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Noncommutative Rational Double Points Daniel Chan Mathematical Sciences Research Institute, Berkeley, California 94720 E-mail: [email protected] Communicated by J. T. Stafford Received November 29, 1999

In this paper we study noncommutative analogues of rational double points. The approach is to consider the action of a finite group G on certain noncommutative analogues of k ww x, y xx which were studied by Artin and Stafford Ž‘‘Regular Local Rings of Dimension 2,’’ manuscript.. An explicit description in terms of generators and relations is given for a large class of such algebras when G is cyclic. Finally, we show that these algebras are AS-Gorenstein of dimension two, have finite representation type and, in many cases, are regular in codimension one. 䊚 2000 Academic Press

1. INTRODUCTION In wASx, Artin and Stafford initiated the local study of smooth noncommutative surfaces by introducing regular rings of dimension two. These noncommutative analogues of the power series ring k ww u, ¨ xx Ž k an algebraically closed field of characteristic zero. are rings of the form B s k ²² u, ¨ ::rŽ r ., where the ‘‘commutation’’ relation r has leading term a quadratic with no linear factors. The simplest types of surface singularities are the rational double points which can be characterised in several ways. Let V s ku q k¨ and let G ; SLŽ V . be a finite subgroup. The rational double points are schemes of the form Spec k ww u, ¨ xxG . Depending on the commutation relation r, G may also act on the regular ring of dimension two, B. We call the resulting invariant ring B G a special quotient surface singularity and this noncommutative analogue of a rational double point is the primary object of study in this paper. We sought to find an explicit description of special quotient surface singularities via generators and relations. The approach was to study carefully the associated graded ring and then use the power series version 725 0021-8693r00 $35.00 Copyright 䊚 2000 by Academic Press All rights of reproduction in any form reserved.

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DANIEL CHAN

of the diamond lemma. If ᒋ denotes the maximal ideal of B then the associated graded ring B s grᒋ B must be either Bq s k ² u, ¨ :rŽ ¨ u y qu¨ . for q g k y 0 of B J s k ² u, ¨ :rŽ ¨ u y u¨ y ¨ 2 .. Conversely any ring whose associated graded ring has this form is regular of dimension two. Let A s B G be a special quotient surface singularity and let F be the filtration on A induced from the ᒋ-adic filtration on B. Then the associated graded ring grF A has the form B G . Artin asked conversely whether or not every complete filtered ring with associated graded ring of the form B G is a special quotient surface singularity. We have a partial answer in the case where G is cyclic. The first step is a classification of all rings which have the correct associated graded ring. Let x, y, z be indeterminates of degree d, 2, d, respectively, and let F be the filtration on k ²² x, y, z :: induced by degree. THEOREM 1.1. relations yx s qxy q xg ,

Suppose A is a quotient of P s k ²² x, y, z :: with defining d

zx s Ž qy q g . ,

zy s qyz q gz,

y d s xz, Ž 1 .

where g g kz q F 3 P. If F also denotes the filtration on A induced from P, then grF A has the form B G for G ; SLŽ V . the diagonal subgroup of order d. Con¨ ersely, e¨ ery k-algebra A complete with respect to a filtration F with grF A , B G and G ; SLŽ V . the diagonal subgroup of order d has the abo¨ e form. It is hoped that these are all special quotient surface singularities but we only know THEOREM 1.2. Let A be the complete local ring in the pre¨ ious theorem with g g F 3 P. Suppose that 1 is the only power of q which is a dth root of unity. Then A is a special quotient surface singularity. Our next goal was to extend well-known properties of commutative rational double points to the noncommutative case. We show that every special quotient surface singularity Ž A, ᒊ . is Auslander᎐Gorenstein of injective dimension two. We also show that in most cases, A is regular in codimension one in the following sense: if T denotes the category of T has finite injecᒊ-torsion modules then the quotient category Mod-ArT tive dimension. The remaining property we examined involves the theory of local cohomology. This has been studied by various authors in the noncommutative graded case and the noncommutative local case has recently been studied by Wu and Zhang and in wCx. In Sections 6, 7, and 8 we sketch enough of the theory Žmainly without proof. to generalise several classical concepts from commutative algebra such as the Cohen᎐Macaulay property. In

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Section 9 we show that like their commutative counterparts, every special quotient surface singularity has only a finite number of isomorphism classes of indecomposable maximal Cohen᎐Macaulay modules. The results in this paper come from my doctoral thesis. It is with pleasure that I thank my thesis supervisor, Mike Artin, for all the help he has given me. 2. THE DIAMOND LEMMA FOR POWER SERIES We will need Bergman’s diamond lemma wB, Theorem 1.2x, both in its usual algebra form and in its power series form. The power series version is well known but not so well documented Žsee wGHx.. We thus include a discussion of it in this section. Throughout this paper we fix a base field k. Recall that a filtration F on an object M in an Abelian category is a set of subobjects F p M 4p g ⺪ satisfying F pq 1 M : F p M for all p. A filtered k-algebra is a k-algebra A with a filtration F by subspaces satisfying the additional conditions Ž F pA.Ž F q A. : F pq q A and F 0A s A. These conditions imply in particular that the F pA are ideals. In this section, A will always denote a filtered k-algebra with filtration F which is complete in the sense that A s p lim ¤ p ArF A. First note the following fact: PROPOSITION 2.1.

Let ⌳ s m␣ 4 ; A. Then the following are equi¨ alent:

i. For e¨ ery p g ⺪, the set ⌳ p s m␣ q F pA ¬ m␣ f F pA4 : ArF pA is a k-basis of ArF pA. ii. The images of the m␣ g ⌳ in the associated graded ring grF A form a basis of grF A. If these hold then e¨ ery element a g A has a unique representation as a con¨ ergent series of the form a s Ý m ␣ g ⌳ c␣ m␣ where c␣ g k. DEFINITION 2.2. A subset ⌳ ; A is said to be a strict topological basis for A if it satisfies the equivalent conditions of the previous proposition. Let X 1 , . . . , X n be a set of indeterminates of degrees d i ) 0 and let ⌫ be the semigroup generated by the X i ’s. We extend the degree map to a semigroup homomorphism deg: ⌫ ª ⺞ and set ⌫d ; ⌫ to be the subset of degree d elements. Let P s k ²² X i :: denote the noncommutative power series ring in the indeterminates X 1 , . . . , X n . We filter P by setting F p P to be the set of power series of degree at least p; i.e., every term of the power series has degree at least p. In general, we shall say that an element ␣ in a filtered object M with filtration F has degree r if x g F r M y F rq1 M. This agrees with the terminology in the previous case where M s P.

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DANIEL CHAN

Let r j , s j g P. Recall that a filtration on a ring determines a topology. We let Ž r j y s j . denote the closed ideal generated by the r j y s j and define the quotient of P with defining relations r j s s j to be PrŽ r j y s j .. This notation should not cause confusion since we will only be interested in complete quotients of P. When the r j are monomials, combinatorial methods can be used to study the quotient PrŽ r j y s j .. We recall some definitions from wB, pp. 180᎐181x. Given any r j and m, mX g ⌫, we can associate to every a g P a new element obtained by replacing every occurrence of the monomial mr j mX with ms j mX and leaving all other monomials alone. We call such elements reductions of a. A sequence of reductions of a is a sequence a s a0 , a1 , . . . such that a iq1 is a reduction of a i . An o¨ erlap ambiguity is a pair ri , r j such that ri s mmX and r j s mX mY for some m, mX , mY g ⌫. We say that the overlap ambiguity can be resol¨ ed if there exists a sequence of reductions a n4 for si mY and a sequence of reductions bn4 for ms j such that for each p g ⺞ we have a n ' bn Žmod F p P . for all n sufficiently large. An inclusion ambiguity is a pair ri , r j such that ri s mr j mX for some m, mX g ⌫. We say that the inclusion ambiguity can be resolved if there are sequences of reductions for si and ms j mX which, as before, are eventually congruent modulo F p P for all p. Using the language of wB, pp. 180᎐181x we may now state the power series version of the diamond lemma: THEOREM 2.3 Ždiamond lemma for power series.. Let P be the noncommutati¨ e power series ring in n indeterminates as abo¨ e and r j g ⌫, s j g P for j in some index set. Let A s PrŽ r j y s j .. Suppose there is a semigroup partial ordering - on ⌫ satisfying m1 - m 2 whene¨ er deg m1 ) deg m 2 and such that the restriction of - to each ⌫d satisfies the descending chain condition. Suppose that each relation r j s s j is compatible with - in the sense that e¨ ery monomial m g ⌫ occurring in s j is less than r j . If all the o¨ erlap and inclusion ambiguities can be resol¨ ed then A has a strict topological basis of the form ⌳ s m ˜ ¬ m g ⌫ has no subword of the form r j 4 , where m ˜ denotes the image of m in A. Proof. Let F be the filtration on A induced by the degree filtration on P. The theorem follows from applying the usual diamond lemma wB, Theorem 1.2x Žor rather its proof. to ArF pA. The strict topological basis provided by this version of the diamond lemma is called the topological Grobner basis. ¨ The two diamond lemmas in tandem provide an effective way of passing between a complete filtered ring and its associated graded ring as the

NONCOMMUTATIVE RATIONAL DOUBLE POINTS

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following proposition shows. To keep notation straight, given r g P let r Ž x . be the element obtained by substituting the elements x s x 1 , . . . , x n for X 1 , . . . , X n , where the x i are elements of the maximal ideal of some complete local ring. In order to describe the associated graded algebra, we need some new indeterminates X i also of degree d i . As before, given an element r g k ² X i : we let r Ž x . denote the element obtained from r by substituting the elements x s x 1 , . . . , x n for X 1 , . . . , X n . PROPOSITION 2.4. Suppose P, A, - , r j , s j are as hypothesised in the pre¨ ious theorem. Let F be the filtration on A induced from the degree filtration on P and e j s deg r j . Consider the degree e j part t j of r j y s j and let r j y s js t j Ž X .. If all o¨ erlap and inclusion ambiguities can be resol¨ ed then the associated graded ring of A is grF A , k ² X i :r Ž r j y s j . . Con¨ ersely, let A be a complete filtered ring with filtration F. Suppose the associated graded ring grF A , k ² X i :rŽr j y s j. for some r jg ⌫ and s jg k ² X i :. Suppose also that there is a semigroup partial ordering - on monomials in X i satisfying the descending chain condition and compatible with the relations r js s j. Let r j s r jŽ X ., s j s s jŽ X .. If all o¨ erlap and inclusion ambiguities can be resol¨ ed, then there exist power series t j in the X i of degree greater than deg r j , such that we ha¨ e an isomorphism A , k ²² X i ::r Ž r j y s j y t j . of filtered algebras. Furthermore, if x i are arbitrary lifts of X i to A, then we may assume by changing the t j , if necessary, that the isomorphism maps x i to X i q Ž r j y s j y t j .. Proof. We attack the forward implication first. Let x i denote the image of X i in A and let x i denote the image of x i in grF A. Choose ⌳ ; ⌫ so that mŽ x . ¬ m g ⌳4 is the topological Grobner basis for A obtained from ¨ the power series version of the diamond lemma. Note first that for m g ⌳, deg m s deg mŽ x . for otherwise we can write mŽ x . as a power series consisting of terms of degree greater than deg m and then rewrite these as a power series with terms in ⌳ of degree greater than deg m. This would contradict the uniqueness of power series representations given in Proposition 2.1. We wish to show that X i ¬ x i induces the desired isomorphism. We first show that the x i satisfy the relations r j y s js 0. By reducing monomials, we may assume that no term of the s j ’s has ri as a subword since reduction does not alter grF A nor the ring k ² X i :rŽr j y s j.. This implies in particular that each term of s j lies in ⌳. There are two cases. If deg r j s deg r j Ž x . s p say, then Žr j y s j.Ž x . is just the image of r j Ž x . y

730

DANIEL CHAN

s j Ž x . in F pArF pq1A and so is zero. If deg r j - deg r j Ž x . then Žr j y s j.Ž x . s r j Ž x . which is zero by the assumption on the degree of r j Ž x .. We thus conclude that X i ¬ x i induces a map : k ² X i :rŽr j y s j. ª grF A. The hypotheses ensure that you can apply the diamond lemma to k ² X i :rŽr j y s j. to obtain the basis mŽ X . ¬ m g ⌳4 . On the other hand, deg m s deg mŽ x . for m g ⌳ shows that grF mŽ x . s mŽ x . so the mŽ x . form a basis for grF A by Proposition 2.1. Thus restricts to a bijection on Grobner bases and so must be an isomorphism as was to be shown. ¨ For the converse, let x i denote arbitrary lifts of X i to A. Then r jŽ x . y s jŽ x . s t j Ž x . for some power series t j of degree greater than deg r j . Since A is complete, we obtain a surjective filtered map : k ²² X i ::rŽ r j ys j y t j . ª A. We apply the usual diamond lemma to grF A and let ⌳ be the resulting Grobner basis. To see that is injective, let a g ¨ k ²² X i ::rŽ r j y s j y t j . which we can write as a nonzero convergent series consisting of terms in ⌳. Computing Ž a. from such a series we see that Ž a. / 0 so is an isomorphism of filtered algebras.

