Generalized F-signature of invariant subrings
aa r X i v : . [ m a t h . A C ] A ug GENERALIZED F -SIGNATURE OF INVARIANT SUBRINGS MITSUYASU HASHIMOTO AND YUSUKE NAKAJIMAA
BSTRACT . It is known that a certain invariant subring R has finite F -representation type. Thus,we can write the R -module e R as a finite direct sum of finitely many R -modules. In such adecomposition of e R , we pay attention to the multiplicity of each direct summand. For themultiplicity of free direct summand, there is the notion of F -signature defined by C. Huneke andG. Leuschke and it characterizes some singularities. In this paper, we extend this notion to nonfree direct summands and determine their explicit values.
1. I
NTRODUCTION
Throughout this paper, we suppose that k is an algebraically closed field of prime charac-teristic p >
0, and V is a d -dimensional k -vector space. Let G ⊂ GL ( V ) be a finite subgroupsuch that the order of G is not divisible by p , and G contains no pseudo-reflections. Let S be a symmetric algebra of V . We denote the invariant subring of S under the action of G by R ≔ S G . Sometimes we denote p e by q . Since char R = p >
0, we can define the Frobenius map F : R → R ( r r p ) and also define the e -times iterated Frobenius map F e : R → R ( r r p e ) for e ∈ N . For any R -module M , we denote the module M with its R -module structure pulled backvia F e by e M . That is, e M is just M as an abelian group, and its R -module structure is given by r · m ≔ F e ( r ) m = r p e m for all r ∈ R , m ∈ M .In our assumption, it is known that the invariant subring R has finite F -representation type(or FFRT for short). The notion of FFRT is defined by K. Smith and M. Van den Bergh [SVdB]. Definition 1.1.
We say that R has finite F-representation type by N if there is a finite set N ofisomorphism classes of finitely generated R-modules, such that for any e ∈ N , the R-module e Ris isomorphic to a finite direct sum of elements of N . More explicitly, finitely many finitely generated R -modules which form such a finite set N are described as follows. Proposition 1.2. ( [SVdB, Proposition 3.2.1] ) Let V = k , V , · · · , V n be a complete set of non-isomorphic irreducible representations of G and we set M i ≔ ( S ⊗ k V i ) G ( i = , , · · · , n ) . ThenR has FFRT by { M (cid:27) R , M , · · · , M n } . From this proposition, we can decompose e R as follows e R (cid:27) R ⊕ c , e ⊕ M ⊕ c , e ⊕ · · · ⊕ M ⊕ c n , e n . Now, we want to investigate the multiplicities c i , e . For the multiplicity of the free direct sum-mand, that is, for the multiplicity c , e , there is the notion of F -signature defined by C. Hunekeand G. Leuschke. Definition 1.3. ( [HL, Definition 9] ) The F-signature of R is s ( R ) ≔ lim e → ¥ c , e p de , if it exists. Mathematics Subject Classification.
Primary 13A35, 13A50.
Key words and phrases. F -signature, finite F -representation type, invariant subrings. Note that K. Tucker showed its existence under more general settings [Tuc, Theorem 4.9] ( see also Proposition 3.1 ) . And it is known that this numerical invariant characterizes somesingularities. For example, s ( R ) = R is regular [HL, Corollary 16] ( see also[Yao2, Theorem 3.1] ) , and s ( R ) > R is strongly F -regular [AL, Theorem 0.2].In our situation, the F -signature of the invariant subring R is determined as follows and it impliesthat R is strongly F -regular. Theorem 1.4. ( [WY, Theorem 4.2] ) The F-signature of the invariant subring R iss ( R ) = | G | . Remark . In [WY], this theorem is proved in terms of minimal relative Hilbert-Kunz multi-plicity. And Y. Yao showed that it coincides with the F -signature [Yao2, Remark 2.3 (4)].Now, we extend this notion to other direct summands. Namely, we investigate the multiplici-ties c i , e ( i = , · · · , n ) and determine the limit lim e → ¥ c i , e p de . In order to determine this limit, we haveto care about the next two problems first. • For each e ∈ N , are the multiplicities c i , e determined uniquely? • Does the limit lim e → ¥ c i , e p de exist?In Section 2, we will show the uniqueness of the multiplicities. In Section 3, we will showthe existence of the limit and determine the limit (see Theorem 3.4).2. U NIQUENESS OF DECOMPOSITION
In this section, we show the uniqueness of the multiplicities. Firstly, we introduce the notionof Frobenius twist ( e.g. [Jan] ) . Definition 2.1.
