Nearly Gorenstein cyclic quotient singularities
aa r X i v : . [ m a t h . A C ] J u l NEARLY GORENSTEIN CYCLIC QUOTIENT SINGULARITIES
ALESSIO CAMINATA AND FRANCESCO STRAZZANTI
Abstract.
We investigate the nearly Gorenstein property among d -dimensional cyclic quotientsingularities k J x , . . . , x d K G , where k is an algebraically closed field and G ⊆ GL( d, k ) is a finitesmall cyclic group whose order is invertible in k . We prove a necessary and sufficient condition tobe nearly Gorenstein that also allows us to find several new classes of such rings. Introduction
Gorenstein rings are among the most important objects in commutative algebra and appear inseveral contexts. On the other hand, despite their celebrated ubiquity [Bas63, Hun99], the class ofGorenstein rings is not so large, compared for instance with that of Cohen-Macaulay rings. In manysignificant cases one encounters Cohen-Macaulay rings which are not Gorenstein rings. For thisreason, many researchers started looking for generalizations of the notion of Gorenstein aiming tofind a class of Cohen-Macaulay rings which is still able to capture some of the interesting properties ofGorenstein rings. In recent years, two of these new classes of rings have drawn particular attention: almost Gorenstein and nearly Gorenstein rings . Almost Gorenstein rings were first defined byBarucci and Fröberg [BF97] for one-dimensional analytically unramified rings and later generalizedby Goto et al. [GMP13, GTT15]. This notion has already been largely investigated and manyproperties are known, see for example [EGI19, GTT16, HJS19, Tan18] and the references therein.On the other hand, nearly Gorenstein rings, which are the object of interest of this paper, havebeen introduced even more recently by Herzog, Hibi, and Stamate [HHS19] in 2019, although theirdefining property was already examined by Ding [Din93], Huneke and Vraciu [HV06], and Striuliand Vraciu [SV11]. Moreover, nearly Gorenstein rings have been studied in several contexts, suchas zero-dimensional schemes [KLL19], affine semigroup rings [HJS19], and affine monomial curves[MS20]. See also [EGI19, DKT20, Kob20, Kum20, Rah20] for other related results.To explain the definition and the motivation of nearly Gorenstein rings we start with a Cohen-Macaulay local ring ( R, m ) which admits a canonical module ω R . The trace of the canonical module,denoted by tr( ω R ) , is defined as the sum of the ideals ϕ ( ω R ) , where the sum is taken over all the R -module homomorphisms ϕ : ω R → R . The importance of tr( ω R ) comes from the fact thatit describes the non-Gorenstein locus of R , since the localization R p at a prime ideal p is notGorenstein if and only if tr( ω R ) ⊆ p . In particular, it follows that R is Gorenstein if and only if tr( ω R ) = R . For this reason, one defines R to be nearly Gorenstein when m ⊆ tr( ω R ) . It is nowclear that a nearly Gorenstein ring is Gorenstein on the punctured spectrum, but the converse doesnot occur in general. Moreover, it also holds that a one-dimensional almost Gorenstein ring is nearlyGorenstein, even though this is no longer true in the higher dimensional case, where the relationbetween these two notions remains unclear. Key words and phrases. nearly Gorenstein, invariant ring, quotient singularity, trace ideal.2020
Mathematics Subject Classification . 13A50, 13H10, 14L30.The second author was partially supported by INdAM, more precisely he was “titolare di una borsa per l’esterodell’Istituto Nazionale di Alta Matematica” and “titolare di un Assegno di Ricerca dell’Istituto Nazionale di AltaMatematica”.
