Dual F-signature of special Cohen-Macaulay modules over cyclic quotient surface singularities
aa r X i v : . [ m a t h . A C ] D ec DUAL F -SIGNATURE OF SPECIAL COHEN-MACAULAY MODULESOVER CYCLIC QUOTIENT SURFACE SINGULARITIES YUSUKE NAKAJIMA
Abstract.
The notion of F -signature was defined by C. Huneke and G. Leuschke and this nu-merical invariant characterizes some singularities. This notion is extended to finitely generatedmodules by A. Sannai and is called dual F -signature. In this paper, we determine the dual F -signature of a certain class of Cohen-Macaulay modules (so-called “special”) over cyclic quo-tient surface singularities. Also, we compare the dual F -signature of a special Cohen-Macaulaymodule with that of its Auslander-Reiten translation. This gives a new characterization of theGorensteiness. Introduction
Throughout this paper, we suppose that k is an algebraically closed field of prime characteristic p >
0. Let ( R, m , k ) be a Noetherian local ring with char R = p >
0. Since char R = p >
0, wecan define the Frobenius map F : R → R ( r r p ). For e ∈ N , we also define the e -times iteratedFrobenius map F e : R → R ( r r p e ). For any R -module M , we denote the module M with its R -module structure pulled back via the e -times iterated Frobenius map F e by e M . Namely, e M is just M as an abelian group, and its R -module structure is defined by r · m := F e ( r ) m = r p e m for all r ∈ R, m ∈ M . We say R is F -finite if R is a finitely generated R -module.In order to investigate the properties of R , C. Huneke and G. Leuschke introduced the notionof F -signature. Definition 1.1 ([HL]) . Let ( R, m , k ) be a reduced F -finite local ring of prime characteristic p >
0. For each e ∈ N , decompose e R as follows e R ∼ = R ⊕ a e ⊕ M e , where M e has no free direct summands. We call a e the e -th F -splitting number of R . Then, the F -signature of R is s ( R ) := lim e →∞ a e p ed , if it exists, where d := dim R .Note that K. Tucker showed its existence in a general situation [Tuc]. As Kunz’s theorem[Kun] shows, this invariant measures the deviation from regularity (see also Theorem 1.4(1) ).For a finitely generated R -module, A. Sannai extended the notion of F -signature as follows. Definition 1.2 ([San]) . Let ( R, m , k ) be a reduced F -finite local ring of prime characteristic p >
0. For a finitely generated R -module M and e ∈ N , we set b e ( M ) := max { n | ∃ ϕ : e M ։ M ⊕ n } , and call it the e -th F -surjective number of M . Then we call the limit s ( M ) := lim e →∞ b e ( M ) p ed Mathematics Subject Classification.
Primary 13A35, 13A50; Secondary 13C14, 16G70.
Key words and phrases. F -signature, dual F -signature, Auslander-Reiten quiver, cyclic quotient surface sin-gularities, special Cohen-Macaulay modules. F -SIGNATURE OF SPECIAL CM MODULES 2 the dual F -signature of M if it exists, where d = dim R . Remark . The dual F -signature of R coincides with the F -signature of R , because the mor-phism e R ։ R ⊕ b e ( R ) is split. Therefore, we use the same notation unless it causes confusion.By using these invariants, we can characterize some singularities. Theorem 1.4 ([HL], [Yao2], [AL], [San]) . Let ( R, m , k ) be a d -dimensional reduced F -finiteNoetherian local ring with char R = p > . Then we obtain (1) R is regular if and only if s ( R ) = 1 , (2) R is strongly F -regular if and only if s ( R ) > .In addition, we suppose R is Cohen-Macaulay with the canonical module ω R , then (3) R is F -rational if and only if s ( ω R ) > , (4) s ( R ) ≤ s ( ω R ) , (5) s ( R ) = s ( ω R ) if and only if R is Gorenstein. As the above theorem shows, the value of s ( R ) and s ( ω R ) contain some information regardingsingularities. How about the value of the dual F -signature for other R -modules ? The value ofthe dual F -signature is not known except in the case of two-dimensional Veronese subrings [San].Therefore, in this paper, we determine the dual F -signature for a certain class of Cohen-Macaulay(= CM) modules (so-called special CM modules) over cyclic quotient surface singularities.The study of special CM modules was started by the work of J. Wunram [Wun1], [Wun2] (thedefinition of special CM modules appears in Section 3). For a finite subgroup G of SL(2 , k ) suchthat the order of G is invertible in k , the McKay correspondence is very famous, that is, there isa one-to-one correspondence between non-trivial irreducible representations of G and irreducibleexceptional curves on the minimal resolution of quotient surface singularity. When we intendto generalize this correspondence to a finite subgroup G of GL(2 , k ), this correspondence is nolonger true. In fact, there are more irreducible representations than exceptional curves. However,if we choose some irreducible representations which are called special, then we again obtain one-to-one correspondence between irreducible special representations of G and exceptional curves[Wun2], and a maximal CM module associated with a special representation is called a specialCM module. For more about the special McKay correspondence, see also [Ish],[Ito] and [Rie]. Remark . All irreducible representations of a finite subgroup of SL(2 , k ) are special, thus wecan recover the McKay correspondence in the original sense from the special one.For a cyclic quotient singularity, a special CM module takes the following simple form. (Formore details on terminologies, see Section 2 and 3.)Suppose R is the invariant subring of S = k [[ x, y ]] under the action of a cyclic group n (1 , a ). Inthis situation, a non-free indecomposable special CM R -module is described as M i t = Rx i t + Ry j t (i.e. it is minimally 2-generated). The following theorem gives the value of the dual F -signature;note that they are all rational. Theorem 1.6 (see Theorem 3.9) . For any non-free indecomposable special CM R -module M i t ,we have s ( M i t ) = min( i t , j t ) + 1 n (if i t = j t ) i t + 12 n (if i t = j t ) . Moreover, by paying attention to special CM modules and their Auslander-Reiten translations,we characterize when the ring is Gorenstein.