3. GROUP ACTIONS ON REGULAR RINGS Regular local rings of dimension two were introduced in wASx. They are noncommutative analogues of an analytic neighbourhood on a smooth surface. In this section we classify certain group actions on such rings. Throughout, we assume that our base field k is algebraically closed of characteristic zero. Let x 1 , . . . , x n be a set of indeterminates of degree one which is totally ordered by a relation - . Let ⌫ be the semigroup generated by the x i ’s. We need two partial orderings on ⌫ in order to apply the two diamond lemmas of the previous section. Both are based on a right to left version of the lexicographic ordering on ⌫ defined as follows: Let a, b g ⌫ be monomials of the same degree. Write a and b as a product of monomials a s aX x i c, b s bX x j c where c has as large a degree as possible. We write a $ b if x i - x j . This is the partial order we use when applying the usual algebra form of the diamond lemma. When using the power series version of the diamond lemma, we refine the order $ by insisting also that a $ b if deg a ) deg b. There should be no confusion in using the same notation to denote these two orders since one will only be used for graded algebras and the other for complete local rings. As usual, we use the language of wB, pp. 180᎐181x. Recall, LEMMA 3.1. The relation $ is a semigroup partial ordering on ⌫ which satisfies the hypotheses of the diamond lemma.

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A monomial x i1 ⭈⭈⭈ x i m is said to be in order or ordered if x i1 G x i 2 G ⭈⭈⭈ G x i m . Otherwise, the monomial is said to be out of order. Let Ž B, ᒋ . denote a complete local k-algebra B with maximal ideal ᒋ. We let grᒋ B denote the associated graded ring of B with respect to the ᒋ-adic filtration. We recall Artin and Stafford’s DEFINITION 3.2 wAS, Definition 1.12x. A complete local ring B with maximal ideal ᒋ is regular of dimension two if grᒋ B , k ² x, y :rŽ Q . where Q is a quadratic form in x, y which is not the product of two linear forms. For the rest of this section, Ž B, ᒋ . will denote a regular complete local ring of dimension two. The next result is due to Artin᎐Stafford and Cartan. As usual, given r an element of a power series ring, we let Ž r . denote the closed ideal generated by r. PROPOSITION 3.3 wAS, Lemma 1.4x. the following,

B , k ²² u, ¨ ::rŽ r . where r is one of

i. Ž q-plane. r s ¨ u y qu¨ y c for some q g k y 04 ii. Ž Jordan plane. r s ¨ u y u¨ y ¨ 2 y c and c is a noncommutati¨ e power series of degree at least three. Furthermore, if there is a finite group G acting on B, then one can assume that G restricts to an action on V s ku q k¨ . Con¨ ersely, e¨ ery ring of the form B , k ²² u, ¨ ::rŽ r ., where r is gi¨ en as in Ži. or Žii., is regular complete local of dimension two. Proof. Suppose there is a finite group action on B. Then by Maschke’s theorem one can lift the action of G on ᒋrᒋ 2 to a vector subspace V of ᒋ so that the natural map V ª ᒋrᒋ 2 is a G-module isomorphism. Since B is regular of dimension two, grᒋ B , k ² u, ¨ :rŽ Q . for some quadratic form Q which is not the product of two linear factors. By changing variables, we may assume that Q is ¨ u y qu¨ or ¨ u y u¨ y ¨ 2 . Let u, ¨ be the unique lifts of u, ¨ to V. We may wish to apply the converse half of Proposition 2.4. Order the variables by ¨ - u and consider the order $ of Lemma 3.1. We express the relations of grᒋ B in the form ¨ u s qu¨ or u¨ q ¨ 2 . In this form, the relations are compatible with $ and there are no overlap or inclusion ambiguities to resolve. Hence, Proposition 2.4 shows that B has the desired presentation. To see that every ring of the form k ²² u, ¨ ::rŽ r . with r as above is regular of dimension two, we need only apply the first half of Proposition 2.4. The power series version of the diamond lemma allows us to read off a strict topological basis for B. LEMMA 3.4.

B has a strict topological basis of the form u i ¨ j ¬ i, j G 04 .

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DANIEL CHAN

Remark. The possible associated graded rings of B, namely, Bq s k ² u, ¨ :rŽ ¨ u y qu¨ . and B J s k ² u, ¨ :rŽ ¨ u y u¨ y ¨ 2 ., can be nicely interpreted using Van den Bergh’s notion of a twisted homogeneous coordinate ring Žsee wAVx.. In fact, Bq , B Ž⺠ 1, O Ž1. . where g AutŽ⺠ 1 . is multiplication by q and B J , B Ž⺠ 1, O Ž1. . where is translation by 1. Fix V ; ᒋ such that the natural map V ª ᒋrᒋ 2 is an isomorphism. We say that an automorphism of B acts linearly on B Žwith respect to V . if the action preserves V. We say that G acts linearly if every g B acts linearly. Since we are interested in invariant rings, we will only study faithful actions of a finite group G on B. If G acts linearly on B, then it will be convenient to identify G with a finite subgroup of GLŽ V .. If, furthermore, we are given a basis u, ¨ for V, then we will also identify GLŽ V . with GL2 Ž k . via the basis u, ¨ . LEMMA 3.5. We continue the notation of Proposition 3.3. Suppose an automorphism of B acts linearly and has order d. Then one of the following must hold: i.

B is a q-plane and

s

ž

r 0

0 , s

/

where is a primiti¨ e dth root of unity and r, s g ⺪. ii. B is a q-plane where q s y1, d is e¨ en, and s Ž 0b 0a . where ab is a d2 th root of unity. iii. B is a q-plane with q s 1. iv.

B is a Jordan plane and s Ž 10 01 ., where is a dth root of unity.

Proof. Since acts linearly, we need only consider the induced automorphism of on grᒋ B. Let u, ¨ be the images of u, ¨ in grᒋ B. For the q-plane where q / 1, the set of normal degree 1 elements Žtogether with 0. is the union of ku and k¨ so these must be -stable. If stabilises each of these lines then we are in case Ži.. If swaps these lines then s Ž 0b 0a . where ab is a d2 -th root of unity. For to preserve the skew commutation relation ¨ u s qu¨ of grᒋ B we must have q s y1. In the Jordan plane case, k¨ is the set of all normal elements of degree 1 Žplus 0. and so must be stable under G. By Maschke’s theorem, there is another -stable vector space of the form k Ž u q ␣ ¨ ., ␣ g k. We may replace u with u q ␣ ¨ since this will not alter the relation ¨ u s u¨ q ¨ 2 . For to preserve this relation, the eigenvectors u¨ and ¨ 2 must have the same eigenvalue. This implies that is scalar on ku q k¨ and we are done.

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Remark. Perhaps the most illuminating way of seeing the classification of actions on grᒋ B is to view grᒋ B as a twisted homogeneous coordinate ring B Ž⺠ 1, O Ž1. .. Now induces an automorphism of ⺠ 1 which commutes with . If is multiplication by q / 1 then it has two fixed points 0 and ⬁. If fixes these then we are in case Ži.. If swaps these, then up to multiplication, is inversion so q s qy1 and we are in case Žii. or Žiii.. In the Jordan case, ⬁ is a fixed point of and thus also of . Hence induces an automorphism of ⺑1 which commutes with translation. The only such maps are translations of which the only one of finite order is the identity. If G is any finite group acting on B, then the ᒋ-adic filtration induces a filtration F on the invariant ring B G. There is a natural graded ring homomorphism grF B G ª Žgrᒋ B . G . In fact we have PROPOSITION 3.6. The natural map grF B G ª Žgrᒋ B . G is an isomorphism. Proof. The morphism is injective because the filtration is induced. To see surjectivity, consider the G-module epimorphism ᒋ p ª ᒋ prᒋ pq 1. By Maschke’s theorem this map splits so fixed elements lift as desired. In commutative algebra, the rational double points are the quotients of Spec k ww u, ¨ xx by a finite subgroup of SL 2 . We seek to classify noncommutative analogues in terms of their associated graded rings. The previous proposition allows us to restrict our attention to the graded case where we have PROPOSITION 3.7. Let u, ¨ be indeterminates and V s ku [ k¨ . Then B s k ² u, ¨ :rŽ ¨ u y qu¨ . and suppose G ; SLŽ V . is a finite group whose action extends to B. Then one of the following holds, i. G is a cyclic group of order d generated by

s

ž

0

0

y1

/

,

where is a primiti¨ e dth root of unity. In this case, B G is the k-algebra on three generators x, y, z subject to the relations yx s qxy, where q s q d.

zx s q d xz,

zy s qyz,

y d s xz,

Ž 2.

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DANIEL CHAN

ii. q s y1 and G is the binary dihedral group ² , ¬ d s 2 , 4 s 1, s y1 :. Up to scaling u and ¨ we ha¨ e

s

ž

0

0

y1

/

Ž a 2 dth root of unity . and s Ž y10 01 .. In this case, B G , k w x, y x. iii. q s 1 and G is either binary dihedral, tetrahedral, octahedral, or icosahedral. Let B s k ² u, ¨ :rŽ ¨ u y u¨ y ¨ 2 . and G ; SLŽ V . be a non-tri¨ ial group whose action extends to B as before. Then G is "14 and B G is the k-algebra on three generators x, y, z subject to the relations yx s xy q xz,

zx s xz q 2 yz q 2 z 2 ,

zy s yz q z 2 ,

y 2 s xz. Ž 3.