For k-vector space V and e ∈ Z , we define k-vector space e V as follows • e V is the same as V as an additive group; • the action of a ∈ k on e V is a · v = a p e v.An element v ∈ V , viewed as an element of e V , is sometimes denoted by e v. Thus a · e v = e ( a p e v ) .By the composition G ֒ → GL ( V ) f → GL ( e V ) , e V is also a representation of G, where f is givenby f ( g )( e v ) = e ( gv ) for g ∈ G and v ∈ V . We call this representation the Frobenius twist of V .Sometimes we denote this representation by V ( − e ) . Let v , · · · , v d be a basis of V . For this basis, we suppose that a representation of G is definedby g · v j = d (cid:229) i = f i j ( g ) v i ( g ∈ G , f i j : G → k ) . Namely, a matrix representation of V is described by (cid:0) f i j ( g ) (cid:1) . Since k is an algebraicallyclosed field, the basis v , · · · , v d also forms a basis of e V , and the action of G on e V is describedas follows g · e v j = e ( g · v j ) = e ( d (cid:229) i = f i j ( g ) v i ) = d (cid:229) i = f i j ( g ) p − e ( e v i ) . ENERALIZED F -SIGNATURE OF INVARIANT SUBRINGS 3 From this observation, a matrix representation of the Frobenius twist e V is described by (cid:0) ( f i j ( g )) p − e (cid:1) ,that is, each component of the matrix representation of e V is the p − e -th power of the originalone.In order to show the uniqueness of the multiplicities, we prove the following. Proposition 2.2.
For e ≥ , c , e , · · · , c n , e ≥ , the following decompositions are equivalent (1) e R (cid:27) M ⊕ c , e ⊕ M ⊕ c , e ⊕ · · · ⊕ M ⊕ c n , e n as R-modules; (2) e S (cid:27) ( S ⊗ k V ) ⊕ c , e ⊕ ( S ⊗ k V ) ⊕ c , e ⊕ · · · ⊕ ( S ⊗ k V n ) ⊕ c n , e as ( G , S ) -modules; (3) e S / m e S (cid:27) V ⊕ c , e ⊕ V ⊕ c , e ⊕ · · · ⊕ V ⊕ c n , e n as G-modules; (4) there exist a i j ∈ q Z ≥ such that e S (cid:27) n M i = c i , e M j = ( S ⊗ k V i )( − a i j ) as q Z -graded ( G , S ) -modules; (5) there exist a i j ∈ q Z ≥ such that e R (cid:27) n M i = c i , e M j = M i ( − a i j ) as q Z -graded R-modules.Remark . A similar correspondence holds for more general situation up to the action of the e -th Frobenius kernel of a group scheme [Has]. For the case of a finite group G , the e -th Frobeniuskernel of G is trivial. Thus, we may ignore it in our context.To prove this proposition, the next theorem plays the central role. This is proved in [Has] formore general settings using some geometric settings. For convenience of the reader, we give ashort and simple proof using a result of Iyama and Takahashi [IT] or Leuschke and Wiegand[LW]. The two-dimensional case is very well-known as a theorem of Auslander [Aus]. See also[Yos, Chapter 10]. Theorem 2.4.
If G contains no pseudo-reflections, then the functor
Ref ( G , S ) → Ref ( R ) ( M M G ) is an equivalence, where Ref ( G , S ) is the category of reflexive ( G , S ) -modules and Ref ( R ) is the category of reflexive R-modules. The quasi-inverse is N ( S ⊗ R N ) ∗∗ .The same functors give an equivalence ∗ Ref ( G , S ) → ∗ Ref ( R ) , where ∗ Ref ( G , S ) is the cat-egory of Z [ / p ] -graded reflexive ( G , S ) -modules and ∗ Ref ( R ) is the category of Z [ / p ] -gradedreflexive R-modules.Proof. Let S ∗ G denote the twisted group algebra. It is L g ∈ G S · g as an S -module (with thefree basis G ), and the multiplication is given by ( sg )( s ′ g ′ ) = ( s ( gs ′ ))( gg ′ ) . A ( G , S ) -moduleand an S ∗ G -module are one and the same thing. As a ( G , S ) -module, S ∗ G and S ⊗ k kG arethe same thing, where kG is the group algebra (the left regular representation) of G over k . SoHom S ( S ∗ G , S ) (cid:27) S ⊗ k k [ G ] (cid:27) S ⊗ k kG (cid:27) S ∗ G , where k [ G ] = ( kG ) ∗ is the k -dual of kG (the leftregular representation).Let us denote