UAL F -SIGNATURE OF SPECIAL CM MODULES 3 Theorem 1.7 (see Theorem 4.2) . Let R be a quotient surface singularity. (Remark that wedon’t restrict to a cyclic case.) Suppose M is an indecomposable special CM R -module. Thenwe have s ( M ) ≤ s ( τ ( M )) . Moreover, if s ( M ) = s ( τ ( M )) for an indecomposable special CM R -module M , then R is Goren-stein. (Note that if R is Gorenstein, then s ( M ) = s ( τ ( M )) holds for all indecomposable MCMmodules.)Remark . Since τ ( R ) ∼ = ω R in our situation, this theorem is an analogue of Theorem 1.4 (4) , (5).But it says that this characterization is obtained by not only the comparison between R and ω R but also the comparison between a special CM module and its AR translation.The structure of this paper is as follows. In order to determine the dual F -signature, we needthe notion of the generalized F -signature and the Auslander-Reiten quiver. Thus, we preparethem in Section 2. In Section 3, we determine the dual F -signature of special CM modulesover cyclic quotient surface singularities and give several examples. In Section 4, we comparea special CM module with its Auslander-Reiten translation by using the dual F -signature andcharacterize the Gorensteiness. Note that the statements appearing in Section 4 hold not onlyfor cyclic quotient surface singularities but also for any quotient surface singularities.2. Preliminary
Generalized F -signature of invariant subrings. Let G be a finite subgroup of GL( d, k )which contains no pseudo-reflections except the identity, and S := k [[ x , · · · , x d ]] be a powerseries ring. We assume that the order of G is coprime to p = char k . We denote the invariantsubring of S under the action of G by R := S G . In order to determine the dual F -signatureof a finitely generated R -module M , we have to know about the structure of e M (for instance,the direct sum decomposition of e M , the asymptotic behavior of the multiplicities of directsummands, etc). To achieve this, we use the results of the generalized F -signature of invariantsubrings [HN].For a positive characteristic Noetherian ring, K. Smith and M. Van den Bergh introducedthe notion of finite F -representation type [SVdB]. This notion is a characteristic p analogueof the notion of finite representation type. The definition of finite F -representation type is thefollowing. Definition 2.1 ([SVdB]) . We say that R has finite F -representation type (or FFRT for short)by N if there exists a finite set N of isomorphism classes of indecomposable finitely generated R -modules, such that for every e ∈ N , the R -module e R is isomorphic to a finite direct sum ofelements of N .For example, a power series ring S has FFRT by { S } (cf. Kunz’s theorem [Kun]) and FFRTis inherited by a direct summand [SVdB]. Thus, an invariant subring R also has FFRT. Moreexplicitly, we have the next proposition. Proposition 2.2 ([SVdB]) . Let V = k, V , · · · , V n − be the complete set of irreducible rep-resentations of G and we set M t := ( S ⊗ k V t ) G ( t = 0 , , · · · , n − . Then R has finite F -representation type by the finite set { M ∼ = R, M , · · · , M n − } . Thus we can write e R as follows. e R ∼ = R ⊕ c ,e ⊕ M ⊕ c ,e ⊕ · · · ⊕ M ⊕ c n − ,e n − . Remark . We can see that each M t is an indecomposable maximal Cohen-Macaulay (= MCM) R -module and M s = M t ( s = t ) under the assumption G contains no pseudo-reflections exceptthe identity. Also, the multiplicities c i,e are determined uniquely in that case. For more details,we refer the reader to [HN, Section 2]. UAL F -SIGNATURE OF SPECIAL CM MODULES 4 Moreover, since an invariant subring R has FFRT, the limit lim e →∞ c t,e p de ( t = 0 , , · · · , n −
1) exists[SVdB], [Yao1]. Therefore we can define the limit s ( R, M t ) := lim e →∞ c t,e p de and call it the generalized F -signature of M t with respect to R . The value of s ( R, M t ) is determined by M. Hashimoto andthe author as follows. Theorem 2.4 ([HN]) . For t = 0 , , · · · , n − , one has s ( R, M t ) = rank R M t | G | Remark . In the case of t = 0 is also due to [WY]. A similar result holds for a finite subgroupscheme of SL [HS].We also obtain the next statement as a corollary. Corollary 2.6 ([HN]) . Suppose an MCM R -module M t decomposes as e M t ∼ = R ⊕ d t ,e ⊕ M ⊕ d t ,e ⊕ · · · ⊕ M ⊕ d tn − ,e n − . Then, for all t, u = 0 , · · · , n − , we obtain s ( M t , M u ) := lim e →∞ d tu,e p de = (rank R M t ) · s ( R, M u ) = (rank R M t ) · (rank R M u ) | G | . Remark . In dimension two, it is known that an invariant subring R is of finite representa-tion type, that is, it has only finitely many non-isomorphic indecomposable MCM R -modules { R, M , · · · , M n − } . From Corollary 2.6, every indecomposable MCM R -module appears in e M t as a direct summand for sufficiently large e . Thus, the additive closure add R ( e M t ) coincideswith the category of MCM R -modules CM( R ). So we apply several results in Auslander-Reitentheory to add R ( e M t ) (see the next subsection).2.2. Auslander-Reiten quiver.