Proof. We suppose first that B s k ² u, ¨ :rŽ ¨ u y qu¨ .. Case Ži.. G is diagonal, q / 1. This is always the case if q / "1 by Lemma 3.5. Then G must be as in Ži. for the diagonal subgroup of SL 2 is isomorphic to kU and the only finite subgroups of kU are cyclic. The invariant ring B G is spanned by the u i ¨ j where d ¬ j y i. Since u and ¨ are normal, we see immediately that B G is generated by u d, u¨ , ¨ d. Let C be the k-algebra on three generators x, y, z with defining relations Ž2.. We set the degree of x and z to be d and the degree of y to be 2 so that C is a graded algebra. Then there is a surjective graded ring homomorphism : C ª B G defined by x ¬ u d, y ¬ cu¨ , z ¬ ¨ d where c s q Ž1yd.r2 . Now the relations Ž2. ensure that the x i y r z j for 0 F r - d span C. Furthermore, the Ž x i y r z j . g ku i dqr ¨ jdqr are linearly independent by Lemma 3.4. Hence is injective and C , B G . Case Žii.. G not diagonal; q s y1. Let H be the subgroup of diagonal elements. As was seen in the previous case, H must be cyclic generated by some automorphism

s

ž

0

0

y1

/

,

where is an eth root of unity and e is the order of H. Let be a non-diagonal element of G which by the lemma has the form Ž 0b 0a .. Its determinant yab s 1 so by scaling u and ¨ if necessary, we may assume a s 1, b s y1. The product of any two off-diagonal elements in G must be diagonal and so lies in H. Hence G is generated by and . Now

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2 s Ž y10 y10 . so e s 2 d for some d. Computing the relations satisfied by and , we see that G is the binary dihedral group. To determine B G , first recall from the previous case that B H has a basis consisting of monomials u i ¨ j where 2 d ¬ j y i. Since has order two modulo H, B G s Žid q .Ž B H .. To compute the latter we consider monomials u i ¨ j with 2 d ¬ j y i so that in particular, i and j have the same parity. Then i

u i ¨ j q Ž u i ¨ j . s u i ¨ j q Ž y1 . ¨ i u j s u i ¨ j q Ž y1 .

iqi j

u j¨ i s ui ¨ j q u j¨ i

forms a basis for B G . We wish to show this element lies in the algebra generated by x s u¨ and y s u 2 d q ¨ 2 d. First note that u 2 and ¨ 2 are central in B. In particular, x and y commute. They are also algebraically independent. We assume j G i, the case i G j being symmetric. Since u jyi is central we see u i ¨ j q u j ¨ i s u i ¨ i Ž ¨ jyi q u jyi .. We show by induction on j y i that this lies in k w x, y x. If j s i then u i ¨ i s "x i so we may assume i s 0 and j s 2 ld. From the binomial theorem, we see that Ž u 2 l d q ¨ 2 l d . y y l is a sum of binomials of the form u 2 m d ¨ 2Ž lym. d q u 2Ž lym. d ¨ 2 m d where 0 - m - l. Hence by the inductive hypothesis, u 2 l d q ¨ 2 l d also lies in k w x, y x so B G , k w x, y x. Case Žiii.. q s 1. This is the classical commutative case for which we refer the reader to wPinkx. Now assume that B s k ² u, ¨ :rŽ ¨ u y u¨ y ¨ 2 .. Lemma 3.5 shows that the only nontrivial group G ; SLŽ V . acting on B is "14 . The invariant ring B G consists of the even degree elements in B and is generated by u 2 , u¨ , ¨ 2 . Let C be the algebra defined by Eq. Ž3.. Checking relations, we see that there is a graded surjective ring homomorphism C ª B G defined by x ¬ 12 Ž u 2 q u¨ . ,

y ¬ u¨ ,

z ¬ 2¨ 2 .

Injectivity is proved as in Ži.. Case Žii. of the proposition is somewhat anomalous. There is no commutative analogue. We have a converse to the previous proposition: PROPOSITION 3.8. The actions of G on V in Proposition 3.7 extend to actions of G on B. This is proved by verifying that G preserves the defining relation vu s quv or vu s uv q v 2 . Whether or not the actions of Proposition 3.7 will induce actions on a regular complete local ring B will depend on what the cubic term c of Proposition 3.3 is. If c s 0 then there will be an induced action on B. We now define the main object of study in this paper.

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DANIEL CHAN

DEFINITION 3.9. A quotient surface singularity is a ring of the form A s B G , where G is a finite group acting on a regular complete local ring B. If G ; SLŽ V . then A is said to be a special quotient surface singularity.

4. THE ASSOCIATED GRADED RING Throughout this section, let k denote an algebraically closed field of characteristic zero. Many properties of filtered rings may be deduced from the associated graded ring. The latter is usually easier to work with. Let B s k²u, v:rŽvu y quv. or k²u, v:rŽvu y uv y v 2 .. Proposition 3.6 states that the associated graded ring of a special quotient surface singularity has the form B G Žwhere G is a finite subgroup of SLŽku q kv... These graded rings were computed in Proposition 3.7. There were two cases when the ring was not commutative: let A q denote the ring defined by the relations Ž2. of the proposition and let A J denote the ring defined by the relations Ž3.. We study these two rings in this section. All rational double points have embedding dimension three. This shows that they are Gorenstein. We would like a noncommutative version of this result, at least in the associated graded case. As in the previous section, let q s q d. The role of the coordinate ring of ambient 3-space will be played by the ring S [ k ² x, y, z :r Ž yx y qxy, zx y q d xz, zy y qyz . for A q and by T [ k ² x, y, z :r Ž yx y xy y xz, zx y xz y 2 yz y 2 z 2 , zy y yz y z 2 . for A J . We study these two rings first. LEMMA 4.1. The ordered monomials x i y j z k ¬ i, j, k G 04 form a k-basis for S and for T. Proof. We wish to apply Bergman’s diamond lemma. Order the variables by z - y - x. The relations in S and T yield reductions which replace the out of order monomials yx, zx, zy with sums of ordered monomials. This reduction is compatible with the order $ of Lemma 3.1 so we may apply Bergman’s diamond lemma. This will show that the ordered monomials form a basis once we resolve the overlap ambiguity for the monomial zyx. Using ¬ to denote reductions, we have in A q , z Ž yx . ¬ qzxy ¬ q dq 1 xzy ¬ q dq 2 xyz

Ž zy . x ¬ qyzx ¬ q

dq 1

yxz ¬ q

dq 2

and xyz,

NONCOMMUTATIVE RATIONAL DOUBLE POINTS

737

which checks out. In A J we have z Ž yx . ¬ zxy q zxz ¬ xzy q 2 yzy q 2 z 2 y q zxz ¬ xyz q xz 2 q 2 y 2 z q 2 yz 2 q 2 zyz q 2 z 3 q zxz

Ž zy . x ¬ yzx q z 2 x ¬ yxz q 2 y 2 z q 2 yz 2 q zxz q 2 zyz q 2 z 3 ¬ xyz q xz 2 q 2 y 2 z q 2 yz 2 q zxz q 2 zyz q 2 z 3 , which agree as well so we are done. Let R be a ring, ␣ an automorphism of R, and ␦ an ␣-derivation of R; i.e., ␦ satisfies the skew-Leibniz rule ␦ Ž rs . s r␦ Ž s . q ␦ Ž r . ␣ Ž s ., r, s g R. Recall that associated to these data, there exists a ring called the Ore extension Rw x; ␣ , ␦ x. As an R-module, this ring is isomorphic to the free module [i G 0 x i R. Multiplication is determined by the rule rx s x ␣ Ž r . q ␦ Ž r . for all r g R. There is a converse result whose proof we omit. LEMMA 4.2. Let R ª RX be a ring extension such that as an R-module, R , [iG 0 x i R for some x g RX . Suppose there exists an automorphism ␣ of R such that rx y x ␣ Ž r . g R for all r g R. Then RX is an Ore extension of R. X

PROPOSITION 4.3. Let R 0 s S or T. Then there exists a chain of subrings R 0 > R1 > R 2 > R 3 s k where R i is an Ore extension of R iq1 for i s 0, 1, 2. Proof. For R 0 s S, this is clear so let R 0 s T. We let R 2 s k w z x which is an Ore extension of k. The subring R1 of R 0 generated by y and z Žover k . is [iG 0 y i R 2 as an R 2-module by Lemma 4.1. The above lemma shows that R1 is an Ore extension of R 2 . Let ␣ in Lemma 4.2 be the automorphism of R1 which maps y ¬ y q z, z ¬ z. This is an automorphism since R1 , k ² y, z :rŽ zy y yz y z 2 .. Applying Lemma 4.2 we conclude that T is an Ore extension of R1. There are surjective ring homomorphisms S ª A q and T ª A J . The kernels of these maps are given by the two-sided ideal generated by y d y xz where d is as in Ž2. of Proposition 3.7 for A q and equals 2 for A J . We wish to show that y d y xz is in fact normal. To prove this, we need a different ordering of the monomials. The following is elementary. LEMMA 4.4. Let ⌫ and ⌫X be semigroups and let - be a semigroup partial ordering on ⌫X which satisfies the descending chain condition. Let f : ⌫ ª ⌫X be a semigroup homomorphism. Then the induced order on ⌫ defined by a - b whene¨ er f Ž a. - f Ž b . is a semigroup partial ordering satisfying the descending chain condition. PROPOSITION 4.5. The monomials x i y j z k ¬ i, k G 0, 0 F j - d4 form a k-basis for A q and A J .

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DANIEL CHAN

Proof. Let ⌫ be the free semigroup generated by x, y, z and let ⌫X be the free semigroup generated by u, ¨ . Let f be the semigroup homomorphism from ⌫ ª ⌫X which maps x ¬ u d, y ¬ u¨ , z ¬ ¨ d. Let $ be the order on ⌫X considered in the previous section which comes from setting X ¨ - u. The partial order - on ⌫ we wish to use is that induced from ⌫ via f. When we apply Bergman’s diamond lemma in this case, we have all the reductions that we used for S and T plus the additional reduction y d ¬ xz. These reductions are all compatible with the new partial order - . The monomials above are precisely the irreducible ones. We need only resolve overlap ambiguities for the monomials zyx, zy d, y d x, and y dq 1. We have already resolved the ambiguity for the monomial zyx in S and T so only the other three need to be resolved. Consider first A q .

Ž zy . y dy1 ¬ qyzy dy 1 ¬ ⭈⭈⭈ ¬ q d y d z ¬ q d xz 2

and

z Ž y d . ¬ zxz ¬ q d xz 2 which agree while y dy 1 Ž yx . ¬ qy dy 1 xy ¬ ⭈⭈⭈ ¬ q d xy d ¬ q d x 2 z

Ž y . x ¬ xzx ¬ q d

and

d 2

x z

which verifies the second monomial. Finally y Ž y d . ¬ yxz ¬ qxyz

and

Ž y d . y ¬ xzy ¬ qxyz,

resolving the last ambiguity. Now consider A J . Here d s 2. We check the monomial zy 2 .

Ž zy . y ¬ yzy q z 2 y ¬ y 2 z q yz 2 q zyz q z 3 ¬ xz 2 q yz 2 q yz 2 q 2 z 3 ¬ xz 2 q 2 yz 2 q 2 z 3 z Ž y 2 . ¬ zxz ¬ xz 2 q 2 yz 2 q 2 z 3 . For the monomial y 2 x we have y Ž yx . ¬ yxy q yxz ¬ xy 2 q xzy q xyz q xz 2 ¬ x 2 z q xyz q xz 2 q xyz q xz 2 ¬ x 2 z q 2 xyz q 2 xz 2

Ž y 2 . x ¬ xzx ¬ x 2 z q 2 xyz q 2 xz 2 , which agree as well so it remains to consider y 3 :

NONCOMMUTATIVE RATIONAL DOUBLE POINTS

y Ž y 2 . ¬ yxz ¬ xyz q xz 2 ,

739

Ž y 2 . y ¬ xzy ¬ xyz q xz 2 ,

which verifies the last overlap check. Recall that for any graded k-module M s [Mn , the Hilbert function of M is h M Ž n. s dim k Mn . PROPOSITION 4.6. The element h s y d y xz is normal in S and T. Hence A q , SrhS and A J , TrhT. Proof. In S, h skew commutes with x, y, and z so the proposition follows. We give a proof for T which works equally well for S. Lemma 4.1 shows that hT Ž n. s 12 Ž n q 2.Ž n q 1. while the previous proposition shows that h A JŽ n. s 2 n q 1. Now h is a regular element since T is an iterated Ore extension of a domain. This gives an exact sequence, h

0 ª T w y2 x ª T ª TrhT ª 0, where the wy2x denotes the shift in grading by two. We deduce that hT r hT Ž n . s hT Ž n . y hT Ž n y 2 . s

1 2

Ž n q 2. Ž n q 1. y n Ž n y 1.

s 2 n q 1. Since A J , TrThT has the same Hilbert function, we must have hT s ThT, which proves the proposition.

5. EXAMPLES OF CYCLIC QUOTIENT SINGULARITIES In this section, we give examples of special quotient surface singularities in the more amenable form of generators and relations. Let B be the associated graded ring of a regular local ring of dimension two and let G be as usual, a subgroup of SLŽ B1 . whose action extends to B. Proposition 3.6 states that for any special quotient surface singularity A, the associated graded ring must have the form B G . In this section, we study such rings. Properties of filtered rings are often determined by their associated graded rings. This suggests that this class of algebras may be of interest in its own right. Throughout this section, we fix an algebraically closed field k of characteristic 0. Unfortunately, we have only obtained results in the case where G is a diagonal and hence cyclic subgroup of SLŽ B1 .. DEFINITION 5.1. A ring A over k, complete with respect to a filtration F, is said to be a singularity of type A dy 1 if the associated graded ring grF A has the form B G for G ; SLŽ B1 ., the diagonal subgroup of order d. If B , k ² u, ¨ :rŽ ¨ u y qu¨ . then we shall say that A is a q-singularity where q s q d and if B , k ² u, ¨ :rŽ ¨ u y u¨ y ¨ 2 . then we shall say that A is a Jordan singularity.

