.[GMP13]
Shiro Goto, Naoyuki Matsuoka and Tran Thi Phuong . Almost Gorenstein rings . Journal of Algebra , 379:355–381, 2013.[GOT + Shiro Goto, Kazuho Ozeki, Ryo Takahashi, Kei-Ichi Watanabe and Ken-IchiYoshida . Ulrich ideals and modules . In
Mathematical Proceedings of the Cambridge Philo-sophical Society , volume 156, pages 137–166. Cambridge University Press, 2014.[GOT + Shiro Goto, Kazuho Ozeki, Ryo Takahashi, Kei-ichi Watanabe and Ken-ichiYoshida . Ulrich ideals and modules over two-dimensional rational singularities . NagoyaMathematical Journal , 221(1):69–110, 2016.[Gro67]
A. Grothendieck . ´El´ements de g´eom´etrie alg´ebrique. IV. ´Etude locale des sch´emas et desmorphismes de sch´emas IV . Inst. Hautes ´Etudes Sci. Publ. Math. , (32):361, 1967. ISSN 0073-8301.[GTT15]
Shiro Goto, Ryo Takahashi and Naoki Taniguchi . Almost gorenstein rings–towards atheory of higher dimension . Journal of Pure and Applied Algebra , 219(7):2666–2712, 2015.[HHS19]
J¨urgen Herzog, Takayuki Hibi and Dumitru I Stamate . The trace of the canonicalmodule . Israel Journal of Mathematics , 233(1):133–165, 2019.[HK71]
J¨urgen Herzog and E Kunz . Der kanonische modul eines Cohen-Macaulay-Rings (sem.,regensburg, 1970/71) . Lecture Notes in Math , 238, 1971.[HS06]
Craig Huneke and Irena Swanson . Integral Closure of Ideals, Rings, and Modules , vol-ume 336 of
London Mathematical Society Lecture Note Series . Cambridge University Press,Cambridge, 2006. ISBN 978-0-521-68860-4; 0-521-68860-4.[HUB91]
Jurgen Herzog, Bernd Ulrich and J¨orgen Backelin . Linear maximal cohen-macaulaymodules over strict complete intersections . Journal of Pure and Applied Algebra , 71(2-3):187–202, 1991.[Kob17]
Toshinori Kobayashi . Syzygies of Cohen-Macaulay modules over one dimensional Cohen-Macaulay local rings . arXiv preprint arXiv:1710.02673 , 2017.[KT19a] Toshinori Kobayashi and Ryo Takahashi . Rings whose ideals are isomorphic to traceideals . Mathematische Nachrichten , 292(10):2252–2261, 2019.[KT19b]
Toshinori Kobayashi and Ryo Takahashi . Ulrich modules over cohen–macaulay localrings with minimal multiplicity . Quarterly Journal of Mathematics , 70(2):487–507, 2019.[KV18]
Andrew R Kustin and Adela Vraciu . Totally reflexive modules over rings that are closeto gorenstein . Journal of Algebra , 2018.[Lin17]
Haydee Lindo . Trace ideals and centers of endomorphism rings of modules over commutativerings . J. Algebra , 482:102–130, 2017. ISSN 0021-8693. doi:10.1016/j.jalgebra.2016.10.026.[Lip71]
Joseph Lipman . Stable ideals and arf rings . American Journal of Mathematics , 93(3):649–685, 1971.[LW12]
Graham J Leuschke and Roger Wiegand . Cohen-Macaulay representations . 181. Amer-ican Mathematical Soc., 2012.[Mai20]
Sarasij Maitra . Partial trace ideals and Berger’s Conjecture . arXiv preprintarXiv:2003.11648 , 2020.[Mas98] VLADIMIR Masek . Gorenstein dimension of modules . arXiv preprint math.AC/9809121 ,1998.[Mor58] Kiiti Morita . Duality for modules and its applications to the theory of rings with minimumcondition . Science Reports of the Tokyo Kyoiku Daigaku, Section A , 6(150):83–142, 1958.
N REFLEXIVE AND I -ULRICH MODULES OVER CURVE SINGULARITIES 25 [NY17] Yusuke Nakajima and Ken-ichi Yoshida . Ulrich modules over cyclic quotient surfacesingularities . Journal of Algebra , 482:224–247, 2017.[PU05]
Claudia Polini and Bernd Ulrich . A formula for the core of an ideal . MathematischeAnnalen , 331(3):487–503, 2005.[Ser65]
Jean-Pierre Serre . Alg`ebre locale. multiplicit´es, volume 11 of cours au coll`ege de france,1957–1958, r´edig´e par pierre gabriel. seconde ´edition, 1965 . Lecture Notes in Mathematics.Springer-Verlag, Berlin , 6, 1965.[Ser97]
Jean-Pierre Serre . Alg`ebre locale, multiplicit´es: cours au Coll`ege de France, 1957-1958 ,volume 11. Springer Science & Business Media, 1997.[Swa80]
Richard G Swan . On seminormality . Journal of Algebra , 67(1):210–229, 1980.[Ulr84]
Bernd Ulrich . Gorenstein rings and modules with high numbers of generators . Mathema-tische Zeitschrift , 188(1):23–32, 1984.[Vit11]
Marie A Vitulli . Weak normality and seminormality . In
Commutative Algebra , pages 441–480. Springer, 2011.
Department of Mathematics, University of Kansas, Lawrence, KS 66045-7523, USA
Email address : [email protected] URL : Department of Mathematics, University of Virginia, 141 Cabell Drive, Kerchof HallP.O. Box 400137 Charlottesville, VA 22904
Email address : [email protected] URL : https://sarasij93.github.io/ Department of Mathematics, University of Kansas, Lawrence, KS 66045-7523, USA
Email address : [email protected] URL ::