In this subsection, we review some results of Auslander-Reiten theory. For details, see some textbooks (e.g.[LW], [Yos]) or [Aus1], [Aus2]. We onlydiscuss such a theory for the case of an invariant subring R = S G in dim R = 2. Definition 2.8 (Auslander-Reiten sequence) . Let R be an invariant subring and M, N beindecomposable MCM R -modules. We call a non-split short exact sequence0 → N f → L g → M → M if for all MCM modules X and for anymorphism ϕ : X → M which is not a split surjection there exists φ : X → L such that ϕ = g ◦ φ .Since R is an isolated singularity, there exists the AR sequence ending in M t for each non-freeindecomposable MCM R -module M t , and it is unique up to isomorphism [Aus2]. Concretely,the AR sequence ending in M t ( t = 0) is0 −→ ( S ⊗ k ( ∧ V ⊗ k V t )) G −→ ( S ⊗ k ( V ⊗ k V t )) G −→ M t = ( S ⊗ k V t ) G −→ , where V is a natural representation of G .In the case of t = 0, there is the exact sequence0 −→ ω R = ( S ⊗ k ∧ V ) G −→ ( S ⊗ k V ) G −→ R = S G −→ k −→ . This exact sequence is called the fundamental sequence of R .We call the left term of these sequences the Auslander-Reiten (= AR) translation of M t , anddenote it by τ ( M t ). Sometimes we denote the middle term of the AR sequence by E M t . It isknown that τ ( M t ) ∼ = ( M t ⊗ R ω R ) ∗∗ , where ( − ) ∗ = Hom R ( − , R ) is the R -dual functor [Aus1]. UAL F -SIGNATURE OF SPECIAL CM MODULES 5 Note that τ ( M t ) = M t − a − and E M t = M t − ⊕ M t − a for t = 0 , , · · · , n − Definition 2.9 (Irreducible morphism) . Let M and N be MCM R -modules. We decom-pose M and N into indecomposable modules as M = ⊕ i M i , N = ⊕ j N j and decompose ψ ∈ Hom R ( M, N ) along this decomposition as ψ = ( ψ ij : M i → N j ) ij . Then we define thesubmodule rad R ( M, N ) ⊂ Hom R ( M, N ) as follows . ψ ∈ rad R ( M, N ) def ⇐⇒ no ψ ij is an isomorphismIn addition, we define the submodule rad R ( M, N ) ⊂ Hom R ( M, N ). The submodule rad R ( M, N )consists of morphisms ψ : M → N such that ψ decomposes as ψ = g ◦ f , M ψ / / f ! ! ❇❇❇❇ NZ g > > ⑥⑥⑥⑥ where Z is an MCM R -module, f ∈ rad R ( M, Z ), g ∈ rad R ( Z, N ). We call a morphism ψ : M → N irreducible if ψ ∈ rad R ( M, N ) \ rad R ( M, N ). SetIrr R ( M, N ) := rad R ( M, N ) (cid:14) rad R ( M, N ) , then Irr R ( M, N ) is a vector space over k .By using these notions, we define the Auslander-Reiten quiver. Definition 2.10 (Auslander-Reiten quiver) . The Auslander-Reiten (= AR) quiver of R isan oriented graph whose vertices are indecomposable MCM R -modules R, M , · · · , M n − withdim k Irr R ( M s , M t ) arrows from M s to M t ( s, t = 0 , , · · · , n − The case of cyclic quotient surface singularities.
Since one of the aims of this paperis to determine the dual F -signature of special CM modules over cyclic quotient surface singu-larities, we restate results in subsection 2.1 and 2.2 for the cyclic case. Thus, we suppose that G is a cyclic group as follows. G := h σ = (cid:18) ζ n ζ an (cid:19) i , where ζ n is a primitive n -th root of unity, 1 ≤ a ≤ n −
1, and gcd( a, n ) = 1. We denote thecyclic group G as above by n (1 , a ). Let S := k [[ x, y ]] be a power series ring and we assume that n is coprime to p = char k . We denote the invariant subring of S under the action of G by R := S G . Since G is an abelian group, any irreducible representations of G are described by V t : σ ζ − tn ( t = 0 , , · · · , n − . Then we set, M t := ( S ⊗ k V t ) G = nX i,j a ij x i y j ∈ S | a ij ∈ k, i + ja ≡ t (mod n ) o , ( t = 0 , , · · · , n − R , and each has rank one.From Corollary 2.6, s ( M t , M u ) = 1 /n ( u = 0 , , · · · , n − e M t on the order of p e , we may consider as e M t ≈ ( R ⊕ M ⊕ · · · ⊕ M n − ) ⊕ p en . (2.1)Also, the AR sequence ending in M t ( t = 0) is0 −→ M t − a − −→ M t − ⊕ M t − a −→ M t −→ . (2.2) UAL F -SIGNATURE OF SPECIAL CM MODULES 6 In the case of t = 0, the fundamental sequence of R is0 −→ ω R −→ M − ⊕ M − a −→ R −→ k −→ . (2.3)Thus, we have τ ( M t ) = M t − a − and E M t = M t − ⊕ M t − a for t = 0 , , · · · , n −
1. It isknown that dim k Irr R ( M s , M t ) is equal to the multiplicity of M s in the decomposition of E M t .Therefore, by (2 .