In this paper, we look at the nearly Gorenstein property for quotient singularities. Let R = k J x , . . . , x d K be a d -dimensional formal power series ring over an algebraically closed field k andlet G be a finite subgroup of GL( d, k ) acting linearly on R . The corresponding invariant ring R G isthe completion at the origin of the coordinate ring of the quotient variety A d k /G , so we will refer toit as a quotient singularity . The study of these objects and their properties lies at the intersectionof several branches of mathematics and has been largely explored both from a geometric and analgebraic point of view. In the modular case, i.e., when the characteristic of k divides the order of thegroup G , even the Cohen-Macaulay property is not fully understood (see e.g. [CW11, Example 8.0.9]or [Kem99]), so we will rather focus on the non-modular situation, that is when char k ∤ | G | . Underthis assumption it is well known that the invariant ring R G is a complete local normal domain and itis Cohen-Macaulay thanks to Hochster-Eagon’s Theorem [HE71]. Moreover, thanks to an old resultof Prill [Pri67] it is not restrictive to assume further that the acting group is small, i.e., it does notcontain pseudo-reflections. In this case, by a result of Watanabe [Wat74a, Wat74b] the Gorensteinproperty of these rings is also well understood. Namely, R G is Gorenstein if and only if the group G is contained in SL( d, k ) . Therefore, it arises as a natural problem to look for a characterizationof the nearly Gorenstein property for these rings. In fact, we investigate precisely this question foran important class of quotient singularities: cyclic quotient singularities, i.e., when the group G iscyclic.For this class, we are able to find a numerical criterion which gives a necessary and sufficientcondition for the ring R G to be nearly Gorenstein. Using this criterion, we identify several familiesof nearly Gorenstein rings. We recall that if G is a cyclic small subgroup of GL( d, k ) of order n with char k ∤ n , we can assume that it is generated by a diagonal matrix φ = diag( λ t , . . . , λ t d ) , where λ isa primitive n -th root of unity in k and t , . . . , t d are positive integers such that gcd( t i , . . . , t i d − , n ) =1 for every ( d − -tuple with distinct integers i , . . . , i d − ∈ { , . . . , d } . We denote the correspondinginvariant ring R G by n ( t , . . . , t d ) . Theorem A (see Proposition 2.4 and Corollaries 2.5, 2.6, 2.9) . Let n, d ≥ and t , . . . , t d ≥ beintegers and assume that at least one of the following holds: • d = 2 ; • n ≤ ; • t ≡ · · · ≡ t d ≡ n ; • t ≡ · · · ≡ t d − ≡ n and t d ≡ − d + 2 mod n .Then, the cyclic quotient singularity n ( t , . . . , t d ) is nearly Gorenstein. In the case t ≡ · · · ≡ t d ≡ n the corresponding invariant ring is a Veronese subalgebra of R . We also notice that, when R is a Gorenstein positively graded k -algebra with positive dimension,Veronese subalgebras of R are known to be nearly Gorenstein by [HHS19, Corollary 4.7]. Moreover,Theorem A says that if the dimension is two or if the order of the group is at most , then cyclicquotient singularities are always nearly Gorenstein. However, as soon as these assumptions aredropped we may find examples of cyclic quotient singularities that are not nearly Gorenstein. Forinstance, the invariant ring (1 , , is not nearly Gorenstein (see Example 2.8). More generally,the numerical criterion we proved can be implemented to find all nearly Gorenstein cyclic quotientsingularities for some values of n and d . For example, see Table 1 for an exhaustive list of non-isomorphic nearly Gorenstein cyclic quotient singularities with small values of n and d .In order to measure the distance of a quotient singularity R G to be Gorenstein or nearly Goren-stein, one can consider its residue which is the length res( R G ) = ℓ R G ( R G / tr( ω R G )) . We havethat res( R G ) = 0 if and only if R G is Gorenstein, and res( R G ) = 1 precisely when R G is nearlyGorenstein, but not Gorenstein. Thus, from Theorem A follows that every two-dimensional cyclic EARLY GORENSTEIN CYCLIC QUOTIENT SINGULARITIES 3 quotient singularity R G has res( R G ) ≤ . However, already in dimension 3 we are able to producecyclic quotient singularities of arbitrarily large residue. Theorem B (see Theorem 2.10) . Let n and m be two coprime positive integers with n ≥ and m < ⌈ n ⌉ . Then the cyclic quotient singularity n (1 , m, n − has residue m . In Section 3 we consider the field of complex numbers C and we turn our attention to thetwo-dimensional case, where the finite small group G ⊆ GL(2 , C ) is not necessarily cyclic. Thenearly Gorenstein property of the corresponding invariant rings was studied by Ding, who gavea complete classification of nearly Gorenstein two-dimensional quotient singularities (see [Din93,Proposition 3.5]). However, somehow a case was left out of Ding’s classification: it is the invariantring of the octahedral group O obtained by adding to the binary octahedral subgroup of SL(2 , C ) a cyclic generator of the form diag( λ, λ ) , where λ ∈ C is a primitive root of unity of order . Weprove that this ring is nearly Gorenstein in Proposition 3.1.The structure of the paper is the following. First, in Section 1 we review some basic definitionsand notations on nearly Gorenstein rings and quotient singularities. Then, in Section 2 we focus onnearly Gorenstein cyclic quotient singularities. In Theorem 2.3 we prove a numerical criterion thatcharacterizes them and we use this to provide several classes of nearly Gorenstein rings as stated inTheorem A. Finally, in Section 3 we study the nearly Gorenstein octahedral singularity mentionedabove which completes Ding’s classification. Acknowledgments.