740

DANIEL CHAN

Let A q and A J be the graded rings of the previous section. They are defined by the relations Ž2. and Ž3. of Proposition 3.7 which we call the standard relations for A q and A J . We wish to determine all singularities of type A dy 1. Let P be the noncommutative power series ring k ²² x, y, z ::. We set the degrees of x and z to be d and the degree of y to be 2. We filter P using the degree as in Section 2. Our first step will be to construct a family of singularities of type A dy 1 depending on a parameter g g kz q F 3 P. Note that when d G 3, then kz ; F 3 P so it plays no role. The inclusion of kz is to account for Jordan singularities in the d s 2 case. THEOREM 5.2. relations yx s qxy q xg ,

Suppose A is a quotient of P s k ²² x, y, z :: with defining d

zx s Ž qy q g . ,

zy s qyz q gz,

y d s xz, Ž 4 .

where g g kz q F 3 P. Then A is a singularity of type A dy 1. It is a Jordan singularity when g f F 3 P and q s 1 and a q-singularity otherwise. N. B. The origin of these relations will be clear from the proof of the next theorem. Proof. We wish to apply Proposition 2.4 to determine the associated graded ring of A. We order monomials as in Proposition 4.5 and note that the relations Ž4. are compatible with this partial order. We need to check overlaps for the monomials y d x, zyx, zy d, and y dq 1. 2

y dy1 Ž yx . ¬ y dy1 x Ž qy q g . ¬ y dy2 x Ž qy q g . ¬ ⭈⭈⭈ ¬ x Ž qy q g .

d

Ž y d . x ¬ xzx ¬ x Ž qy q g . d , verifying the overlap check for y d x. The overlap check for zy d is similar so we consider zyx. z Ž yx . ¬ zx Ž qy q g . ¬ Ž qy q g .

dq 1

Ž zy . x ¬ Ž qy q g . zx ¬ Ž qy q g .

and dq 1

,

which also agree. Finally, we have

Ž y dq 1 . y ¬ xzy ¬ qxyz q xgz

and

y Ž y dq 1 . ¬ yxz ¬ qxyz q xgz.

This verifies the hypotheses for Proposition 2.4. Thus if x, y, z are the images of x, y, z in grF A and g is the image of g in F 2 PrF 3 P, then grF A is the k-algebra with generators x, y, z and defining relations yx s qxy s xg ,

d

zx s Ž qy q g . ,

zy s qyz q gz,

y d s xz.

741

NONCOMMUTATIVE RATIONAL DOUBLE POINTS

These are in fact defining relations for A q or A J too. To see this, observe first that when g g F 3 P so that g s 0, then the above equations reduce to the standard relations ŽŽ2. of Proposition 3.7. for A q . Suppose now that g s az for some nonzero a g k. If q s 1 then the linear change of variables x ¬ ay2 x, y ¬ ay1 y, z ¬ z gives the standard relations ŽŽ3. of Proposition 3.7. for A J . Similarly, when q / 1 the linear change of variables x ¬ x q aŽ q q 1. Ž q y 1.

y1

2

y q q Ž q q 1. Ž q y 1.

y ¬ y q aŽ q y 1.

y1

x,

y2

z,

z¬z

gives the standard relations ŽŽ2. of Proposition 3.7. for A q . This finishes the proof of the theorem. We have a converse result. THEOREM 5.3. Any singularity A of type A dy 1 is isomorphic to the quotient of P s k ²² x, y, z :: with defining relations Ž4. of Theorem 5.2. More precisely, let x, y, z be generators of grF A satisfying the standard relations for A q or A J . Then there exist lifts x, y, z of x, y, z to A such that the relation y d s xz holds. Suppose now that x, y, z are any lifts of x, y, z to A such that y d s xz. If A is a q-singularity then there exists g g F 3A such that the other relations in Ž4. also hold. If A is a Jordan singularity then there exists h g F 3A such that the relations in Ž4. hold with g s z q h. Proof. Suppose first that A is a q-singularity. We saw in Proposition 4.5 that the hypotheses for Bergman’s diamond lemma were satisfied for grF A. Hence, we may apply Proposition 2.4 which shows that A is a quotient of P with defining relations yx s qxy q zX ,

zx s q d y d q yX ,

zy s qyz q xX ,

y d s xz q wX , Ž ).

where xX , zX g F dq3A, yX , wX g F 2 dq1A. Observe that in the last relation, we can eliminate the higher order term as follows. First note that Propositions 4.5 and 2.1 yield the topological Grobner basis x i y j z k ¬ i, k G 0, ¨ X 0 F j - d4 for A. Hence, since the degree r of w is at least 2 d q 1, we can write wX s xz˜ q ˜ xz with ˜ x, ˜ z g F rydA. Replacing x with x 1 s x q ˜ x and z with z1 s z q ˜ z we find y d s x 1 z1 y wY , where wY s ˜˜ xz g F 2 ry2 dA, which is deeper in the filtration than wX . Iterating this procedure produces the desired elimination so henceforth we shall assume that wX s 0.

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DANIEL CHAN

We now determine some consequences of associativity.

Ž y d . x s xzx s q d xy d q xyX s q d x 2 z q xyX . This also equals dy1

Ž y . . . Ž y Ž yx . . . . . . s q d x 2 z q Ý q i y dy1yi zX y i . is0

Similarly, qxyz q zX z s yxz s y Ž y d . s Ž y d . y s xzy s qxyz q xxX . This yields two ‘‘diamond lemma’’ equations, xyX s

dy1

Ý q i y dy1yi zX y i

Ž 5.

is0

zX z s xxX .

Ž 6.

Writing our xX and zX in terms of the Grobner basis and comparing ¨ coefficients in Eq. Ž6., we see we must have xX s gz and zX s xg for some g g F 3A. Note that by induction, y i x s x Ž qy q g . i so Ž5. gives xyX s

dy1

Ý q i x Ž qy q g . dy1yi gy i .

is0

Since A is a domain, we may left cancel by x and then use the noncommutative binomial theorem to deduce yX s Ž qy q g . y q d y d , d

proving the theorem when A is a q-singularity. Suppose now that A is a Jordan singularity. As in the previous case, we may apply Proposition 2.4 to deduce that A is a quotient of P with defining relations yx s xy q xz q zX ,

zx s y 2 q 2 yz q 2 z 2 q yX ,

zy s yz q z 2 q xX ,

y d s xz q wX . As before, we may change variables to eliminate wX . We consider some consequences of associativity. xyz q xz 2 q zX z s yxz s y 3 s xzy s xyz q xz 2 q xxX .

NONCOMMUTATIVE RATIONAL DOUBLE POINTS

743

Hence we get xxX s zX z as in Eq. Ž6.. It is thus possible to find h g F 3 P such that xX s hz, zX s xh. This gives all the relations in Ž4. except the second one. We determine yX by considering the monomial zy 2 . z Ž y 2 . s zxz s y 2 z q 2 yuz 2 q 2 z 3 q yX z

Ž zy . y s yzy q z 2 y q hzy s y 2 z q yz 2 q yhz q zyz q z 3 q zhz q hyz q hz 2 q h 2 z s y 2 z q yz 2 q yhz q yz 2 q z 3 q hz 2 q z 3 q zhz q hyz q hz 2 q h 2 z. On comparing these two equations we see that yX s yh q 2 hz q zh q hy q h 2 . Substituting back in gives zx s y 2 q 2 yz q 2 z 2 q yh q 2 hz q zh q hy q h2 . We compare this with the right hand side of the corresponding relation in Ž4.: 2 Ž y q z q h . s y 2 q yz q zy q z 2 q yh q hz q zh q hy q h2

s y 2 q 2 yz q 2 z 2 q yh q 2 hz q zh q hy q h2 . Since these coincide, we are done. It is hoped that every singularity A of type A dy 1 is a special quotient surface singularity; that is, ‘‘Spec A’’ possesses a smooth d-fold cover. We have only found d-fold covers for certain q-singularities. Fortunately, these already include many interesting examples of quotient singularities. Recall that q g k is said to be generic with respect to dth roots of unity if 1 is the only power of q which is a dth root of unity. This is equivalent to the fact that either q is not a root of unity or q is a primitive nth root of unity where n is coprime to d. Note for such q that the geometric sums 1 li Ý dy are non-zero for every l g ⺪. To simplify notation, we will supis0 q press the P in F p P and denote it by just F p. THEOREM 5.4. Let q g k be generic with respect to dth roots of unit and let A be a q-singularity of type A dy 1. Let q be any dth root of q which is also generic with respect to dth roots of unity. Ž N. B. Such a q always exists.. Then there exists a complete regular local ring of dimension two, B s k ²² u, ¨ ::rŽ ¨ u y qu¨ y c ., such that A , B G where G is the cyclic group ² : of order d which acts on B ¨ ia : u ¬ ⑀ u, ¨ ¬ ⑀y1 ¨ for any primiti¨ e dth root of unity ⑀ .

744

DANIEL CHAN

Proof. By Theorem 5.3 we may assume A has the relations Ž4.. We consider g in the relations as a noncommutative power series in x, y, z. There are many choices for g and we pick one at random. We set P s k ²² u, ¨ :: where u, ¨ have degree one and let G act by : u ¬ ⑀ u, ¨ ¬ ⑀y1 ¨ . Let F be the natural filtration on P induced by degree. We need to find x, y, z, c g P G such that, modulo the relation ¨ u s qu¨ q c, the variables x, y, z satisfy relations Ž4. and generate P G . It turns out that we may choose y s q Ž1yd.r2 u¨ where q Ž1yd.r2 denotes either of the two square roots of q Ž1yd.. The key reduction in the proof of the theorem is the following lemma. LEMMA 5.5. Let y s q Ž1yd.r2 u¨ , x 0 s u d, z 0 s ¨ d, c 0 s 0. It suffices to find inducti¨ ely x i , z i , c i g P G such that the equations hold yx i ' qx i y q x i g i

modŽ ri . q ku d ¨ 2q i q F dq 3qi

z i y ' qyz i q g i z i

modŽ ri . q uF 1q i ¨ d q F dq 3qi

y d ' x i zi

Ž †.

modŽ ri . .

where ri s ¨ u y qu¨ y c i , g i s g Ž x i , y, z i ., and also x iq1 ' x i , z iq1 ' z i mod F dq 1qi and c iq1 ' c i mod F 3q i. The congruences hold when i s 0. Proof. We need to deal simultaneously with all the regular complete local rings PrŽ ri . so it is convenient to work in P as follows. Recall from Lemma 3.4 that the diamond lemma on PrŽ ri . yields a topological Grobner basis consisting of the monomials u j ¨ l 4 . The diamond lemma ¨ actually gives more. It gives a projection i g End k Ž P . onto the completion of the linear span of the Grobner basis such that, for a, b g P, a ' b ¨ modŽ ri . if and only if i Ž a. s i Ž b .. This projection comes from the reduction system ‘‘replace the submonomial ¨ u with qu¨ q c i ’’ Žsee wB, p. 180x.. Also, for any p g ⺞, the congruence a ' b modŽ ri . q F p is equivalent to i Ž a. ' i Ž b . mod F p since the i ’s do not decrease degree. In general, the computation of i Ž a. involves an infinite number of replacements ¨ u ¬ qu¨ q c i but the computation of i Ž a. modulo F p stops after a finite number of replacements. Let x, z, c be the limits of x i 4 , z i 4 , c i 4 , and g End k P, the projection associated to c. We wish to show that B s PrŽ r . is the sought for ‘‘smooth d-fold cover’’ of the theorem. Note that since c i , c g P G , G acts on PrŽ ri . and PrŽ r ..

NONCOMMUTATIVE RATIONAL DOUBLE POINTS

745

Theorem 5.3 applied to Ž PrŽ ri .. G shows that there exists ˜ g g P G such that yx i ' qx i y q x i ˜ g

modŽ ri .

z i y ' qyz i q ˜ gz i

modŽ ri .

y ' x i zi d

Ž ).

modŽ ri . .