2) and (2 . M t − to M t , and from M t − a to M t for t = 0 , , · · · , n −
1. Namely, we have dim k Irr R ( M t − , M t ) = 1 and dim k Irr R ( M t − a , M t ) = 1. Remark . Since S ∼ = R ⊕ M ⊕· · ·⊕ M n − , each MCM R -modules M t is an R -submodule of S ,and we can take a morphism · x (resp. · y ) as a basis of 1-dimensional vector space Irr R ( M t − , M t ) (resp.Irr R ( M t − a , M t )). M t − = (cid:8) f ∈ S | σ · f = ζ t − n f (cid:9) x −→ M t = (cid:8) f ∈ S | σ · f = ζ tn f (cid:9) M t − a = (cid:8) f ∈ S | σ · f = ζ t − an f (cid:9) y −→ M t = (cid:8) f ∈ S | σ · f = ζ tn f (cid:9) Example 2.12.
Let G = (1 ,
3) be a cyclic group of order 7. Irreducible representations of G are V t : σ ζ − t ( t = 0 , · · · , , where ζ is a primitive 7-th root of unity. Then the AR quiver of R is described as follows. Forsimplicity, we only describe subscripts as vertices, and all common numbers are identified. / / / / / / · · · / / / / / / O O / / O O / / O O / / · · · / / O O / / O O / / O O O O / / O O / / O O / / · · · / / O O / / O O / / O O ... O O ... O O ... O O ... ... O O ... O O ... O O O O / / O O / / O O / / · · · / / O O / / O O / / O O O O / / O O / / O O / / · · · / / O O / / O O / / O O O O / / O O / / O O / / · · · / / O O / / O O / / O O Remark . For each diagram a / / bc O O / / d O O , if b = 0 then 0 → M c → M a ⊕ M d → M b → M b , and any diagram commutes a y / / (cid:9) bc x O O y / / d x O O by Remark 2.11.3. Dual F -signature of Special CM modules In this section, we introduce the notion of special CM modules and determine the dual F -signature of them. Firstly, we recall the definition of special CM modules over an invariantsubring R , and the properties of them. Definition 3.1 ([Wun2]) . For an MCM R -module M , we call M special if ( M ⊗ R ω R ) (cid:14) tor isalso an MCM R -module.In other words, let ϕ be the natural morphism M ⊗ R ω R → ( M ⊗ R ω R ) ∗∗ , then M ⊗ R ω R (cid:14) Ker ϕ is also an MCM R -module if and only if M is a special CM R -module. In that case, we havethe following (cf. [Rie, Lemma 9]), M ⊗ R ω R (cid:14) Ker ϕ ∼ = τ ( M ) ∼ = ( M ⊗ R ω R ) ∗∗ . UAL F -SIGNATURE OF SPECIAL CM MODULES 7 Therefore, M is a special CM R -module if and only if ϕ is a surjection. Furthermore, there areseveral characterizations of special CM modules as follows (see [IW, Theorem 2.7 and 3.6]). Proposition 3.2.
Suppose that M is an MCM R -module. Then the following are equivalent. (1) M is a special CM module, (2) Ext R ( M, R ) = 0 , (3) (Ω M ) ∗ ∼ = M where Ω M is the syzygy of M . Suppose M is a special CM R -module, then we have the following exact sequence by thecondition (3). Here, µ R ( M ) is the number of minimal generators of M .0 → M ∗ ∼ = Ω M → R ⊕ µ R ( M ) → M → . Thus, we have µ R ( M ) = 2 rank R M . The converse is true if rank R M = 1 (cf. [Wun2, Theo-rem 2.1]). If rank R M >
1, the converse is no longer true (cf. [Nak, Example A.5] and [IW]).Since each MCM module over cyclic quotient surface singularities has rank one, a special CMmodule is minimally 2-generated (see Theorem 3.5).For a cyclic group G = n (1 , a ), we can describe special CM-modules as follows.Firstly, we consider the Hirzebruch-Jung continued fraction expansion of n/a , na = α − α − · · · − α r := [ α , α , · · · , α r ] , and then we introduce the notion of i -series and j -series (cf. [Wem], [Wun1]). Definition 3.3.
For n/a = [ α , α , · · · , α r ], we define the i -series and the j -series as follows. i = n, i = a, i t = α t − i t − − i t − ( t = 2 , · · · , r + 1) ,j = 0 , j = 1 , j t = α t − j t − − j t − ( t = 2 , · · · , r + 1) . Remark . By the construction method of the i -series and the j -series, it is easy to see · i t ≡ j t a (mod n ) , · i = n > i = a > i > · · · > i r = 1 > i r +1 = 0 , · j = 0 < j = 1 < j = α < · · · < j r < j r +1 = n. By using the i -series and the j -series, we can characterize special CM R -modules. Theorem 3.5 ([Wun1]) . For a cyclic group G = n (1 , a ) with n/a = [ α , α , · · · , α r ] , specialCM R -modules are M i t ( t = 0 , , · · · , r ) . Moreover, minimal generators of M i t are x i t and y j t for t = 1 , · · · , r . Example 3.6.