This work began at the Institute of Mathematics of the University of Barcelona(IMUB). The authors would like to express their gratitude to the IMUB for providing a fruitful workenvironment. 1.
Preliminaries
In this section we recall some basic definitions and standard facts on nearly Gorenstein rings andquotient singularities.1.1.
Nearly Gorenstein rings.
Let ( R, m ) be a Cohen-Macaulay local ring which admits a canon-ical module ω R . The trace of the canonical module, denoted by tr( ω R ) , is the sum of the ideals ϕ ( ω R ) for any R -module homomorphism ϕ : ω R → R . In other words, we have tr( ω R ) = X ϕ ∈ Hom R ( ω R ,R ) ϕ ( ω R ) . The trace of ω R describes the non-Gorenstein locus of R . In fact, given a prime ideal p ⊆ R , then R p is not Gorenstein if and only if tr( ω R ) ⊆ p (cf. [HHS19, Lemma 2.1]). In particular, since tr( ω R ) is an ideal, one has that R is Gorenstein if and only if tr( ω R ) = R . Definition 1.1 (Herzog, Hibi, Stamate [HHS19]) . R is called nearly Gorenstein if m ⊆ tr( ω R ) .It is immediately clear from the definition that Gorenstein rings are nearly Gorenstein and that R is nearly Gorenstein but not Gorenstein if and only if tr( ω R ) = m . In order to give a measure tothe distance of a ring to be Gorenstein or nearly Gorenstein, one defines the residue of R as res( R ) = ℓ R ( R/ tr( ω R )) ∈ N ∪ {∞} . The ring R is Gorenstein if and only if res( R ) = 0 and it is nearly Gorenstein if and only if res( R ) ≤ .If there exists a canonical module ω R that is also an ideal of R we say that ω R is a canonical idealof R . In this case there is a useful formula to find its trace. We denote the total ring of fractions of R by Q ( R ) . ALESSIO CAMINATA AND FRANCESCO STRAZZANTI
Lemma 1.2. [HHS19, Lemma 1.1]
Let ( R, m ) be a local domain with a canonical ideal ω R . Then,the trace ideal of the canonical module of R is equal to tr( ω R ) = ω R ( R : Q ( R ) ω R ) . In particular, if ω R is a canonical ideal, then it is included in tr( ω R ) because ∈ ( R : Q ( R ) ω R ) .1.2. Quotient singularities.
Let k be an algebraically closed field and let G be a finite subgroupof GL( d, k ) such that the order | G | of G is coprime with the characteristic of k . We consider apower series ring R = k J x , . . . , x d K over k . The group G acts linearly on R with the action on thevariables x , . . . , x d given by matrix multiplication. We denote by R G the ring of invariants underthis action and we will call it also (non-modular) quotient singularity .We recall that an element σ ∈ G is called pseudo-reflection if the fixed subspace { v ∈ k d : σv = v } has dimension d − . We will always assume that the acting group G is small , i.e., that it doesnot contain pseudo-reflections. This is not restrictive in our setting. In fact, by a theorem of Prill[Pri67] if G is not small we can replace R by another power series ring S and find a small finite lineargroup H such that R G ∼ = S H . This is essentially a consequence of the Chevalley–Shephard–ToddTheorem which implies that the ring of invariants of a finite group generated by pseudo-reflectionsacting on a power series ring is again a regular local ring.Under the previous assumptions, the quotient singularity R G is a Cohen-Macaulay completelocal normal domain of dimension d . Moreover, Watanabe [Wat74a, Wat74b] proved that R G isGorenstein if and only if G ⊆ SL( d, k ) . In this case R G is called special quotient singularity . If G is a cyclic group, then R G is called cyclic quotient singularity . The Kleinian singularities are thetwo-dimensional complex special quotient singularities C J x , x K G .In order to study the nearly Gorenstein property of quotient singularities, it is important tounderstand their canonical module. To this purpose, we introduce the following definition. Definition 1.3.