We compare the first congruence with the first congruence of Ž†.. Note that the lowest degree term of i Ž x i Ž g i y ˜ g .. is the product of u d with the lowest degree term of i Ž g i y ˜ g .. Hence

i Ž g i y ˜ g. ' 0

mod k¨ 2q i q F 3q i .

Similarly, comparing the second congruences of Ž). and Ž†. we find

i Ž g i y ˜ g. ' 0

mod uF 1q i q F 3q i .

Now the intersection of k¨ 2q i q F 3q i and uF 1q i q F 3q i is F 3q i so in fact gi ' ˜ g modŽ ri . q F 3q i and the first two congruences of Ž†. become yx i ' qx i y q x i g i

modŽ ri . q F dq 3qi

Ž 7.

z i y ' qyz i q g i z i

mod Ž ri . q F dq 3qi .

Ž 8.

We can now show that A , B G . It is clear that B G is Žtopologically. generated by the images of x, y, z in B so it suffices to show they satisfy the desired defining relations. Now let g also denote g Ž x, y, z .. Modulo F dq 3qi we have 0 ' i Ž yx i y qx i y y x i g i . ' i Ž yx y qxy y xg . ' Ž yx y qxy y xg . . The last congruence follows from the fact that if W is of degree G d q 2 then ŽW . y i ŽW . g F dq 3qi since c ' c i mod F 3qi. Hence in B we have yx s qxy q xg which is the first of the defining relations in Ž4.. Similarly we have y d s xz so we may apply Theorem 5.3 to find relations yx s qxy q xg X ,

zx s Ž qy q g X . , d

zy s qyz q g X z.

Since B is a domain we must have g s g X , verifying the lemma. We now return to the proof of the theorem for which we need to verify the inductive step in the lemma. We may assume the congruences Ž7., Ž8. and the last congruence of Ž†. hold for some particular value of i and set c iq1 s c i q ⌬ c, x iq1 s x i q ⌬ x, z iq1 s z i q ⌬ z, where ⌬ c g F 3q i P G and ⌬ x, ⌬ z g F dq 1qi P G are to be solved for. We may replace the last congru-

746

DANIEL CHAN

ence in Ž†. with the weaker congruence y d ' x i zi ,

modŽ ri . q F 2 dq1qi

Ž 9.

since, by the argument in Theorem 5.3, we can alter x i and z i in degrees G d q 1 q i to obtain y d ' x i z i modŽ ri .. For the rest of the proof, it may be helpful to keep in mind that we are only interested in ⌬ c modulo F 4q i and ⌬ x, ⌬ z modulo F dq 2qi and that the congruences Ž†. we wish to solve lead to linear congruences. We consider the first congruence in Ž). and determine the effect of replacing c i with c iq1. As has been noted already, any term W of degree G d q 3 satisfies i ŽW . ' iq1ŽW . mod F dq 4qi. This accounts for all terms in the congruence except the lowest degree terms of yx i and x i y. The lowest degree term of x i y is u dq 1 ¨ which is already in lexicographic order so we deal with the former,

i Ž Ž q Ž1yd.r2 u¨ . u d . d

s qu d Ž q Ž1yd.r2 u¨ . q q Ž1yd.r2 i

ž žÝ

Ý q jy1 u j c i u dyj

js1

/

d

s qu d Ž q Ž1yd.r2 u¨ . q q Ž1yd.r2 i

q jy1 u j c iq1 u dyj

js1

/

d

y q Ž1yd.r2 i

žÝ

q jy1 u j Ž ⌬ c . u dyj

js1

/

' iq1 Ž Ž q Ž1yd.r2 u¨ . u d . d

y q Ž1yd.r2 iq1

ž

Ý q jy1 u j Ž ⌬ c . u dyj js1

/

mod F dq4qi ,

which yields d

yx i ' qx i y q x i ˜ g q q Ž1yd.r2

Ý q jy1 u j Ž ⌬ c . u dyj

modŽ riq1 . q F dq4qi .

js1

We carry out a similar computation with the other congruences in Ž).. The second is similar to the first so we consider the last. Set t Ž n. s 12 nŽ n q 1.,

747

NONCOMMUTATIVE RATIONAL DOUBLE POINTS

the nth triangular number. Again, the key term will be the lowest degree one,

i Ž q Ž1yd.r2 u¨ .

ž

d

ytŽ dy1.

i Ž u Ž qu¨ q c i .

/ sq

dy1

s qytŽ dy1. i u Ž qu¨ q c i .

ž

¨

dy 1

. y Ž qu¨ .

dy1

qq dy1 u Ž u¨ . s qyt Ž dy1. i

dy 1

¨ ¨

/

dy1

žÝ

q tŽ dy1.ytŽ j. u dyj

js1 j

j

/

= Ž qu¨ q c i . y Ž qu¨ . ¨ dyj q u d ¨ d . Now the i ’s preserve left multiplication by u and right multiplication by ¨ so we may concentrate on the square bracketed term, j

i Ž qu¨ q c i . y Ž qu¨ .

j

j

s i Ž qu¨ q c iq1 y ⌬ c . y Ž qu¨ . j

' i Ž qu¨ q c iq1 . y Ž qu¨ .

j

j

j

y i q jy1

Ý Ž u¨ . ly1 Ž ⌬ c . Ž u¨ . jyl ls1

mod F 2 jq2qi j

' iq1 Ž qu¨ q c iq1 . y Ž qu¨ .

j

j

y iq1 q

jy1

Ý Ž u¨ . ly1 Ž ⌬ c . Ž u¨ . jyl ls1

mod F 2 jq2qi . Substituting back in and reversing the above computation for i q 1 instead of i gives

i Ž q Ž1yd.r2 u¨ .

ž

d

/

' iq1 Ž q Ž1yd.r2 u¨ .

ž

dy1

y iq1

d

/

j

Ý Ý q jy1ytŽ j. u dyj Ž u¨ . ly1 Ž ⌬ c . Ž u¨ . jyl ¨ dyj

js1 ls1

mod F 2 dq2qi .

748

DANIEL CHAN

Hence, dy1

y d ' x i zi q

j

Ý Ý q jy1ytŽ j. u dyj Ž u¨ . ly1 Ž ⌬ c . Ž u¨ . jyl ¨ dyj

js1 ls1

modŽ riq1 . q F 2 dq2qi Incorporating also the effect of changing x i , z i to x iq1 , z iq1 we find d

yx iq1 ' qx iq1 y q x iq1 ˜ g q q Ž1yd.r2

Ý q jy1 u j Ž ⌬ c . u dyj js1

q yŽ ⌬ x. y qŽ ⌬ x. y

modŽ riq1 . q F dq 4qi d

z iq1 y ' qyz iq1 q ˜ gz iq1 q q Ž1yd.r2

Ý q jy1 ¨ dyj Ž ⌬ c . ¨ j

Ž ‡.

js1

q Ž ⌬ z . y y qy Ž ⌬ z . j

dy1

y d ' x iq1 z iq1 q

modŽ riq1 . q F dq 4qi

Ý Ý q jy1ytŽ j. u dyj Ž u¨ . ly1 Ž ⌬ c . Ž u¨ . jyl ¨ dyj

js1 ls1

y Ž ⌬ x . z iq1 y x iq1 Ž ⌬ z .

modŽ riq1 . q F 2 dq2qi .

Let ⌬ g s i Ž g i y ˜ g . which, as was noted in the proof of the lemma, lies in F 3q i P G. Note that since g g F 3 we have g iq1 y g i g F 4q i. Also,

iq1 Ž g iq1 y ˜ g . ' i Ž g iq1 y ˜ g.

mod F 4q i

Hence on comparing the congruences Ž‡. with the desired ones Ž†. and Ž9., we see the theorem amounts to solving d

x iq1 Ž ⌬ g . ' q Ž1yd.r2

Ý q jy1 u j Ž ⌬ c . u dyj q y Ž ⌬ x . y q Ž ⌬ x . y js1

modŽ riq1 . q ku d ¨ 3q i q F dq 4qi

Ž 10 .

d

Ž ⌬ g . z iq1 ' q Ž1yd.r2

Ý q jy1 ¨ dyj Ž ⌬ c . ¨ j q Ž ⌬ z . y y qy Ž ⌬ z . js1

modŽ riq1 . q uF 2q i ¨ d q F dq 4qi dy1

0'

Ž 11 .

j

Ý Ý q jy1ytŽ j. u dyj Ž u¨ . ly1 Ž ⌬ c . Ž u¨ . jyl ¨ dyj y Ž ⌬ x . z iq1

js1 ls1

y x iq1 Ž ⌬ z .

modŽ riq1 . q F 2 dq2qi .

Ž 12 .

NONCOMMUTATIVE RATIONAL DOUBLE POINTS

749

The above congruences are linear in ⌬ x, ⌬ z, ⌬ c, ⌬ g and admit the trivial solution when ⌬ g g F 4q i. Hence, we may assume ⌬ g s u  ¨ ␥ where  q ␥ s 3 q i. There are two cases: Case 1.  ) 0. Let ⌬ x s 0 and ⌬ c s ␣ u  ¨ ␥ where ␣ g k is to be solved for. Note that ⌬ c g F 3q i P G since ⌬ g g F 3q i P G. Since d

iq1

ž

Ý q jy1 u j Ž ⌬ c . u dyj js1

d

'␣

/ ž

Ý q jy1q ␥ Ž dyj. js1

/

u dq  ¨ ␥

mod F dq4qi ,

solving Ž10. depends on the invertibility of Ý djs1 q jy1q ␥ Ž dyj.. But our hypothesis on q guarantees this geometric sum is non-zero. Now iq1 of the double sum term in Ž12. is a multiple of u dq y1 ¨ dq ␥y1 modulo F 2 dq2qi so we may solve Ž12. by setting ⌬ z to be an appropriate multiple of u y1 ¨ dq ␥y1. Note that ⌬ z g F dq 1qi P G as was required. Finally, to check Ž11. we need only observe that iq1 of every term lies in uP¨ d. Case 2.  s 0. Let ⌬ z s 0. As in the previous case we can solve Ž11. to find ⌬ c s ␣ ¨ ␥ and Ž12. to obtain ⌬ x as some multiple of u dy 1 ¨ ␥y1. The verification of Ž10. is as in the previous case. This completes the proof of the theorem.

6. LOCAL DUALITY In this section, we give an expose ´ of the theory of local duality for noncommutative complete local rings. The results in this section were obtained by Wu and Zhang wWZx and in a weaker form in wC, Sects. 2 and 3x. We will content ourselves with stating the main results of the theory since proofs will be given elsewhere and are, for the most part, simple translations of the results for noncommutative graded algebras Žsee wY1, VdBx.. Throughout this section, we will be concerned with a Noetherian local k-algebra A with maximal ideal ᒊ which is complete with respect to the ᒊ-adic filtration. We shall also assume that the residue field Arᒊ is naturally isomorphic to k. Wu and Zhang omit this last assumption and also only assume that A is one-sided Noetherian. Local duality involves two duality functors. The first of these is Matlis duality which swaps the category of Noetherian right A-modules with the category of artinian left A-modules. Let Ak denote the Ž A, A.-bimodule p Ž . lim ª p Hom k Arᒊ , k , that is, the continuous dual of A.

750

DANIEL CHAN

PROPOSITION 6.1 wC, Lemma 1.1.16x. The bimodule Ak is the injecti¨ e hull of Arᒊ as a left module and as a right module. Hence, it makes sense to define DEFINITION 6.2. The Matlis dual of a module M is M k[ Hom AŽ M, A . k.