Let G = (1 ,
3) be a cyclic group of order 7. The Hirzebruch-Jung continuedfraction expansion of 7 / − − / , , , and the i -series and the j -series are described as follows. i = 7 , i = 3 , i = 2 , i = 1 , i = 0 ,j = 0 , j = 1 , j = 3 , j = 5 , j = 7 . Thus, special CM modules are
R, M , M , M and these are described explicitly R = k [[ x , x y, xy , y ]] M = Rx + Ry M = Rx + Ry M = Rx + Ry.
UAL F -SIGNATURE OF SPECIAL CM MODULES 8 We now show, using AR theory, how to investigate possible surjections e M ։ M ⊕ b e . / / y + y + y + ?>=<89:; O O / / O O / / O O / / O O / / x K S O O / / O O / / O O / / O O / / x K S O O / / O O / / O O / / O O / / O O Figure 1 • / / y / / y / / y / / ?>=<89:; • O O / / • O O / / O O / / O O / / x O O • O O / / • O O / / • O O / / • O O / / x O O • O O / / • O O / / • O O / / • O O / / • O O Figure 2
We take the MCM R -module M as an example. From the AR quiver around the vertex (cid:13) ,we can see that there are several morphisms ending in (cid:13) and obtain minimal generators x and y by following the morphisms described by double arrows in Figure 1.Since each diagrams a / / bc O O / / d O O are commute, morphisms from vertices which are denoted by • in Figure 2 to (cid:13) go through “0”(that is , “ R ”). Thus, the image of each morphism • → (cid:13) isin m M where m is the maximal ideal of R . By Nakayama’s lemma, such a morphism doesn’tcontribute to construct a surjection. Thus we may ignore them. Also, there are morphisms fromvertices which are denoted by ⋆ in Figure 3 to (cid:13) . Minimal generators of M ⋆ are generated bymorphisms from 0 (which are located outside of dotted area in Figure 3) to ⋆ . Considering thecomposition of such a morphism and ⋆ → (cid:13) R → M ⋆ → M (1 δ x m y m δ ) , where δ is a minimal generator of M ⋆ and m ≥ , m ≥
1. Then it is easy to see that theimage of the morphism ⋆ → (cid:13) is in m M . Thus we may ignore them. y / / y / / y / / ?>=<89:; / / ❴❴❴❴❴ ⋆ x O O / / ⋆ x O O / / x O O x O O O O ✤✤✤✤ O O ✤✤✤✤✤✤ Figure 3 y / / y / / y / / ?>=<89:; x O O x O O Figure 4
Thus, in order to investigate a surjection e M ։ M ⊕ b e , we need only discuss the MCM R -modules located in the horizontal direction from M to R and the vertical direction from M to R (Figure 4).In general, the number of minimal generators of a special CM R -module M i t is two andminimal generators take a form like x i t , y j t by Theorem 3.5. Thus, it is equivalent to there is no“0” in dotted vertices area of Figure 5. By the above arguments, in order to construct a surjection e M i t ։ M ⊕ b e i t , we may only discuss horizontal direction arrows from R to M i t and verticaldirection arrows from R to M i t . We consider sets of subscripts of vertices F t = { , , · · · , i t − } and G t = { i t − a, · · · , i t − j t a ≡ } as in Figure 5. It is easy to see that |F t | = i t , |G t | = j t . UAL F -SIGNATURE OF SPECIAL CM MODULES 9 GF ED G t ≡ i t − j t a y / / i t − ( j t − a y / / · · · y / / i t − a y / / i t − a y / / i t • O O / / • O O / / · · · / / • O O / / • O O / / i t − x O O ... O O ... O O ... ... O O ... O O ... x O O • O O / / • O O / / · · · / / • O O / / • O O / / x O O • O O / / • O O / / · · · / / • O O / / • O O / / x O O BCED F t Figure 5
To determine the dual F -signature of special CM R -modules, we prepare some notations andlemmas.For the i -series ( i , · · · , i r ) associated with n (1 , a ) and any β ∈ Z ≥ with 0 ≤ β ≤ n −
1, thereare unique non-negative integers d , · · · , d r ∈ Z ≥ such that β = d i + h , h ∈ Z ≥ , ≤ h < i ,h t = d t +1 i t +1 + h t +1 , h t +1 ∈ Z ≥ , ≤ h t +1 < i t +1 , ( t = 1 , · · · , r − ,h r = 0 . Thus, we can describe β as follows, β = d i + d i + · · · + d r i r . For such β , there is the unique integer e β ∈ Z ≥ such that a e β ≡ β (mod n ) , ≤ e β ≤ n − Lemma 3.7 ([Wun1]) . Let e β be the same as above. Then e β is described as e β = d j + d j + · · · + d r j r , where ( j , · · · , j r ) is the j -series associated with n (1 , a ) . Lemma 3.8.
Let the notation be the same as above, then F t ∩ G t = { } as a set of subscriptsof vertices.Proof. It is trivial that 0 ∈ F t ∩ G t by the definition of F t and G t . Thus, it suffices to showthere is no pair ( m , m ) ∈ Z > such that m ≡ m a (mod n ), where 1 ≤ m ≤ i t − ≤ m ≤ j t −
1. Assume that there exists such a pair ( m , m ). Then there are non-negativeintegers d , · · · , d r such that m = d i + d i + · · · + d r i r . Since 1 ≤ m ≤ i t − i t > i t +1 (cf. Remark 3.4), d = · · · = d t = 0 and there exists λ such that t + 1 ≤ λ ≤ r and d λ = 0.From Lemma 3.7, we obtain m = d j + d j + · · · + d r j r . Thus, m = d t +1 j t +1 + · · · + d r j r ≥ j λ > j t . This contradicts m ≤ j t − (cid:3) We are now ready to state the main theorem.