Let R = k J x , . . . , x d K be a power series ring and let G be a finite small subgroupof GL( d, k ) such that char k ∤ | G | . We say that an element f ∈ R is a G -canonical element of R if σ ( f ) = det σ · f for all σ ∈ G .The previous definition is motivated by the following result due to Singh [Sin70] and Watanabe[Wat74b, Theorem ′ ] (see also [Sta78] or [BH98, Theorem 6.4.9] for an alternative proof). Theorem 1.4 (Singh-Watanabe) . Let R and G be as above and let f ∈ R be a G -canonical element,then f R ∩ R G is a canonical ideal of R G . Cyclic quotient singularities
In this section we focus on the nearly Gorenstein property for cyclic quotient singularities. Weconsider a formal power series ring R = k J x , . . . , x d K over an algebraically closed field k and a finitesmall cyclic group G ⊆ GL( d, k ) such that | G | = n is not zero in k .Since G is a finite cyclic group, we can assume that it is generated by a diagonal matrix φ of theform φ = diag( λ t , . . . , λ t d ) = λ t . . . λ t . . . ... ... . . . ... . . . λ t d where λ is a primitive n -th root of unit in k and t , . . . , t d ≥ are integers. If t d ≡ n ,one can set S = k J x , . . . , x d − K and H ⊆ GL( d − , k ) the group generated by diag( λ t , . . . , λ t d − ) ,then R G is nearly Gorenstein if and only if S H is Gorenstein, by [HHS19, Proposition 4.5]. Forthis reason, we will assume without loss of generality that t , . . . , t d n . In this case, EARLY GORENSTEIN CYCLIC QUOTIENT SINGULARITIES 5 the lack of pseudo-reflections in G is equivalent to the condition gcd( t i , . . . , t i d − , n ) = 1 for every ( d − -tuple with distinct integers i , . . . , i d − ∈ { , . . . , d } . With these conventions, we denotethe cyclic quotient singularity R G by n ( t , . . . , t d ) . We point out that this notation is not unique.For instance (1 , ,
2) = (2 , , are equal because they are invariant rings with respect to thesame group. Since the action of G on R is diagonal, the k -algebra n ( t , . . . , t d ) can be generatedby monomials, more precisely one can choose a (non-minimal) system of generators as follows R G = k J x α . . . x α d d | α + · · · + α d ≤ n and α t + · · · + α d t d ≡ n K . Proposition 2.1.
Let R and G be as above. A G -canonical element of R is given by f = x x . . . x d .Moreover, we have tr( ω R G ) = ( f R ) G ( R : Q ( R ) f R ) G . Proof.
Since G is generated by φ = diag( λ t , . . . , λ t d ) , to prove that f is a G -canonical element it isenough to observe that φ ( f ) = λ t . . . λ t d f = det( φ ) f .We now prove that tr( ω R G ) = ( f R ) G ( R : Q ( R ) f R ) G . First, notice that by Lemma 1.2 andTheorem 1.4 we have tr( ω R G ) = ( f R ∩ R G )( R G : Q ( R G ) ( f R ∩ R G )) . So, since f R ∩ R G = ( f R ) G , itis enough to prove that R G : Q ( R G ) ( f R ∩ R G ) = ( R : Q ( R ) f R ) G . The inclusion ( R : Q ( R ) f R ) G ⊆ ( R G : Q ( R G ) ( f R ∩ R G )) is clear. Conversely, consider an element a/b ∈ ( R G : Q ( R G ) ( f R ∩ R G )) with gcd( a, b ) = 1 . By hypothesis gcd( t , . . . , t d − , n ) = 1 and sothere exist a , . . . , a d − positive integers such that a t + · · · + a d − t d − ≡ n . Therefore,there exists a positive r such that ( a t + · · · + a d − t d − ) r + t + · · · + t d − + t d ≡ n. This implies that h = ( x a x a . . . x a d − d − ) r f ∈ R G , because φ ( h ) = ( Q d − i =1 λ ra i t i Q di =1 λ t i ) h = h . Itfollows that ah/b ∈ R G , then b is a monomial and x d does not divide b . Since we can repeat thesame reasoning with respect to every variable, we get that b is squarefree and, therefore, b divides f . This means that af /b ∈ R and, then, a/b ∈ ( R : Q ( R ) f R ) G as required. (cid:3) Lemma 2.2.