When the module M is Noetherian or Artinian, then the Matlis dual is also given by the continuous dual Žsee wC, Proposition 1.1.21x.. The topology on an Artinian module is discrete so the Matlis dual is just the k-linear dual. If M is a bimodule, then there may be some ambiguity as to whether the Matlis dual refers to the left or right module structure. In this case, we will write out the functor in full, as Hom AŽy, E . or Hom A⬚Žy, E . accordingly. Let N Ž A. denote the category of right Noetherian A-modules. Similarly, we let AŽ A. denote the category of right Artinian modules. We will drop the argument A when it is understood. Let A⬚ denote the opposite ring of A. Then we may identify N Ž A⬚. and AŽ A⬚. with the category of left Noetherian and Artinian modules. We let C Ž A. denote the Gabriel product AŽ A. . N Ž A. which is the category of modules which are extensions of Noetherian modules by Artinian modules. Let B ª C be a ring homomorphism and let K be a full Abelian subcategory of Mod-B. Suppose that the full subcategory of C-modules which are in Obj K when considered as B-modules is closed under extensions. Then we let DUK Ž B . denote the full subcategory of the derived U category DU Ž B . consisting of complexes with cohomology in K Žhere denotes q, y, b or can be omitted.. The unadorned tensor symbol m will always mean mk . We obtain the following version of Matlis duality. THEOREM 6.3 ŽMatlis duality.. The Matlis dual defines a duality between N Ž A. and AŽ A⬚. which extends to a duality between C Ž A. and C Ž A⬚.. It hence induces a duality between D C Ž A. and D C Ž A⬚.. Let B and C be q Ž . Ž . k-algebras and M g Dy C Ž A . B⬚ m A and N g D C Ž A . C⬚ m A . Then there is a natural isomorphism RHom A Ž M, N . , RHom A⬚ Ž N k , M k . in DqŽ C⬚ m B .. The proof of this theorem can be found in wWZ, Proposition 3.3Ž3.; C, Theorem 1.1.25x. We consider now the second duality functor which occurs in local duality. Morally speaking, it is a duality between Noetherian left modules and Noetherian right modules.

NONCOMMUTATIVE RATIONAL DOUBLE POINTS

751

We need some notation first. Let A ª B be a morphism of k-algebras. We denote the restriction of scalars functor by res BA : Mod-B ª Mod-A. We will omit the subscript or superscript when there is no confusion. The restriction functor is exact and so extends to a functor from DŽ B . ª DŽ A.. In particular, if A e denotes the enveloping ring A⬚ m A then given M g DŽ Ae . it makes sense to consider A M and MA . Mimicking wY1, Definition 3.3x we make the DEFINITION 6.4. An element in D b Ž A e . is said to be a dualising complex for A if i. A and A have finite injective dimension; i.e., Ext Ai Žy, . s 0 i Ž s Ext A⬚ y, . for all i c 0. ii. A and A have Noetherian cohomology. iii. The canonical morphisms A ª RHom AŽ , . and A ª RHom A⬚Ž , . are isomorphisms in DŽ A e .. The simplest example of a dualising complex occurs when A itself has finite injective dimension as a right module and as a left module. Then s A is a dualising complex. Dualising complexes induce the aforementioned duality as follows: PROPOSITION 6.5 wY1, Proposition 3.5x. Let C be a k-algebra. If is a dualising complex then the functors RHom AŽy, . and RHom A⬚Žy, . define a duality between DNbŽ A.Ž C⬚ m A. and DNbŽ A⬚.Ž A⬚ m C .. If a dualising complex exists, then it is unique up to the Picard group of invertible bimodules. Recall that an invertible bimodule is a bimodule L for which there exists an inverse bimodule Ly1 ; i.e., L mA Ly1 , A , Ly1 mA L. These are easily classified as follows. Given an algebra automorphism of A, let AŽ . denote the Ž A, A.-bimodule which is isomorphic to A as a right module and whose left module structure is given by a . x s Ž a. x for a, x g A where on the left we have scalar multiplication and on the right we have multiplication in A. Then AŽ . is an invertible bimodule and every invertible bimodule has this form. The uniqueness result for dualising complexes can now be precisley stated. Below, we use w d x to denote a shift by d places of a complex. THEOREM 6.6 wY1, Theorem 3.9x. Let be a dualising complex for A. Then X is a dualising complex for A if and only if , mA Lw d x in DŽ Ae . for some in¨ ertible bimodule L and d g ⺪. In commutative algebra, local duality relates the two dualities using local cohomology. Let ⌫ᒊ denote the ᒊ-torsion functor for right modules and ⌫ᒊ ⬚ the ᒊ-torsion functor for left modules. The ith local cohomology

752

DANIEL CHAN

group of a right module M is defined to be H i Ž R⌫ᒊ Ž M .. and is denoted by Hᒊi Ž M .. Grothendieck’s local duality formula in the commutative case is k

R⌫ᒊ Ž M . s RHom A Ž M, . , where is the dualising complex which is chosen to be normalised Žsee wH1x for the definition.. Together with condition 3 for a complex to be dualising we see that for such an we have R⌫ᒊ Ž . , A k. In noncommutative algebra, Yekutieli discovered that even this relationship may fail for every dualising complex of a given ring. He thus introduced DEFINITION 6.7 wY1, Definition 4.1x. A dualising complex is said to be balanced if R⌫ᒊ Ž . , A k, R⌫ᒊ ⬚Ž . in DŽ A e .. Theorem 6.6 ensures uniqueness of balanced dualising complexes. The existence question for balanced dualising complexes is tied to local duality. To state the main result, we need some definitions. Following wAZ, Definition 3.7x, we define DEFINITION 6.8. A satisfies if R⌫ᒊ Ž M . has Artinian cohomology for every Noetherian right A-module M. By wH1, Chapter 1, Proposition 7.3x, this is equivalent to the fact that R⌫ᒊ is a functor from DNq to Dq A . We let cd denote the cohomological dimension of a functor. The key result is THEOREM 6.9. The following are equi¨ alent: i. A and A⬚ satisfy and cd ⌫ᒊ , cd ⌫ᒊ ⬚ are finite. ii. Local duality: There exists g D b Ž Ae . satisfying conditions Ži., Žii. of Definition 6.4 and the following: gi¨ en any k-algebra C and complexes M g DŽ C⬚ m A. and N g DŽ A⬚ m C ., Hom A Ž R⌫ᒊ Ž M . , A k . s RHom A Ž M, . , Hom A⬚ Ž R⌫ᒊ ⬚ Ž N . , A k . s RHom A⬚ Ž N, . hold in DŽ A⬚ m C . and DŽ C⬚ m A.. iii. A has a balanced dualising complex. If these conditions hold then is the balanced dualising complex and equals Hom A Ž R⌫ᒊ Ž A . , A k . s Hom A⬚ Ž R⌫ᒊ ⬚ Ž A . , A k . . This theorem is proved in wWZ, Theorem 3.6; C, Theorem 1.2.6x. We sometimes want to know when a dualising complex can be balanced, that is, when there exists an invertible bimodule L and an

NONCOMMUTATIVE RATIONAL DOUBLE POINTS

753

integer d such that mA Lw d x is a balanced dualising complex for A. The answer is given by PROPOSITION 6.10 wY1, Proposition 4.4 and Theorem 4.8x. Let be a dualising complex for A. Then for d g ⺪, the following conditions are equi¨ alent: i. RHom AŽ k, . , k wyd x in DŽ A e .. ii. RHom A⬚Ž k, . , k wyd x in DŽ A e .. iii. A has a balanced dualising complex of the form mA Lw d x for some in¨ ertible bimodule L. The first two conditions of the proposition are generalisations of the Gorenstein condition of Artin and Schelter Žsee wASchx.. One immediate corollary of this proposition is COROLLARY 6.11. Suppose that A has finite injecti¨ e dimension as a left module and as a right module and that RHom AŽ k, A. , k wyd x for some d g ⺪. Then A has a balanced dualising complex of the form Lw d x where L is an in¨ ertible bimodule. Furthermore, d is the injecti¨ e dimension of A as a right module or a left module. Proof. The first assertion follows from the fact that A is a dualising complex since it has finite injective dimension as a right and left module. The second assertion requires considering minimal injective resolutions which we define in DEFINITION 6.12. Let M g DŽ A.. A minimal injecti¨ e resolution of M is a quasi-isomorphism M ª I where I is a complex of injectives with the property that the inclusion of cocyles Z j ¨ I j is essential. The existence of such a resolution for M g DqŽ A. is guaranteed by wY1, Lemma 4.2x. We wish to determine the shape of a minimal injective resolution of a balanced dualising complex , say as a right module. Since A has finite injective dimension, we know it must have the form 0 ª yd ª ydq1 ª ⭈⭈⭈ ª e ª 0 for some d, e g ⺪. We need to introduce some notation. DEFINITION 6.13. For M g DŽ A., we let supŽ M . s sup i ¬ H i Ž M . / 04 and infŽ M . s inf i ¬ H i Ž M . / 04 . We define the injecti¨ e dimension of to be id [ yinfŽ .. By the proof of wH1, Chap. I, Proposition 7.6x and local duality, we see that es

sup MgMod-A

sup RHom A Ž M, . 4 s

sup MgMod-A

ysup R⌫ᒊ Ž M . k 4 s 0.

754

DANIEL CHAN

Minimality of the resolution ensures that there is non-zero cohomology at the yd term so we have in fact d s id . Furthermore, local duality shows that cd ⌫ᒊ s s

sup MgMod-A

sup

yinf R⌫ᒊ Ž M . k 4 yinf RHom A Ž M, . 4 s d.

MgMod-A

The above computation works equally well for a minimal left injective resolution of so we obtain PROPOSITION 6.14. A minimal injecti¨ e resolution of a balanced dualising complex for A has the form 0 ª yd ª ydq1 ª ⭈⭈⭈ ª 0 ª 0, where d s id s cd ⌫ᒊ s cd ⌫ᒊ ⬚. Conclusion of Proof of Corollary 6.11. Observe that the invertible bimodule L is isomorphic to A as a left module and as a right module so the resolution of the proposition is also the minimal injective resolution for A. This shows that d s id A A s id A A , giving the final assertion of the corollary. DEFINITION 6.15. If A satisfies the conditions of Corollary 6.11 then A is said to be AS᎐Gorenstein. If, furthermore, A has finite global dimension then A is said to be AS-regular. We have the following important example. PROPOSITION 6.16 wASx. The complete local ring A is regular of dimension two if and only if it is an AS-regular domain of global dimension two.

7. SOME CLASSICAL RESULTS We include in this section several generalisations of classical results in commutative algebra. These results are well known in the graded case and in the local case, have been proved by Wu and Zhang and in wC, Chap. 1, Sect. 4x. Again, for the most part, proofs will be omitted as they will appear elsewhere. Throughout this section, let Ž A, ᒊ . and Ž B, ᒋ . be Noetherian complete local rings with residue field k as in the previous section. PROPOSITION 7.1. Suppose there is a local homomorphism A ª B such that B is a finite A-module on the left and on the right. Let res denote the restriction functor from B-modules to A-modules. Then there is a natural

NONCOMMUTATIVE RATIONAL DOUBLE POINTS

755

isomorphism res( R⌫ᒋ , R⌫ᒊ (res of functors from DqŽ B . ª DqŽ A. or from DqŽ B e . ª DqŽ B⬚ m A.. Proof. Note first that res( ⌫ᒋ , ⌫ᒊ (res since B is finitely generated over A. Because restriction is an exact functor, it suffices by Grothendieck’s theorem to show that restriction maps any B-injective I to a ⌫ᒊ -acyclic. Now by wG, Chap. 3, Corollaire 2 of Proposition 6x, we are reduced to two cases: Ži. I is ᒋ-torsionfree and Žii. I is a direct sum of B k ’s. In the first case we consider the spectral sequence Ext iB Ž Tor jA Ž Arᒊ p , B . , I .

«

p Ext iqj A Ž Arᒊ , I . .

Since I is B-injective, the sequence collapses to Hom B Ž Tor jA Ž Arᒊ p , B . , I . , Ext Aj Ž Arᒊ p , I . . Now Tor jA Ž Arᒊ p , B . is finite dimensional over k so since we are assuming that I is ᒋ-torsionfree, the left hand side must be zero. Taking direct limits, we see that R⌫ᒊ Ž I . s 0. In the second case, R⌫ᒊ Ž I . s I by wVdB, Lemma 4.4x so I is again ⌫ᒊ -acyclic and the proposition follows. The following corollary was pointed out to me by Ammon Yekutieli. It follows directly from local duality and the above proposition. A full proof can be found in wC, Corollary 1.4.2x. COROLLARY 7.2 Žadjunction formula for finite morphisms.. Let Ž A, ᒊ . and Ž B, ᒋ . be as in the pre¨ ious proposition. If A has a balanced dualising complex A , then B has a balanced dualising complex and it is gi¨ en by

B , RHom A Ž B, A .

in D Ž A⬚ m B .