UAL F -SIGNATURE OF SPECIAL CM MODULES 10 Theorem 3.9.
Let the notation be the same as above, then for any non-free special CM R -module M i t one has s ( M i t ) = min( i t , j t ) + 1 n (if i t = j t ) i t + 12 n (if i t = j t ) . Proof.
In order to determine the value of the dual F -signature of M i t , we have to find themaximum number b e such that there is a surjection e M i t ։ M ⊕ b e i t . Note that we may consider e M i t as e M i t ≈ ( R ⊕ M ⊕ · · · ⊕ M n − ) ⊕ p en by (2.1), hence we may assume the number of each indecomposable MCM module in e M i t is the same on the order of p e . Let F t , G t be the sets of vertices as in Figure 5. By theabove observations, MCM modules which contribute to construct a surjection are M i t itselfand modules corresponding to elements in F t or G t . Since an indecomposable MCM modulewhich is not isomorphic to R and M i t could construct at most one generator of M i t , we shouldfirst combine MCM modules corresponding to elements in F t \ { } with those in G t \ { } forconstructing surjections efficiently, and then we should use R and M i t . Therefore, in whatfollows, we will find disjoint sets of summands of e M i t which surject onto M i t as much aspossible along this strategy.Firstly, we show the case of i t > j t . Thus, |F t | > |G t | . We choose elements f and g from F t \ { } and G t \ { } respectively, and consider corresponding indecomposable MCM R -modules M f and M g . Here, we remark that f = g by Lemma 3.8. Then we can construct a surjection M f ⊕ M g ։ M i t . / / · · · / / g / / · · · / / i t ... O O f O O ... O O O O Then we consider the sets F t \ { , f } and G t \ { , g } . Similarly, we choose elements f and g from the sets F t \{ , f } and G t \{ , g } respectively, and construct a surjection M f ⊕ M g ։ M i t .By repeating the same process, we finally arrive at G t \ { , g , · · · , g j t − } = ∅ and have j t − ∈ G t (that is R ), we construct a surjection by combining R and an indecomposable MCM module corresponding to an element f ′ ∈ F t \ { , f , · · · , f j t − } 6 = ∅ . In addition, there is a trivial surjection M i t ։ M i t . Thus, through these processes, we couldobtain disjoint sets of summands { M f , M g } , · · · , { M f jt − , M g jt − } , { M f ′ , R } , { M i t } which surject onto M i t . Thus, we have a surjection( R ⊕ M ⊕ · · · ⊕ M n − ) ⊕ p en ։ M ⊕ p en ( j t +1) i t . Therefore the dual F -signature of M i t is s ( M i t ) = j t + 1 n . UAL F -SIGNATURE OF SPECIAL CM MODULES 11 Similarly, we obtain s ( M i t ) = i t + 1 n for the case of i t < j t .For the case of i t = j t , we repeat the same process until we have F t \ { , f , · · · , f i t − } = ∅ and G t \ { , g , · · · , g j t − } = ∅ , and then we have a surjection( M ⊕ · · · ⊕ M i t − ⊕ M i t +1 ⊕ · · · ⊕ M n − ) ⊕ p en ։ M ⊕ p en ( i t − i t . In addition, there is a trivial surjection M i t ։ M i t . For now, we don’t use R , and by using twofree summands we also construct the surjection: R ⊕ R ( x it y jt ) ։ M i t . Thus, the dual F -signature of M i t is s ( M i t ) = i t − n + 1 n + 12 n = 2 i t + 12 n . (cid:3) Example 3.10.
Let the notation be as in Example 3.6. Then, the dual F -signature of specialCM modules are s ( M ) = 27 , s ( M ) = 37 , s ( M ) = 27 . Next, we give an example for the case i t = j t . Example 3.11.
Let G = (1 ,
5) be a cyclic group of order 8. The Hirzebruch-Jung continuedfraction expansion of 8 / − − / , , , and the i -series and the j -series are described as follows. i = 8 , i = 5 , i = 2 , i = 1 , i = 0 ,j = 0 , j = 1 , j = 2 , j = 5 , j = 8 . Thus, special CM modules are
R, M , M , M . In this case, we have i = j and there aresurjections as follows. 0 y / / y / / x O O x O O M ։ M M ⊕ M ։ M R ⊕ R ։ M . Thus, the dual F -signature of M is s ( M ) = 18 + 18 + 116 = 516 . Example 3.12.
Let G = n (1 , n − ⊂ SL(2 , k ) be a cyclic group of order n , that is, Dynkintype A n − . The Hirzebruch-Jung continued fraction expansion of n/ ( n −
1) is nn − − − · · · − / , , · · · , | {z } n − ] , and the i -series and the j -series are described as follows, i = n, i = n − , i = n − , · · · , i n − = 1 , i n = 0 ,j = 0 , j = 1 , j = 2 , · · · , j n − = n − , j n = n. UAL F -SIGNATURE OF SPECIAL CM MODULES 12 Namely, i t = n − t, j t = t ( t = 1 , , · · · , n − G = n (1 , n − ⊂ SL(2 , k ) are special. Thus, any M t is a special CM moduleand the dual F -signature of M t is obtained by Theorem 3.9. s ( M i t ) = n + j t n = t + 1 n (if t < n )1 n + t − n + 12 n = 2 t + 12 n (if t = n )1 n + i t n = n − t + 1 n (if t > n ) . For other Dynkin types (i.e. D n , E , E , E ), see [Nak].4. Comparing with the Auslander-Reiten translation
In this section, we compare the dual F -signature of a special CM module with its AR trans-lation. It will give us a characterization of Gorensteiness (see Theorem 4.2). As we mentionedin Section 1, it is an analogue of Theorem 1.4 (4) , (5).The statements appearing in this section are valid for any quotient surface singularities.Therefore, we suppose that G is a finite subgroup of GL(2 , k ) which contains no pseudo-reflections except the identity, and S := k [[ x, y ]] be a power series ring. We assume that theorder of G is coprime to p = char k . We denote the invariant subring of S under the action of G by R := S G . Let V = k, V , · · · , V n be the complete set of irreducible representations of G and set indecomposable MCM R -modules M t := ( S ⊗ k V t ) G ( t = 0 , , · · · , n ). Lemma 4.1.
