DUAL F-SIGNATURE OF COHEN-MACAULAY MODULES OVER QUOTIENT SURFACE SINGULARITIES

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1 DUL F-SIGNTURE OF COHEN-MCULY MODULES OVER QUOTIENT SURFCE SINGULRITIES YUSUKE NKJIM. INTRODUCTION Thrughut this paper, we suppse that k is an algebraically clsed field f prime characteristic p >. Let R be a Netherian ring f prime characteristic p >, then we can define the Frbenius mrphism F : R R (r r p ). Fr e N, we als define the e-times iterated Frbenius mrphism F e : R R (r r pe ). Fr any R-mdule M, we define the R-mdule F e M via F e as fllws. That is, F e M is just M as an abelian grup, and its R-mdule structure is defined by r m F e (r)m = r pe m (r R, m M). We say R is F-finite if F R is a finitely generated R-mdule. Fr example, if R is an essentially f finite type ver a perfect field r cmplete Netherian lcal ring with a perfect residue field k, then R is F-finite. In this article, we nly discuss such rings, thus the F-finiteness is always satisfied. In psitive characteristic cmmutative algebra, we understand the prperties f R thrugh the structure f F e M. Fr this purpse, several numerical invariants are defined. Firstly, we intrduce the ntin f F-signature defined by C. Huneke and G. Leuschke. Definitin. ([HL]). Let (R,m,k) be a d-dimensinal reduced F-finite Netherian lcal ring f prime characteristic p >. Fr each e N, we decmpse F e R as fllws F e R R a e M e, where M e has n free direct summands. We call a e the e-th F-splitting number f R. Then, the a e limit s(r) lim is called the F-signature f R. e ped nd K. Tucker shwed its existence [Tuc]. By Kunz s therem, R is regular if and nly if F e R is a free R-mdule f rank p ed [Kun]. Thus, rughly speaking, the F-signature s(r) measures the deviatin frm regularity. The next therem cnfirms this intuitin. Therem. ([HL], [Ya], [L]). Let (R,m,k) be a d-dimensinal reduced F-finite Netherian lcal ring with charr = p >. Then we have () R is regular if and nly if s(r) =, () R is strngly F-regular if and nly if s(r) >. This ntin is extended fr a finitely generated R-mdule as fllws. Definitin. ([San]). Let (R,m,k) be a d-dimensinal reduced F-finite Netherian lcal ring f prime characteristic p >. Fr a finitely generated R-mdule M and e N, we set b e (M) max{n ϕ : F e M M n }, THIS PPER IS N NNOUNCEMENT OF OUR RESULT ND THE DETILED VERSION WILL BE SUBMITTED TO SOMEWHERE.

2 b e (M) and call it the e-th F-surjective number f M. Then we call the limit s(m) lim e p ed F-signature f M if it exists. dual Remark.. Since the mrphism F e R R b e(r) splits, if M is ismrphic t the basering R, then the dual F-signature f R in sense f Definitin. cincides with the F-signature f R. Thus, we use the same ntatin unless it causes cnfusin. Remark.. Since the m-adic cmpletin cmmutes with F e ( ), we can easily reduce the case f cmplete lcal ring in Definitin. and.. Thus, we may assume that the Krull-Schmidt cnditin hlds fr R. Just like the F-signature, the dual F-signature als characterizes sme singularities. Therem. ([San]). Let (R, m, k) be a d-dimensinal reduced F-finite Chen-Macaulay lcal ring with charr = p >. Then we have () R is F-ratinal if and nly if s(ω R ) >, () s(r) s(ω R ), () s(r) = s(ω R ) if and nly if R is Grenstein. In this way, the value f s(r) and s(ω R ) characterize sme singularities. Nw we have sme questins. Let M be a finitely generated R-mdule which may nt be R r ω R. Then Des the value f s(m) have sme pieces f infrmatin abut singularities What des the explicit value f s(m) mean Is there any cnnectin between s(m) and ther numerical invariants Hwever, it is difficult t try these questins fr nw. Because the value f dual F-signature is nt knwn and we dn t have an effective methd fr determining it except nly a few cases. Fr example, the case f tw-dimensinal Vernese subrings is studied in [San, Example.7]. Thus, in this article, we investigate the dual F-signature fr Chen-Macaulay (=CM) mdules ver tw-dimensinal ratinal duble pints. Therefre, in the rest f this article, we suppse that G is a finite subgrup f SL(,k) and the rder f G is cprime t p = chark. We remark that G cntains n pseud-reflectins in this situatin and it is well knwn that a finite subgrup f SL(,k) is cnjugate t ne f the type s-called ( n ),(D n ),(E ),(E 7 ) r (E 8 ). We dente the invariant subring f S k[[x,y]] under the actin f G by R S G and the maximal ideal f R by m. In this situatin, the invariant subring R is Grenstein by [Wat]. We call R (r equivalently Spec R) ratinal duble pints (r Du Val singularities, Kleinian singularities, DE singularities in the literature). Let V = k,v,,v n be the full set f nn-ismrphic irreducible representatins f G. We set M t (S k V t ) G (t =,,,n). Under the assumptin G cntains n pseud-reflectins, we can see that each M t is an indecmpsable maximal Chen-Macaulay (=MCM) R-mdule (rank R M t = dim k V t ) and M s M t (s t). Fr mre details refer t [HN, Sectin ]. In this article, we will investigate the value f s(m t ). In rder t determine the dual F- signature, we have t understand the fllwing tpics; () The structure f F e M t, namely What kind f MCM appears in F e M t as a direct summand The asympttic behavir f F e M t n the rder f p e. () Hw d we cnstruct a surjectin F e M t M b e t