B , RHom A⬚ Ž B, A .

in D Ž B⬚ m A . .

The next result, communicated to me by James Zhang, is also a direct consequence of local duality. It is proved in wC, Corollary 1.4.4x. COROLLARY 7.3. Let Ž A, ᒊ . and Ž B, ᒋ . be as in Proposition 7.1 and suppose further that the map A ª B is split as a morphism of right A-modules and of left A-modules. If B has a balanced dualising complex B , then A has a balanced dualising complex A which is a direct summand of B when considered as an object in DŽ A. or DŽ A⬚..

8. REFLEXIVE AND COHEN᎐MACAULAY MODULES In this section, we collect some well-known results about reflexive modules and introduce Cohen᎐Macaulay modules. Throughout, let A

756

DANIEL CHAN

denote a Noetherian ring and let D denote the A-linear dual Hom AŽy, A. or Hom A⬚Žy, A. as the case may be. DEFINITION 8.1. A finitely generated A-module is said to be reflexi¨ e if the natural morphism M ª DDŽ M . is an isomorphism. Recall that if A is a semiprime Goldie and QŽ A. is its ring of fractions, then a module M is said to be torsionfree if the natural morphism M ª M mA QŽ A. is an injection, and torsion if the map is zero. There are many ways to characterise torsionfree modules. The following proposition is well known. Lacking a good reference, we include a proof here. PROPOSITION 8.2. Let A be a Noetherian domain and let M be a finitely generated right A-module. Then ker Ž M ª DD Ž M . . s ker Ž M ª M mA Q Ž A . . Hence, the following are equi¨ alent: i. M is torsionfree. ii. The natural isomorphism M ª DDŽ M . is an injection. iii. M can be embedded in a finitely generated free module F. In Žiii., the free module F may be chosen so that FrM is torsion. Proof. We first prove equality of the kernels which will give the equivalence of Ži. and Žii.. Let m g kerŽ M ª M mA QŽ A.. and f g Hom AŽ M, A.. Consider the commutative diagram A

6

fmA Q Ž A .

QŽ A .

6

6

M mA QŽ A.

f

6

M

Since the vertical morphism on the right is injective, f Ž m. s 0 so kerŽ M ª M mA QŽ A.. : kerŽ M ª DDŽ M ... Suppose now that m f kerŽ M ª M mA QŽ A... Since QŽ A. is a skew-field, we have an isomorphism M mA QŽ A. , QŽ A. n for some n. Further, as M is finitely generated, we have a commutative diagram of the form

NONCOMMUTATIVE RATIONAL DOUBLE POINTS

757

in Mod-A where x is left multiplication by some x g QŽ A.. We must have f Ž m. / 0 so we can find a homomorphism g: A n ª A such that the composite Ž g ( f .Ž m. / 0. Hence m f kerŽ M ª DDŽ M .. and the kernels must coincide. If Ži. holds, then f above gives the desired embedding of Žiii. with A nrf Ž M . torsion. Conversely, Žiii. implies Ži. since free modules are torsionfree. The next proposition gives alternative characterisations of reflexivity. PROPOSITION 8.3. Let A be a Noetherian domain and let M be a finitely generated right A-module. The following are equi¨ alent: i. M is reflexi¨ e. ii. There exists a finitely generated free module F and a torsionfree module G such that M fits into an exact sequence 0 ª M ª F ª G ª 0. iii. There exists a reflexi¨ e module F and a torsionfree module G such that M fits into an exact sequence 0 ª M ª F ª G ª 0. Proof. Žadapted from wH2, Proposition 1.1x.. We shall only prove the hard implication Žiii. « Ži.. Note that M is torsionfree. Consider the dual exact sequence 0 ª D Ž G . ª D Ž F . ª D Ž M . ª N ª 0, where N is the appropriate submodule of Ext 1AŽ G, A.. Dualising again we obtain the commutative diagram

where the bottom row is just a complex which is exact at the DDŽ M . term. We first show that N is torsion. Indeed, by the previous proposition, we can find a finitely generated free module P containing G with PrG torsion. Since QŽ A. is flat over A we have Q Ž A . mA Ext 1A Ž G, A . , Q Ž A . mA Ext 2A Ž PrG, A . , Ext 2QŽ A. Ž PrG mA Q Ž A . , Q Ž A . . s 0. N being a submodule of Ext 1AŽ G, A. must be torsion. Hence Proposition 8.2 shows that DŽ N . s 0 and so the map DDŽ M . ª DDŽ F . is injective.

758

DANIEL CHAN

The previous proposition also shows that M ª DDŽ M . and G ª DDŽ G . are injections. A diagram chase now completes the proof. PROPOSITION 8.4. Let A, B be Noetherian domains, M a finitely generated right A-module, and L a Ž B, A.-bimodule such that B L is reflexi¨ e. Then Hom AŽ M, L. is also a reflexi¨ e B-module. Proof. Consider a presentation Am ª An ª M ª 0. Taking homomorphisms into L gives the exact sequence 0 ª Hom A Ž M, L . ª Ln ª Lm . Since Ln is reflexive by assumption and every submodule of Lm is torsionfree, Hom AŽ M, L. is reflexive by Proposition 8.3. Following wY2, Definition 1.4x, we consider the DEFINITION 8.5. Let Ž A, ᒊ . be a local ring. Given a finitely generated A-module M, we define the canonical dimension of M to be ␦ Ž M . s supŽ R⌫ᒊ Ž M .. and the depth of M to be depthŽ M . s infŽ R⌫ᒊ Ž M ... DEFINITION 8.6. Let Ž A, ᒊ . be a local ring such that cd ⌫ᒊ is finite. Let M be a finitely generated A-module. M is said to be Cohen᎐Macaulay if ␦ Ž M . s depthŽ M .. If depthŽ M . s cd ⌫ᒊ then M is said to be maximal Cohen᎐Macaulay. If Ž A, ᒊ . is a complete local ring with a balanced dualising complex , then by local duality we may use RHom AŽy, . instead of R⌫ᒊ to test a module is Cohen᎐Macaulay. A module M is Cohen᎐Macualay if and only if its dual RHom AŽ M, . has only one non-zero cohomology group. DEFINITION 8.7. Let A be a complete local ring with balanced dualising complex . We say that A is AS᎐Cohen᎐Macaulay if is isomorphic to a shift of a bimodule. Examples of AS᎐Cohen᎐Macaulay rings include AS᎐Gorenstein rings Žsee Definition 6.15.. In fact, these are precisely the AS᎐Cohen᎐Macaulay rings whose dualising complexes are isomorphic to a shift of an invertible bimodule. Remark. If A is AS᎐Cohen᎐Macaulay then, as an A-module, A is maximal Cohen᎐Macaulay since a minimal injective resolution for its dual has the form 0 ª yd ª ydq1 ª ⭈⭈⭈ ª 0 ª 0,

NONCOMMUTATIVE RATIONAL DOUBLE POINTS

759

where d s cd ⌫ᒊ by Proposition 6.1. We refer to d simply as the dimension of A. Applying the long exact sequence in local cohomology to the exact sequence in Proposition 8.3Žiii. yields another classic result from commutative algebra. PROPOSITION 8.8. Let A be an AS᎐Cohen᎐Macaulay domain of dimension two. If M is reflexi¨ e then it is maximal Cohen᎐Macaulay. There is a natural noncommutative analogue of the Auslander᎐Buchsbaum formula. In the graded case, this has already been established by Jørgensen in wJx. The local case is proved in wWZ, Theorem 5.8; C, Theorem 1.5.9x. THEOREM 8.9 ŽAuslander᎐Buchsbaum formula.. Let A be a complete local ring with balanced dualising complex and let M be a finitely generated module of finite projecti¨ e dimension. Then the projecti¨ e dimension is gi¨ en by pd M s depth Ž A . y depth Ž M . . COROLLARY 8.10. Let A be an AS-regular ring Ž Definition 6.15.. Then e¨ ery maximal Cohen᎐Macaulay module is free.

9. FINITENESS OF REPRESENTATION TYPE We assume throughout this section that the base field k is algebraically closed of characteristic zero. Quotient surface singularities have balanced dualising complexes. In fact, Proposition 7.3 shows that PROPOSITION 9.1. E¨ ery quotient surface singularity is AS᎐Cohen᎐Macaulay of dimension two. We next establish a noncommutative analogue of the fact that rational double points are Gorenstein. For this, we will need to introduce the Auslander condition. DEFINITION 9.2. A Noetherian ring A satisfies the Auslander condition if for every finitely generated left or right module M, every i g ⺞, and every submodule N : Ext iAŽ M, A. we have inf Ž RHom A Ž N, A . . G i. DEFINITION 9.3. A Noetherian ring A is said to be Auslander᎐Gorenstein if it has finite injective dimension as a left and right module and

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satisfies the Auslander condition. If, furthermore, A has finite global dimension then A is said to be Auslander regular. For us, the relevance of this notion is given by Levasseur’s THEOREM 9.4 wL, Theorem 6.3x. Let A be a Noetherian complete local ring with residue field k. If A is Auslander᎐Gorenstein then it is AS᎐Gorenstein. The proof in wLx is given for graded rings but works equally well in the local case. We need two criteria for testing to see if a ring is Auslander᎐Gorenstein. THEOREM 9.5 wBj, Theorem 3.9x. Let Ž A, F . be a complete filtered ring such that grF A is Noetherian. Suppose that grF A is Auslander᎐Gorenstein of injecti¨ e dimension d. Then A is Auslander᎐Gorenstein of injecti¨ e dimension F d. THEOREM 9.6 wL, Theorem 3.6Ž2.x. Let A be a Noetherian ⺞-graded algebra. Let x g A be an homogeneous regular normal element of positi¨ e degree. Then ArxA is Auslander᎐Gorenstein of injecti¨ e dimension d if and only if A is Auslander᎐Gorenstein of injecti¨ e dimension d q 1. Let S and T be the rings defined in Section 4. COROLLARY 9.7. The rings S and T are Auslander regular of global dimension three. Proof. Since S and T are three-fold iterated Ore extensions ŽLemma 4.3., Hilbert’s syzygies theorem guarantees that S and T have global dimension three. We need to show they are Auslander᎐Gorenstein. Now, x, y, and z are normal in S so applying Theorem 9.6 three times shows that S is Auslander᎐Gorenstein. Also, z is normal in T and TrzT , k w x, y x which is commutative Gorenstein so Levasseur’s theorem again shows that T is Auslander᎐Gorenstein. Let B s k ² u, ¨ :rŽ ¨ u y qu¨ . or k ² u, ¨ :rŽ ¨ u y u¨ y ¨ 2 .. Let A s B G Žwhere G is a finite subgroup of SLŽ ku q k¨ ... PROPOSITION 9.8. Let Ž A, F . be a complete filtered ring whose associated graded ring is isomorphic to A. Then A is Auslander᎐Gorenstein of injecti¨ e dimension F 2 and hence AS᎐Gorenstein. If A is a special quotient surface singularity then the injecti¨ e dimension actually equals two. Proof. By Proposition 3.7 A is either commutative or equals A q or A J of the previous section. In the commutative case, it is well known that A is Gorenstein of injective dimension two and hence Auslander᎐Gorenstein.