Let h = x a . . . x a d d be a monomial of R G and let f = x x . . . x d . Then, h ∈ tr( ω R G ) if and only if one of the following two conditions holds:(1) a i > for all i = 0 , . . . , d ;(2) h = x a σ (1) · · · x a j σ ( j ) with j < d , where σ is a permutation of { , . . . , d } and there exist in-tegers b , . . . , b j such that < b k ≤ a k + 1 for every k ∈ { , . . . , j } and P jk =1 b k t σ ( k ) ≡− P dk = j +1 t σ ( k ) mod n .Proof. If a i > for every i , we observe that h ∈ f R ⊆ tr( ω ( R G )) because ∈ ( R : Q ( R ) f R ) G .Therefore, without loss of generality we suppose that h = x a . . . x a j j for some j < d . We recallthat by Proposition 2.1 tr( ω R G ) = ( f R ) G ( R : Q ( R ) f R ) G . Moreover, we observe that ( f R ) G and ( R : Q ( R ) f R ) G are generated by monomials and quotient of monomials respectively because f is amonomial and G is cyclic. Therefore, since h is a monomial, we have that h ∈ tr( ω R G ) if and onlyif there is an equality(1) h = ( x b . . . x b d d ) x c . . . x c d d x e . . . x e d d with x b . . . x b d d ∈ ( f R ) G and x c . . . x c d d /x e . . . x e d d ∈ ( R : Q ( R ) f R ) G , where we assume that thefraction is irreducible. Since f = x . . . x d , it follows that b i ≥ and e i ≤ for every i = 1 , . . . , d . ALESSIO CAMINATA AND FRANCESCO STRAZZANTI
Moreover, for every k = j + 1 , . . . , d we have a k = 0 which implies b k = e k = 1 and c k = 0 .We also note that a i = b i + c i − e i for every i and, therefore, ≤ b i ≤ a i + 1 . Since h ∈ R G ,if x b . . . x b j j x j +1 . . . x d is invariant under the action of G , also x c . . . x c j j /x e . . . x e j j x j +1 . . . x d isinvariant. Recall that R G = k J x α . . . x α d d | α + · · · + α d ≤ n and α t + · · · + α d t d ≡ n K .Then, it is possible to write h as in (1) if and only if there exist integers b , . . . , b j such that ≤ b i ≤ a i + 1 and P ji =1 b i t i + P dk = j +1 t k ≡ n . (cid:3) Observing that R G is nearly Gorenstein if and only if the conditions of Lemma 2.2 hold for everygenerator of the maximal ideal of R G we get the following criterion. Theorem 2.3.
The ring R G is nearly Gorenstein if and only if for every < i < d , every permu-tation σ of { , . . . , d } and every i -tuple ( a , . . . , a i ) of positive integers such that a + · · · + a i ≤ n and a t σ (1) + · · · + a i t σ ( i ) ≡ n , there exist integers b , . . . , b i such that P ij =1 b j t σ ( j ) ≡− P dk = i +1 t σ ( k ) mod n and < b j ≤ a j + 1 for every j ∈ { , . . . , i } . We want to use the previous theorem to find examples of nearly Gorenstein cyclic quotientsingularities. First, we recall that by Watanabe’s Theorem a n ( t , . . . , t d ) -singularity is Gorensteinif and only if the acting group G is contained in SL( d, k ) which is in turn equivalent to the condition t + · · · + t d ≡ n . For instance, for each dimension d the singularity n (1 , . . . , , t d ) with t d ≡ − d + 1 mod n is Gorenstein. In a similar fashion, we can obtain examples of nearly Gorensteincyclic quotient singularities in every dimension. Proposition 2.4.