3 T shw the frmer ne, we need the ntin f generalized F-signature. S we review it in Sectin. fter that we will use the ntin f the uslander-reiten quiver t shw the latter prblem. Thus, we give a brief summary f the uslander-reiten thery in Sectin. In Sectin, we actually determine the value f dual F-signature f CM mdules. Since the strategy fr determining the dual F-signature is almst the same fr all the DE cases, we will give a cncrete explanatin nly fr the case f D. In Sectin, we give the cmplete list f the value f dual F-signature fr all the DE cases.. GENERLIZED F-SIGNTURE OF INVRINT SUBRINGS Firstly it is knwn that R is f finite CM representatin type, that is, it has nly finitely many nn-ismrphic indecmpsable MCM mdules {R,M,,M n }. Since F e R is an MCM R-mdule, we can describe as F e R R c,e M c,e M c n,e n. Since the Krull-Schmidt cnditin hlds fr R, the multiplicities c t,e are determined uniquely. Fr understanding the asympttic behavir f the multiplicity c t,e, we cnsider the limit c t,e s(r,m t ) lim e p e (t =,,,n). We call it generalized F-signature f M t. In ur situatin, this limit exists [SVdB, Ya]. nd the value f this limit is knwn as fllws. Therem.. ([HS, Lemma.], see als [HN, Therem.]) Fr t =,,, n, we have s(r,m t ) = rank R M t G = dim kv t G Remark.. In the case f t =, we have s(r,r) = s(r) and the abve result is als due t [HL, Example 8], [WY, Therem.]. s a crllary, we als have the next statement. Crllary.. ([HN, Crllary.]) Suppse an MCM R-mdule F e M t decmpses as fllws. F e M t R dt,e M dt,e M dt n,e n. Then, fr all s,t =,,n, we have ds,e t s(m t,m s ) lim e p e = (rank R M t ) s(r,m s ) = (rank R M t ) (rank R M s ). G Remark.. s Crllary. shws, every indecmpsable MCM R-mdules appear in F e M t as a direct summand fr sufficiently large e >>. Therefre, the additive clsure add R (F e M t ) cincides with the categry f MCM R-mdules CM(R). S we can apply several results scalled uslander-reiten thery t add R (F e M t ). We discuss it in the next sectin.. REVIEW OF USLNDER-REITEN THEORY Frm Nakayama s lemma, when we discuss the surjectivity f F e M t Mt b, we may cnsider each MCM mdule as a vectr space after tensring the residue field k. Thus, we want t knw a basis f M t mm t (i.e. minimal generatrs f M t ). Fr sme m N, we have

4 x ^ 8 g e z : % ' M t Hm R (R,M t ) mm t nn split {R R m M t }. Frm this bservatin, we identify a minimal generatr f M t with a mrphism frm R t M t which desn t factr thrugh free mdules except the starting pint. In rder t find such mrphisms, we will use the ntin f uslander-reiten (=R) quiver. S we review sme results f uslander-reiten thery in this sectin. Fr mre details, see sme textbks (e.g. [Ys]). In rder t define the R quiver, we intrduce the ntin f irreducible mrphism. Definitin. (Irreducible mrphism). Suppse M and N are MCM R-mdules. We decmpse M and N int indecmpsable mdules as M = i M i, N = j N j and als decmpse ψ Hm R (M,N) alng the abve decmpsitin as ψ = (ψ i j : M i N j ) i j. Then we define submdule rad R (M,N) Hm R (M,N) as ψ rad R (M,N) de f n ψ i j is an ismrphism. Furthermre, we define submdule rad R (M,N) Hm R(M,N). The submdule rad R (M,N) cnsists f mrphisms ψ : M N such that ψ decmpses as ψ = f g, where f rad R (M,Z), g rad R (Z,N) and Z is an MCM R-mdule. We say that a mrphism ψ : M N is irreducible if ψ rad R (M,N) \ rad R (M,N). In this setting, we define the k-vectr space Irr R(M,N) as Irr R (M,N) rad R (M,N) rad R (M,N). By using this ntin, we define the R quiver. Definitin. (uslander-reiten quiver). The R quiver f R is an riented graph whse vertices are indecmpsable MCM R-mdules {R,M,,M n } and draw dim k Irr R (M s,m t ) arrws frm M s t M t (s,t =,,,n). In ur situatin, the R quiver f R cincides with the McKay quiver f G by [us], s we can describe it frm representatins f G (fr the definitin f McKay quiver, refer t [Ys, (.)]). nd mre frtunately, the R quiver f R cincides with the extended Dynkin diagram crrespnding t a finite subgrup f SL(,k) after replacing each edges by arrws. This is a kind f McKay crrespndence. Therefre the uslander-reiten quiver f R is the left hand side f the fllwing, ( n ) n n n n (D n ) n n