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In the other cases, Proposition 4.6 shows that A is a quotient of the domain S or T by a normal homogeneous element so by Theorem 9.6, A is Auslander᎐Gorenstein of injective dimension two. Bjork’s Theorem 9.5 ¨ and Proposition 9.1 now yield the proposition. DEFINITION 9.9. A complete local ring A with balanced dualising complex is said to have finite representation type if the number of isomorphism classes of indecomposable maximal Cohen᎐Macaulay modules is finite. The proof for the next theorem is taken from wAus, Proposition 2.1x. It is included here so the reader can see that all the elements of the proof have indeed been established in the noncommutative case. THEOREM 9.10. Let B be a regular complete local ring of dimension two and let A s B G Ž G ; SL 2 as before. be a special quotient surface singularity. Then A has finite representation type and, in fact, the indecomposable maximal Cohen᎐Macaulay modules are precisely the indecomposable summands of B. Proof. Note that the category of Noetherian A-modules is Krull᎐ Schmidt by the usual argument Žsee wC, Corollary 1.6.2x for example.. Hence it suffices to show that any indecomposable maximal Cohen᎐Macaulay module M is a direct summand of B and that B is maximal Cohen᎐Macaulay as an A-module. Let M X s Hom AŽ M, A.. Since A is AS᎐Gorenstein, A is a dualising complex for A. Furthermore, since M is maximal Cohen᎐Macaulay, RHom AŽ M, A. , Hom AŽ M, A. so M is reflexive. Now 0 ª A ª B is split as a sequence of left A-modules so we have a split sequence of right A-modules: 0 ª M s Hom A⬚ Ž M X , A . ª Hom A⬚ Ž M X , B . . Proposition 8.4 shows that Hom A⬚Ž M X , B . is a reflexive B-module so by Proposition 8.8, it is maximal Cohen᎐Macaulay and hence free over B ŽCorollary 8.10.. This shows that M is a summand of B. That B is maximal Cohen᎐Macaulay over A follows from Proposition 7.1 and the fact that B is maximal Cohen᎐Macaulay over B.

10. REGULARITY IN CODIMENSION ONE Every commutative quotient singularity is regular in codimension one. In this section we prove an analogue of this fact for certain special quotient surface singularities.

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Let A be a ring. If A is commutative, then regularity in codimension one is traditionally defined by looking at the localisations at height one primes of A. Unfortunately, noncommutative geometry tends not to be local so this definition fails for noncommutative rings. An alternative way to define regularity in codimension one in the commutative case is to consider the scheme X consisting of Spec A with all the points of codimension two or greater removed. Then A is regular in codimension one if X is smooth. This suggests DEFINITION 10.1. Let Ž A, ᒊ . be a two dimensional complete local ring with balanced dualising complex and let T denote the category of ᒊ-torsion A-modules. Then A is said to be regular in codimension one if the T has finite injective dimension. quotient category Mod-ArT T is sometimes called the punctured spectrum The category Mod-ArT since, when A is commutative, it corresponds to the category of quasicoherent sheaves on Spec A with the closed point removed. Let B be a ring and let G be a finite group which acts on B. Let A s B G. There is a standard technique for studying the invariant ring which involves establishing a Morita equivalence between A and the skew group ring S s G) B Žsee wMont; McR, Chap. 7, Sect. 8x for details.. We need a modification of the theory. First recall that as a right B-module S s [ g G B while multiplication is given by the commutation rule b s Ž Ž b ... Let e be the idempotent < G1 < Ý in S so that eS is a projective S-module. There is a ring isomorphism A ª eSe given by a ¬ eae s ae s ea. The Morita context in the classical case is given by

: eS mS Se ª eSe , A,

: Se mA eS ª S,

where the maps are appropriate restrictions of the multiplication map in S. The map is always an isomorphism but will not be in the cases we are interested in. Given an ideal J of some ring, let J-tors denote the category of J-torsion modules. The version of Morita equivalence we want is PROPOSITION 10.2. Let J be an ideal of the ring B such that SJ s SJ and let I be an ideal of the ring A s B G . Suppose that Ž). I n : J and J n : IB for some n g ⺞. If SrSeS is J-torsion then ymA eS and ymS Se induce in¨ erse equi¨ alences between the quotient categories ŽMod-A.rŽ I-tors. and ŽMod-S .rŽ J-tors.. Proof. We first show that the map of the Morita context is an isomorphism modulo J-torsion. By hypothesis, it is surjective so we need only show that K [ ker is J-torsion. Now Se is a projective left

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S-module so we have an exact sequence 0 ª K mS Se ª Se mA eS mS Se ª Se. The last map is an isomorphism since : eS mS Se ª eSe is. Hence K mS Se s 0. This shows that K , KrŽ K mS Se mA eS . , K mS Ž SrSeS . which is J-torsion so is indeed an isomorphism modulo J-torsion. We now show that ymS Se induces a functor from ŽMod-S .rŽ J-tors. ª ŽMod-A.rŽ I-tors.. By wG, Chap. III, Sect. 1, Corollaire 2x, it suffices to show that the composite ymS Se

Mod-A ª Ž Mod-A . r Ž I-tors .

6

Mod-S

is exact and maps J-torsion modules to 0. Exactness holds since Se is projective so we need to verify that M mS Se is I-torsion whenever M is a J-torsion S-module. This follows from condition Ž). which guarantees that

Ž M mS Se . I n s im M mS BeI n s im M mS BI n e : im M mS Je s im MJ mS Se, where the image is taken in M mS Se. Now we show that ymA eS induces a functor on quotient categories by showing that the composite ymA eS

Mod-S ª Ž Mod-S . r Ž J-tors .

6

␣ : Mod-A

is exact and annihilates I-torsion modules. We know that ␣ is right exact so consider an injection M ª N of A-modules. Let K [ kerŽ M mA eS ª N mA eS .. Since Se is a projective S-module we have an exact sequence, 0 ª K mS Se ª M mA eS mS Se ª N mA eS mS Se. Since : eS mA Se ª A is an isomorphism, the map on the right is injective and K mS Se s 0. By the argument in the first paragraph, this shows that K is J-torsion. Hence ␣ is exact. Now let M be an I-torsion module. Then

Ž M mA eS . J n : im M mA eSIB s im M mA IeS s im MI mA eS so M mA eS is J-torsion so ymA eS induces a functor on quotient categories as desired. To see that the functors ymA eS and ymS Se induce inverse equivalences between quotient categories, we need to show that the composites

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ymA eS mS Se and ymS Se mA eS are naturally equivalent to the identity on the quotient categories. This follows from the fact that and are isomorphisms modulo torsion. Suppose now that the base field k is algebraically closed of characteristic zero. For the rest of the section, let Ž B, ᒋ . denote a regular complete local ring of dimension two. The first step toward proving regularity in codimension one is PROPOSITION 10.3. Let Ž B, ᒋ . be a regular complete local ring of dimension two and let S be the corresponding skew group ring. The quotient category ŽMod-S .rŽ ᒋ-tors. has injecti¨ e dimension at most two. This follows from Maschke’s theorem and wG, Chap. III, Sect. 3, Corollaire 2x. Let u, ¨ be topological generators for B as in Proposition 3.3. As in Section 2.1, we let V s ku q k¨ and identify GLŽ V . with GL2 using the basis u, ¨ . Let , be primitive dth roots of unity for some d g ⺞ and s Ž 0 0 .. We first prove regularity in codimension one for the following quotient singularities. PROPOSITION 10.4. Suppose that the action of on V extends to B and let G be the group generated by . Then SeS contains Sᒋ 2 d. Hence the quotient singularity A s B G is regular in codimension one in this case. Proof. To show SeS = Sᒋ 2 d it suffices to show that u i ¨ 2 dyi g SeS for 0 F i F 2 d. We will prove this in the case where i G d. The case i F d is similar. For any 0 F l - d, SeS contains zl [ ul

Ý 0Fr-d

r u iyl ¨ 2 dyi s

Ý

r r l u i ¨ 2 dyi .

0Fr-d

Since is a primitive dth root of unit, we see that SeS also contains 1 dy 1 i 2 dyi as was to be shown. Hence SeS = Sᒋ 2 d and we may d Ý ls0 z l s u ¨ T is naturally equivalent invoke Proposition 10.2 to conclude that Mod-ArT to ŽMod-S .rŽ ᒋ-tors.. This has finite injective dimension by the previous proposition so A is regular in codimension one. For other examples, we need to reduce the question to the graded case. Let B s grᒋ B and ᒋ s grᒋ ᒋ. Note that there is an ᒋ-adic filtration on S arising from the right B-module structure. Let S denote the associated graded ring grᒋ S which is isomorphic to G) B. We omit the proof of the following. LEMMA 10.5.

If SrSeS is ᒋ-torsion then SrSeS is ᒋ-torsion.

Recall that g GL2 is a pseudo-reflection if / 1 and 1 is an eigenvalue for .

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PROPOSITION 10.6. Suppose G is a finite subgroup of GL2 without pseudo-reflections which acts linearly on B and B s grᒋ B is commutati¨ e. Then B G is regular in codimension one. Proof. By the previous lemma, it suffices to show that SrSeS is ᒋ-torsion. This is the commutative case where the result is well known. It is a consequence of the fact that the map Spec B ª Spec B G is ´ etale away from the vertex ᒋ. We have thus shown that every special quotient surface singularity is regular in codimension one with the possible exception of the rings B G where G is non-diagonal and grᒋ B is not commutative. The result is not known in this case but the method of proof above certainly fails Žsee wC, Claim 2.5.7x for details..

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M. Artin and J. T. Stafford, ‘‘Regular Local Rings of Dimension 2,’’ manuscript. M. Artin and W. Schelter, Graded algebras of global dimension 3, Ad¨ . in Math. 66 Ž1987., 172᎐216. wAusx M. Auslander, Rational singularities and almost split sequences, Trans. Amer. Math. Soc. 293 Ž1986., 511᎐531. wAVx M. Artin and M. Van den Bergh, Twisted homogeneous coordinate rings, J. Algebra 133 Ž1990., 249᎐271. wAZx M. Artin and J. Zhang, Noncommutative projective schemes, Ad¨ . in Math. 109 Ž1994., 228᎐287. wBx G. Bergman, The diamond lemma in ring theory, Ad¨ . in Math. 29 Ž1978., 178᎐218. wBjx Bjork, Dubreil᎐Mal¨ The Auslander condition on Noetherian rings, in ‘‘Seminaire ´ liavin 1987᎐88,’’ Lecture Notes in Math., Vol. 1404, Springer-Verlag, Berlin, 1989. wCx D. Chan, ‘‘Noncommutative Rational Double Points,’’ MIT doctoral thesis, 1999. wGx P. Gabriel, Des Categories Abeliennes, Bull. Soc. Math. France 90 Ž1962., 323᎐448. ´ ´ wGHx L. Gerritzen and R. Holtkamp, On Grobner bases of noncommutative power series ¨ rings, Indag. Math. 4, 503᎐519. wH1x R. Hartshorne, ‘‘Residues and Duality,’’ Lecture Notes in Math., Vol. 20, SpringerVerlag, BerlinrHeidelbergrNew York, 1966. wH2x R. Hartshorne, Stable reflexive sheaves, Math. Ann. 254 Ž1980., 121᎐176. wJx P. Jorgensen, Non-commutative graded homological identities, J. London Math. Soc. 57 Ž1998., 336᎐350. wLx T. Levasseur, Some properties of non-commutative regular graded rings, Glasgow Math. J. 34 Ž1992., 277᎐300. wMcRx J. C. McConnell and J. C. Robson, ‘‘Noncommutative Noetherian Rings,’’ Wiley, ChichesterrNew YorkrBrisbanerTorontorSingapore, 1987. wMontx S. Montgomery, ‘‘Fixed Rings of Finite Automorphism Groups of Associative Rings,’’ Lecture Notes in Math., Vol. 818, Springer-Verlag, BerlinrHeidelbergrNew York, 1980. wPinkx H. Pinkham, Singularites sur les Singularites ´ de KleinᎏI, II, in ‘‘Seminaire ´ ´ des Surfaces,’’ Lecture Notes in Math., Vol. 777, Springer-Verlag, BerlinrHeidelbergr New York, 1980.

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M. Van den Bergh, Existence theorems for dualizing complexes over non-commutative graded and filtered rings, J. Algebra 195 Ž1997., 662᎐679. Q. S. Wu and J. Zhang, Dualizing complexes over noncommutative local rings, submitted for publication. A. Yekutieli, Dualizing complexes over noncommutative graded algebras, J. Algebra 153 Ž1992., 41᎐84. A. Yekutieli, The residue complex of a noncommutative graded algebra, J. Algebra 186 Ž1996., 522᎐543.

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