Let d ≥ and n ≥ be integers such that gcd( − d + 2 , n ) = 1 . Choose aninteger t d ≥ such that t d ≡ − d + 2 mod n . Then, the quotient singularity n (1 , . . . , , t d ) is nearlyGorenstein, but not Gorenstein.Proof. As usual let R = k J x , . . . , x d K and consider the group G generated by diag( λ, . . . , λ, λ t d ) for a primitive n -th root of unity λ ∈ k , so that R G = n (1 , . . . , , t d ) . It is clear that R G is notGorenstein, since · · · +1 − d +2 = 1 n . We prove that R G is nearly Gorenstein by usingTheorem 2.3. Consider < i < d , a permutation σ of { , . . . , d } and a i -tuple ( a , . . . , a i ) of positiveintegers such that a + · · · + a i ≤ n and a t σ (1) + · · · + a i t σ ( i ) ≡ n . If t σ (1) = · · · = t σ ( i ) = 1 ,then we have a + · · · + a i = n . Therefore, the sum P j ≤ i b j for < b j ≤ a j + 1 runs over all possibleresidues modulo n , thus there exist b j ’s such that P ij =1 b j t σ ( j ) ≡ − P dk = i +1 t σ ( k ) mod n is satisfied.Suppose now that t σ (1) = · · · = t σ ( i − = 1 and t σ ( i ) = t d ≡ − d + 2 mod n . We distinguish twopossibilities. If i = 1 , then we have ( − d + 2) a ≡ n and, since − d + 2 is coprime with n ,we obtain a ≡ n , which forces a = n being a ≤ n . Therefore, < b ≤ a + 1 rangesover all possible residues modulo n and we conclude as before. Suppose now that i > . We choose b i = a i + 1 , b i − = a i − , and b j = a j + 1 for all j = 1 , . . . , i − . Then, we obtain b + · · · + b i − + b i t d ≡ a + · · · + a i − + ( − d + 2) a i + ( i − · − d + 2) ≡ − ( d − i ) mod n. Hence, R G is nearly Gorenstein. (cid:3) The case d = 2 was left out from the previous proposition, but in fact two-dimensional cyclicquotient singularities are always nearly Gorenstein. Corollary 2.5. If R = k J x , x K and G is cyclic, then R G is nearly Gorenstein.Proof. Let a ≤ n be such that a t σ (1) ≡ n . Since there are no pseudo-reflections in G , wehave gcd( t σ (1) , n ) = 1 and, then, n divides a . In particular, a = n . Therefore, there is a solutionof the equation b t σ (1) ≡ − t σ (2) mod n such that < b ≤ n = a and Theorem 2.3 implies that R G is nearly Gorenstein. (cid:3) EARLY GORENSTEIN CYCLIC QUOTIENT SINGULARITIES 7
Now, we focus on groups with small order. We recall that for n = 2 and d even the ring R G is Gorenstein since G ⊆ SL( d, k ) . More generally, we prove that for n ≤ it is always nearlyGorenstein. Corollary 2.6.
Let G be a cyclic group of order at most , then R G is nearly Gorenstein.Proof. We prove only the case of order , since the case when | G | = 2 can be done in the sameway. So, assuming | G | = 3 , we will prove that R G is nearly Gorenstein by using Theorem 2.3. Let ( a , . . . , a i ) be positive integers such that a t σ (1) + · · · + a i t σ ( i ) ≡ for a permutation σ of { , . . . , d } . We need to find positive integers b j ≤ a j + 1 such that P ij =1 b j t σ ( j ) ≡ − P dk = i +1 t σ ( k ) mod 3 . If − P dk = i +1 t σ ( k ) ≡ , it is enough to set b j = a j for every j = 1 , . . . , i . If − P dk = i +1 t σ ( k ) ≡ and there exists ≤ p ≤ i such that t σ ( p ) ≡ , then we canset b p = a p + 1 and b j = a j for ≤ j ≤ i , j = p . Assume now that t σ ( j ) ≡ for every j = 1 , . . . , i . If i = 1 , then a has to be equal to and we can put b = 2 , otherwise it is enoughto set b = a + 1 , b = a + 1 and b j = a j for ≤ j ≤ i . The case − P dk = i +1 t σ ( k ) ≡ isanalogous to the previous one. (cid:3) As soon as the dimension of R is bigger than and the order of G is greater than , it is possibleto find cyclic quotient singularities R G that are not nearly Gorenstein. In order to exhibit someexamples we state a necessary condition which follows immediately from Theorem 2.3. Remark 2.7.
Let n ( t , . . . , t d ) be nearly Gorenstein. If gcd( t σ (1) , . . . , t σ ( i ) , n ) = m > for some i > and some permutation σ of { , . . . , d } , then t σ ( i +1) + · · · + t σ ( d ) ≡ m . Indeed, if wechoose a = n , Theorem 2.3 implies that there exists b such that b t σ (1) ≡ − P dk =2 t σ ( k ) mod n .Therefore, it is enough to consider this congruence modulo m . Example 2.8. (1) Let gcd( n, t ) = m > , with gcd( m + 1 , n ) = 1 and let t = t = · · · = t d − and t d − = t d = m + 1 . Therefore, gcd( t , . . . , t d − , n ) = m > , but t d − + t d ≡ m .Hence, R G is not nearly Gorenstein by the previous remark. For instance, (4 , ,
5) = (1 , , isnot nearly Gorenstein.(2) Let t = 1 , t = n − , t = n − . We have t + t ≡ n , but there are no < b , b ≤ such that b − b ≡ n . Hence, Theorem 2.3 implies that n ( t , t , t ) is not nearly Gorenstein.In particular (1 , , is not nearly Gorenstein. We also notice that in this case, if n > is odd,we have gcd( t σ (1) , . . . , t σ ( i ) , n ) = 1 for every i and, therefore, the converse of Remark 2.7 does nothold.Another interesting class of nearly Gorenstein quotient singularities is given by Veronese subal-gebras, which are obtained when t = t = · · · = t d = 1 . See [HHS19, Corollary 4.8] for a proof inthe positively graded case. Corollary 2.9.