5 O O O a O } =! a } =! (E ) (E 7 ) (E 8 ) where a vertex t crrespnds the MCM R-mdule M t and the right hand side f the figure means rank R M t.. DUL F-SIGNTURE OVER RTIONL DOUBLE POINTS The strategy fr determining the dual F-signature is almst the same fr all the DE cases. S frm nw n, we will explain the methd f determining it by using the fllwing example. ( ) ( ) ζ ζ Example.. The binary dihedral grup G D = ζ, is the type D ζ in the list [Ys, (.)] and G =. Fr the invariant subring under the actin f G, the R quiver takes the frm f D, with relatins b a B C c a =, C c + D d + E e =, B b =, d D =, a + b B + c C =, e E = In rder t find mrphisms frm R t M t which desn t factr thrugh free mdules except the starting pint, we define the stable categry CM(R) as fllws. The bjects f CM(R) are same as thse f CM(R) and the mrphism set is given by d E D e (.) Hm R (X,Y ) Hm R (X,Y ) P(X,Y ), X,Y CM(R) where P(X,Y ) is the submdule f Hm R (X,Y ) cnsisting f mrphisms which factr thrugh a free R-mdule. By the prperty f R quiver, we can see that mrphisms frm R t nn-free indecmpsable M t CM(R) (n the R quiver f D ) always g thrugh the vertex M at the beginning. Thus, the cmpsitin f R a M and nn-zer elements f Hm R (M,M t ) are exactly what we wanted. Therefre we will find nn-zer elements f Hm R (M,M t ). Fr this purpse we

6 rewrite the R quiver as a repetitin f the riginal ne. a b c e d B C D E Since this quiver has relatins, it seems t be difficult t extract nn-zer mrphisms frm the abve picture. But there is a useful technique s-called cunting argument f R quiver. By using such a technique, we can extract desired mrphisms. This methd first appeared in the wrk f Gabriel [Gab] and it is als used fr classifying special CM mdules ver qutient surface singularities [IW]. Fr the details f this kind f cunting argument, see [Gab, Iya, IW]. fter applying such a technique, we btain the fllwing picture. nd paths n this quiver represent nn-zer mrphisms in Hm R (M,M t ). The fllwing quiver is the cmpsitin f R a M and nn-zer element f Hm R (M,M t ) fr t =,,, (the expnent f each vertex implies the multiplicity). (.) Thus, we identify paths n this quiver with minimal generatr f each MCM R-mdule M t. Fr example, minimal generatrs f M are identified with and (.) Of curse, there are several paths frm R t M nt nly the abve nes. But they are same up t mdul radical because this quiver has relatins. When we cnsider a surjectin, it desn t matter if we identify them. By using these results, we try t determine s(m ) as an example. Firstly, we suppse an MCM R-mdule F e M decmpses as F e M R d,e M d,e M d n,e n. Frm Crllary., we have the fllwing fr all s =,,,n, d s,e s(m,m s ) = lim e p e = (rank R M )(rank R M s ) = rank R M s. G