Let M t be an MCM R -module as above. Then we have e M t ≈ ( R ⊕ d ,t ⊕ M ⊕ d ,t ⊕ · · · ⊕ M ⊕ d n,t n ) ⊕ p en ≈ e τ ( M t ) (4.1) on the order of p e ( e ≫ ), where d i,t = (rank R M t ) · (rank R M i ) and τ stands for the ARtranslation. Furthermore, we have R ⊕ d ,t ⊕ M ⊕ d ,t ⊕ · · · ⊕ M ⊕ d n,t n ∼ = τ ( R ) ⊕ d ,t ⊕ τ ( M ) ⊕ d ,t ⊕ · · · ⊕ τ ( M n ) ⊕ d n,t . Proof.
From Corollary 2.6, we may write e M t ≈ ( R ⊕ d ,t ⊕ M ⊕ d ,t ⊕ · · · ⊕ M ⊕ d n,t n ) ⊕ p en , e τ ( M t ) ≈ ( R ⊕ d ′ ,t ⊕ M ⊕ d ′ ,t ⊕ · · · ⊕ M ⊕ d ′ n,t n ) ⊕ p en , where d ′ i,t = (rank R τ ( M t )) · (rank R M i ). Since rank R M t = rank R τ ( M t ), it follows that d i,t = d ′ i,t ( i = 0 , , · · · , n ). This implies (4.1).Since the AR translation τ gives a bijection from the set of finitely many indecomposableMCM R -modules to itself, we set τ ( M i ) = M σ ( i ) ( i = 0 , , · · · , n ) where σ is an element ofsymmetric group S n +1 . Then we have R ⊕ d ,t ⊕ M ⊕ d ,t ⊕ · · · ⊕ M ⊕ d n,t n = M ⊕ d σ (0) ,t σ (0) ⊕ M ⊕ d σ (1) ,t σ (1) ⊕ · · · ⊕ M ⊕ d σ ( n ) ,t σ ( n ) , and d σ ( i ) ,t = (rank R M t ) · (rank R M σ ( i ) ) = (rank R M t ) · (rank R τ ( M i ))= (rank R M t ) · (rank R M i ) = d i,t . Thus, M ⊕ d σ (0) ,t σ (0) ⊕ M ⊕ d σ (1) ,t σ (1) ⊕ · · · ⊕ M ⊕ d σ ( n ) ,t σ ( n ) = τ ( R ) ⊕ d ,t ⊕ τ ( M ) ⊕ d ,t ⊕ · · · ⊕ τ ( M n ) ⊕ d n,t . UAL F -SIGNATURE OF SPECIAL CM MODULES 13 (cid:3) Theorem 4.2.
For any indecomposable special CM R -module M t , we have s ( M t ) ≤ s ( τ ( M t )) . Moreover, if s ( M t ) = s ( τ ( M t )) for an indecomposable special CM R -module M t , then R isGorenstein. (Note that if R is Gorenstein, then s ( M ) = s ( τ ( M )) holds for all indecomposableMCM modules.)Proof. From Lemma 4.1, we may write e M t ≈ e τ ( M t ) ≈ ( R ⊕ d ,t ⊕ M ⊕ d ,t ⊕ · · · ⊕ M ⊕ d n,t n ) ⊕ p en when we discuss the asymptotic behavior on the order of p e where d i,t = (rank R M t ) · (rank R M i ).Let b e := b e ( M t ) be the e -th F -surjective number of M t , hence there exists a surjection e M t ։ M ⊕ b e t . Since M t is special, the number of minimal generators of M t is equal to u := 2 rank R M t .Thus, there exists a surjection R ⊕ b e u ։ M ⊕ b e t which induces the following commutative diagram. e M t / / / / M ⊕ b e t R ⊕ b e u O O : : : : ✉✉✉✉✉✉ Applying the functor ( −⊗ R ω R ) ∗∗ to this commutative diagram, then we obtain the commutativediagram. e τ ( M t ) ≈ ( e M t ⊗ R ω R ) ∗∗ ψ / / τ ( M t ) ⊕ b e ω ⊕ b e uR O O ψ ✐✐✐✐✐✐✐✐✐✐✐✐✐✐ Note that the morphism ψ is surjective because the surjection R ⊕ b e u ։ M ⊕ b e t induces ω ⊕ b e uR / / / / ∼ = (cid:15) (cid:15) ( M t ⊗ R ω R ) ⊕ b e ⊕ ϕ (cid:15) (cid:15) (cid:15) (cid:15) (( ω R ) ∗∗ ) ⊕ b e u ψ / / (( M t ⊗ R ω R ) ∗∗ ) ⊕ b e and ϕ : M t ⊗ R ω R → ( M t ⊗ R ω R ) ∗∗ is surjective. This implies ψ is also surjective, and we have s ( M t ) ≤ s ( τ ( M t )).If R is Gorenstein, then M ∼ = τ ( M ) for all indecomposable MCM modules. Thus s ( M ) = s ( τ ( M )) holds. In the rest, we assume that R is not Gorenstein, and hence R = ω R . Thereforewe also have M = τ ( M ) for all indecomposable MCM modules. For any indecomposable specialCM module M t , we have the following surjection by the same way as above ω ⊕ b e uR −→ e τ ( M t ) ≈ R ⊕ d ,t ⊕ n M i =1 M ⊕ d i,t i ! ⊕ p en ψ − ։ τ ( M t ) ⊕ b e . In this surjection, morphisms which go through R don’t contribute to construct a surjection byNakayama’s lemma. Thus, in addition to a surjection ω ⊕ b e uR − ։ τ ( M t ) ⊕ b e , we also construct asurjection R ⊕ p en d ,t − ։ τ ( M t ) ⊕ p en d ,tv , where v is the number of minimal generators of τ ( M t ). Therefore, we obtain b e ( τ ( M t )) ≥ b e + d ,t p e vn UAL F -SIGNATURE OF SPECIAL CM MODULES 14 where b e ( τ ( M t )) is the e -th F -surjective number of τ ( M t ). Thus, s ( τ ( M t )) ≥ s ( M t ) + d ,t vn > s ( M t ) . (cid:3) Acknowledgements.