The Veronese subalgebras of R are nearly Gorenstein.Proof. Let < i < d and let a , . . . , a i be positive integers such that a + · · · + a i = n . Let kn < d ≤ ( k + 1) n for some non-negative integer k .Assume first that d − i − kn ≥ . Then, we have ≤ d − i − kn ≤ n − i that implies i ≤ n − ( d − i − kn ) ≤ n = P ij =1 a j . Therefore, there exist b , . . . , b i such that ≤ b j ≤ a j and P ij =1 b j t σ ( j ) ≡ n − ( d − i − kn ) ≡ − P dk = i +1 t σ ( k ) mod n .Assume now that d − i − kn < . It follows that < kn − d + i < i and then kn − d + i X j =1 ( a j + 1) t σ j + i X l = kn − d + i +1 a l t σ l ≡ n + kn − d + i ≡ − ( d − i ) ≡ − d X k = i +1 t σ ( k ) mod n. ALESSIO CAMINATA AND FRANCESCO STRAZZANTI
Hence, the claim follows by Theorem 2.3. (cid:3)
In Table 1 we present an exhaustive list of non-isomorphic cyclic quotient singularities for d = 3 and ≤ n ≤ , and for d = 4 and ≤ n ≤ . Moreover, by using the numerical criterion ofTheorem 2.3, we report if they are Gorenstein, nearly Gorenstein or not.Ring Is nearly Gor. Ring Is nearly Gor. Ring Is nearly Gor. (1 , , NG (1 , , G (1 , , NG (1 , , not NG (1 , , NG (1 , , NG (1 , , G (1 , , NG (1 , , not NG (1 , , NG (1 , , NG (1 , , not NG (1 , , G (1 , , NG (1 , , G (1 , , not NG (1 , , not NG (1 , , not NG (1 , , NG (1 , , NG (1 , , not NG (1 , , NG (1 , , G (1 , , NG (1 , , not NG (1 , , G (1 , , not NG (1 , , not NG (1 , , , G (1 , , , not NG (1 , , , not NG (1 , , , NG (1 , , , not NG (1 , , , G (1 , , , G (1 , , , NG (1 , , , G (1 , , , NG (1 , , , not NG (1 , , , NG (1 , , , NG (1 , , , not NG (1 , , , NG (1 , , , G (1 , , , G (1 , , , NG (1 , , , not NG (1 , , , G (1 , , , not NG (1 , , , not NG (1 , , , G (1 , , , not NG (1 , , , NG (1 , , , not NG (1 , , , not NG (1 , , , not NG (1 , , , not NG (1 , , , NG (1 , , , not NG (1 , , , G (1 , , , not NG (1 , , , not NG (1 , , , not NG (1 , , , not NG (1 , , , not NG (1 , , , G (1 , , , not NG (1 , , , G (1 , , , G Table 1.
Cyclic quotient singularities for d = 3 and ≤ n ≤ , and for d = 4 and ≤ n ≤ . G means Gorenstein and NG means nearly Gorenstein.We conclude this section by recalling that the residue of a local ring is a measure of how far isa ring from being nearly Gorenstein. In the next theorem we show that we have cyclic quotientsingularities of arbitrarily large residue already in dimension . Theorem 2.10.
Let R = k J x, y, z K and let n and m be two coprime positive integers with n ≥ and m < ⌈ n ⌉ . Consider the group G generated by diag( λ, λ m , λ n − ) , where λ is a primitive n -throot of unit in k . Then, res( R G ) = m . In particular, R G is nearly Gorenstein if and only if m = 1 . EARLY GORENSTEIN CYCLIC QUOTIENT SINGULARITIES 9
Proof.