7 Thus, we may cnsider F e M (R M M M M M )pe, n the rder f p e in this case. When we try t cmpute dual F-signature, the part f (p e ) is harmless. S we identify F e M with ( R M M M M M ) and cnsider a surjectin R M M M M M M b. s we shwed befre, minimal generatrs f M are identified with paths in (.). We dente the left (resp. right) f them by g (resp. g ). In rder t cnstruct a surjectin, we pay attentin t g. We can see that M can generate g thrugh the mrphism M B M. a " min. gen. f M + B = g : a! B Similarly, R can generate g thrugh the mrphism R B a M. Mrever, M clearly generate M g thrugh the identity map (M M ). nd we have n ther such MCMs. Cllectively, MCM mdules which generate the minimal generatr g are {R,M,M }. Thus, the value f s(m ) can take s(m ) = (In rder t shw the rati f s(m ) t the rder f G, we dn t reduce the fractin). In this way, we btain the upper bunds f s(m ). Next, we will shw that we can actually cnstruct a surjectin + + R M M M M M M s we shwed befre, each minimal generatr f M is identified with a mrphism frm t in (.). Cnsidering the cmpsitin f such mrphisms and B, we have g and g at the same time. Namely, we have the surjectin M B M. Mrever, there is the surjectin M M clearly. In this way, M and M can generate g and g at the same time. But R can t generate g and g at the same time (nly generate either g r g ). We will use it fr generating g and use the remaining MCMs {M,M,M } fr generating g. (Frm the picture (.), we read ff that these remaining MCMs generate g.) Fr example, we have R M M. Thus, we cnclude s(m ) =... SUMMRY OF THE VLUE OF DUL F-SIGNTURE By using similar methd, we can btain the dual F-signature fr all the DE cases. Finally, we give the cmplete list f the dual F-signature. Therem.. The fllwing is the Dynkin diagram Q and crrespnding values f dual F- signature (In rder t shw the rati f dual F-signature t the rder f G clearly, we dn t reduce fractins ). () Type n 7

8 n is an even number (i.e. n = r) n : r r + n n n + n + r + n + r + n + n + n + n is an dd number (i.e. n = r ) n : r r r + n n n + n + r n + r + (n + ) r n + n + n + () Type D n n is an even number (i.e. n = r) n D n : m r r + n n (n ) (n ) (n ) (n ) m (n ) r (n ) r (n ) r (n ) (n ) (n ) n is an dd number (i.e. n = r ) n D n : m r r n n (n ) (n ) (n ) (n ) m (n ) r (n ) r (n ) r (n ) (n ) (n ) () Type E E :

9 () Type E E 7 : () Type E 8 8 E 8 : REFERENCES [L] I. berbach and G. Leuschke, The F-signature and strngly F-regularity, Math. Res. Lett. (), -. [us] M. uslander, Ratinal singularities and almst split sequences, Trans. mer. Math. Sc. 9 (98), n., -. [us] M. uslander, Islated singularities and existence f almst split sequences, Prc. ICR IV, Springer Lecture Ntes in Math. 78 (98), 9-. [R] M. uslander and I. Reiten, lmst split sequences fr Ratinal duble pints, Trans. mer. Math. Sc. (987), n., [Gab] P. Gabriel, uslander-reiten sequences and representatin-finite algebras, Lecture Ntes in Math. 8, Representatin thery I (Prceedings, Ottawa, Carletn Univ., 979), Springer, (98), 7, [HS] N. Hara and T. Sawada, Splitting f Frbenius sandwiches, RIMS Kôkyûrku Bessatu B (), -. [HN] M. Hashimt and Y. Nakajima, Generalized F-signature f invariant subrings, arxiv:.9. [HL] C. Huneke and G. Leuschke, Tw therems abut maximal Chen-Macaulay mdules, Math. nn. (), n., 9-. [Kun] E. Kunz, Characterizatins f regular lcal rings fr characteristic p, mer. J. Math. (99), [Iya] O. Iyama, τ-categries I: Ladders, lgeb. Represent. Thery 8 (), n., 97-. [IW] O. Iyama and M. Wemyss, The classificatin f special Chen Macaulay mdules, Math. Z. (), n., -8. [San]. Sannai, Dual F-signature, t appear in Internatinal Mathematics Research Ntices, arxiv:.8. [SVdB] K. E. Smith and M. Van den Bergh, Simplicity f rings f differential peratrs in prime characteristic, Prc. Lndn Math. Sc. () 7 (997), n., -. [Tuc] K. Tucker, F-signature exists, Invent. Math. 9 (), n., 7-7. [Wat] K. Watanabe, Certain invariant subrings are Grenstein. I, Osaka J. Math. (97), -8. [WY] K. Watanabe and K. Yshida, Minimal relative Hilbert-Kunz multiplicity, Illinis J. Math. 8 (), n., 7-9. [Ya] Y. Ya, Mdules with Finite F-Representatin Type, J. Lndn Math. Sc. 7 (), n., -7. [Ya] Y. Ya, Observatins n the F-signature f lcal rings f characteristic p, J. lgebra 99 (), n., [Ys] Y. Yshin, Chen-Macaulay mdules ver Chen-Macaulay rings, Lndn Mathematical Sciety Lecture Nte Series,, Cambridge University Press, Cambridge, (99). GRDUTE SCHOOL OF MTHEMTICS, NGOY UNIVERSITY, CHIKUS-KU, NGOY, -8 JPN address: mzmath.nagya-u.ac.jp 9