The author is deeply grateful to Professor Mitsuyasu Hashimoto forgiving him valuable advice and encouragements. He also thanks Akiyoshi Sannai for giving somecomments about the dual F -signature and thanks Professor Ken-ichi Yoshida for suggestingthe comparison between special CM modules and other modules (Section 4 is based on hissuggestion). Finally, the author would also like to thank the referee for his/her careful readingand useful comments.The author is supported by Grant-in-Aid for JSPS Fellows (No. 26-422). References [AL] I. Aberbach and G. Leuschke,
The F -signature and strongly F -regularity , Math. Res. Lett. (2003), 51–56.[Aus1] M. Auslander, Rational singularities and almost split sequences , Trans. Amer. Math. Soc. (1986), no.2, 511–531.[Aus2] M. Auslander,
Isolated singularities and existence of almost split sequences , Proc. ICRA IV, SpringerLecture Notes in Math. (1986), 194–241.[HL] C. Huneke and G. Leuschke,
Two theorems about maximal Cohen-Macaulay modules , Math. Ann. (2002), no. 2, 391–404.[HN] M. Hashimoto and Y. Nakajima,
Generalized F -signature of invariant subrings , J. Algebra, (2015),142–152.[HS] N. Hara and T. Sawada, Splitting of Frobenius sandwiches , RIMS Kˆokyˆuroku Bessatsu
B24 (2011), 121–141.[Ish] A. Ishii,
On the McKay correspondence for a finite small subgroup of
GL(2 , C ), J. Reine Angew. Math. (2002), 221–233.[Ito] Y. Ito, Special McKay correspondence , S´emin. Congr. (2002), 213–225.[IW] O. Iyama and M. Wemyss, The classification of special Cohen Macaulay modules , Math. Z. (2010), no.1, 41–83.[Kun] E. Kunz,
Characterizations of regular local rings for characteristic p , Amer. J. Math. (1969), 772–784.[LW] G. Leuschke and R. Wiegand, Cohen-Macaulay Representations , vol. 181 of Mathematical Surveys andMonographs, American Mathematical Society (2012).[Nak] Y. Nakajima,
Dual F -signature of Cohen-Macaulay modules over rational double points , Algebr. Represent.Theory, (2015), 1211–1245.[Rie] O. Riemenschneider, Special representations and the two-dimensional McKay correspondence , HokkaidoMath. J. (2003), no. 2, 317–333.[San] A. Sannai, On Dual F -Signature , Int. Math. Res. Not. IMRN, Vol. 2015, no. 1, pp. 197–211.[SVdB] K. E. Smith and M. Van den Bergh, Simplicity of rings of differential operators in prime characteristic ,Proc. Lond. Math. Soc. (3) (1997), no. 1, 32–62.[Tuc] K. Tucker, F -signature exists , Invent. Math. (2012), no. 3, 743–765.[Wem] M. Wemyss, Reconstruction algebras of type A , Trans. Amer. Math. Soc. (2011), 3101–3132.[Wun1] J. Wunram,
Reflexive modules on cyclic quotient surface singularities , Lecture Notes in Mathematics,Springer-Verlag (1987), 221–231.[Wun2] J. Wunram,
Reflexive modules on quotient surface singularities , Math. Ann. (1988), no. 4, 583–598.[WY] K. Watanabe and K. Yoshida,
Minimal relative Hilbert-Kunz multiplicity , Illinois J. Math. (2004), no.1, 273–294.[Yao1] Y. Yao, Modules with Finite F -Representation Type , J. Lond. Math. Soc. (2) (2005), no. 2, 53–72.[Yao2] Y. Yao, Observations on the F -signature of local rings of characteristic p , J. Algebra (2006), no. 1,198–218.[Yos] Y. Yoshino, Cohen-Macaulay modules over Cohen-Macaulay rings , London Mathematical Society LectureNote Series, , Cambridge University Press, Cambridge, (1990).
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