In order to compute res( R G ) = ℓ R G ( R G / tr( ω R G )) we count how many monomials of themaximal ideal m of R G are not in tr( ω R G ) . We fix f = xyz and we recall that tr( ω R G ) =( f R ) G ( R : Q ( R ) f R ) G by Proposition 2.1. Let g = x a y b z c ∈ m . If a, b, c > , then we canwrite g = f x a − y b − z c − ∈ ( f R ) G ⊆ tr( ω R G ) . If b = c = 0 , then g = x a ∈ m implies that n divides a . Then, the condition of Lemma 2.2 is satisfied for g , since b gives all possible residues modulo n for < b ≤ a + 1 . Therefore, x a ∈ tr( ω R G ) . Similarly, one obtains that y b , z c ∈ tr( ω R G ) because gcd( m, n ) = gcd( n − , n ) = 1 .It remains to check the monomials of the form x a y b , x a z c , y b z c ∈ m with a, b, c > . We useagain the criterion of Lemma 2.2. If y b z c ∈ m , then mb + ( n − c ≡ n . It follows that mb + ( n − c + 1) ≡ n − ≡ − − t mod n , therefore y b z c ∈ tr( ω R G ) . If x a y b ∈ m , then a + mb ≡ n implies ( a + 1) + mb ≡ ≡ − ( n −
1) = − t mod n , thus x a y b ∈ tr( ω R G ) .Finally, consider a monomial x a z c ∈ m . If a ≥ n or c ≥ n , then x a y b ∈ tr( ω R G ) because x n and y n are in tr( ω R G ) , therefore we may assume a, c < n . Since x a z c ∈ m , we have a + ( n − c ≡ n ,thus a ≡ c mod n which implies a = c . By Lemma 2.2, x a z c ∈ tr( ω R G ) if and only if there exist < b , b ≤ a + 1 such that b − b ≡ − m mod n . We notice that b − b ∈ { a, a − , . . . , − a + 1 , − a } and, so, there are exactly m − monomials in m of the form x a z a that do not satisfy this criterion: x a z a with ≤ a ≤ m − . Hence, dim R G / m ( R G / tr( ω R G )) = dim R G / m ( m / tr( ω R G )) + 1 = m . (cid:3) An addendum to Ding’s classification in dimension two
Let C be the field of complex numbers and set R = C J u, v K . In his paper [Din93], Ding classifiednearly Gorenstein quotient singularities R G , where G is a finite small subgroup of GL(2 , C ) . Hisresult relies on the well-known classification of such subgroups which goes back to Klein [Kle84](see also [Bea10, Bri67, Rie77]). However, a nearly Gorenstein quotient singularity was left out ofhis classification. We are going to describe it.We consider the octahedral group O generated by matrices φ = (cid:18) λ λ (cid:19) , ψ = (cid:18) ζ ζ − (cid:19) , τ = (cid:18) ii (cid:19) , η = 1 √ (cid:18) ζ ζ ζ ζ (cid:19) , where λ and ζ are primitive roots of unity in C of orders and respectively. In other words, O is the extension of the binary octahedral subgroup of SL(2 , C ) of order generated by ψ, τ, η withthe cyclic group of order generated by φ . Proposition 3.1.
The quotient singularity R O is nearly Gorenstein.Proof. We consider the polynomials g = ( u v − uv ) ,g = uv ( u − v )( u − u v − u v + v ) ,g = u + 14 u v + v , which are generating invariants for the Gorenstein singularity E (see e.g. [LW12, §6.16]) and inparticular invariants for the action of ψ , τ , and η . By [Rie77, Satz 6], a minimal set of generatorsfor the maximal ideal of R O is given by z = g , z = g g , z = g g , z = g g , z = g g , z = g g g . Now, we consider the polynomial f = g . It is invariant for the action of the matrices ψ, τ, η ∈ SL(2 , C ) , and φ ( f ) = λ f = det( φ ) f . Therefore, f is a O -canonical element and, thus, ω R O =( f R ) O is a canonical ideal of R O by Theorem 1.4. We have that z , z , z and z are multiples of f , then they are in tr( ω R O ) . Moreover, the fact z z = z z = g g ∈ ( R O : Q ( R O ) ( f R ∩ R O )) implies that z = z z z , z = z z z ∈ tr( ω R O ) as well. Hence, R O is nearly Gorenstein. (cid:3) We also point out that Ding erroneously included cyclic quotient singularities n (1 , n − in his listof nearly Gorenstein not Gorenstein quotient singularities. Indeed, it is well known that n (1 , n − is Gorenstein. References [BF97]
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Alessio Caminata, Institut de Mathématiques, Université de Neuchâtel, Rue Emile-Argand 11,CH-2000 Neuchâtel, Switzerland
E-mail address : [email protected] Francesco Strazzanti, Dipartimento di Matematica, Università di Bologna, Piazza di Porta SanDonato 5, 40126 Bologna, Italy
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