by S ′ the R -module S with the trivial G -module structure. Note that S ′ → ( S ⊗ k k [ G ]) G given by s (cid:229) g ∈ G gs ⊗ e g is an isomorphism, where { e g | g ∈ G } is the dual basisof k [ G ] , dual to G , which is a basis of kG . Note that g ′ e g = e g ′ g .For M ∈ Ref ( G , S ) , M G is certainly reflexive. Indeed, there is a presentation ( S ∗ G ) u → ( S ∗ G ) v → M ∗ → . (2.1)Applying ( ? ) G ◦ Hom S ( ? , S ) , 0 → M G → ( S ′ ) v → ( S ′ ) u MITSUYASU HASHIMOTO AND YUSUKE NAKAJIMA is exact. As it is easy to see that S ′ satisfies the ( S ) -condition as an R -module (that is, for P ∈ Spec R , if depth R P ( S ′ P ) <
2, then S ′ P is a maximal Cohen–Macaulay R P -module), so is M G ,and it is reflexive.On the other hand, it is obvious that ( S ⊗ R N ) ∗∗ is a reflexive ( G , S ) -module, since it is a dualof some S -finite ( G , S ) -module.Let u : N → (( S ⊗ R N ) ∗∗ ) G be the map given by u ( n ) = l ( ⊗ n ) , where l : S ⊗ R N → ( S ⊗ R N ) ∗∗ is the canonical map. We show that u is an isomorphism. To verify this, since both N and (( S ⊗ R N ) ∗∗ ) G are reflexive, it suffices to show that u P : N P → ((( S ⊗ R N ) ∗∗ ) G ) P (cid:27) (( S P ⊗ R P N P ) ∗∗ ) G is an isomorphism for P ∈ Spec R with dim R P ≤ N P is a freemodule, and we may assume that N P = R P by additivity. This case is trivial.Let e : ( S ⊗ R M G ) ∗∗ → M be the composite ( S ⊗ R M G ) ∗∗ a ∗∗ −−→ M ∗∗ l − −−→ M , where a : S ⊗ R M G → M is given by a ( s ⊗ m ) = sm . We show that e is an isomorphism. Since ( S ⊗ R M G ) ∗ and M are reflexive, it suffices to show that a ∗ : M ∗ → ( S ⊗ R M G ) ∗ is an isomor-phism. By the five lemma and the existence of the presentation of the form (2.1), we mayassume that M = S ⊗ k k [ G ] . Then a ∗ is identified with the map S ∗ G (cid:27) ( S ⊗ k k [ G ]) ∗ a ∗ −→ ( S ⊗ R ( S ⊗ k k [ G ]) G ) ∗ (cid:27) ( S ⊗ R S ′ ) ∗ (cid:27) Hom R ( S ′ , S ) . It is easy to see that this map is given by sg ( s ′ s ( gs ′ )) . This is an isomorphism by [IT,Theorem 4.2] or [LW, Theorem 5.12].As u and e are isomorphisms, M M G and N ( S ⊗ R N ) ∗∗ are quasi-inverse to each other,and hence they are category equivalences.The graded version is proved similarly. (cid:3) By using this theorem, we give the proof of Proposition 2.2.
Proof of Proposition 2.2.
The equivalences of ( ) and ( ) , ( ) and ( ) follow from Theorem 2.4,and ( ) is obtained by applying ( − ⊗ S k ) to ( ) . If we forget the grading from ( ) , then we obtain ( ) . ( ) Thm . . ⇐⇒ ( ) ⊗ S k = ⇒ ( ) ~ww forgetgrading ( ) ⇐⇒ Thm . . ( ) So we will show ( ) ⇒ ( ) . If we consider e S / m e S as a q Z -graded G -module, then we canwrite e S / m e S (cid:27) n M i = c i , e M j = V i ( − a i j ) for some a i j ∈ q Z ≥ . Then as in the proof of [SVdB, Proposition 3.2.1], we have e S (cid:27) S ⊗ k ( e S / m e S ) , and (4) follows. (cid:3) Especially, the decomposition ( ) appears in Proposition 2.2 is unique. Thus, we obtain thenext statement as a corollary. ENERALIZED F -SIGNATURE OF INVARIANT SUBRINGS 5 Corollary 2.5.
Each M i is indecomposable and the multiplicities c i , e are determined uniquely. In Proposition 2.2 and Corollary 2.5, the condition “ G contains no pseudo-reflections” is es-sential. If G contains a pseudo-reflection, then there is a counter-example as follows. Example 2.6.
Let S = k [ x , y ] be a polynomial ring, where ( char k , | G | ) = . Set G = (cid:10) s = (cid:18) (cid:19) (cid:11) , that is G is a symmetric group S , and, V = k , V = sgn are irreducible represen-tations of G. (Note that s is a pseudo-reflection.) Then, R ≔ S G (cid:27) k [ x + y , xy ] . Since R is apolynomial ring, e R (cid:27) R ⊕ p e . On the other hand,M ≔ ( S ⊗ k V ) G = { f ∈ S | s · f = ( sgn s ) f } = ( x − y ) R (cid:27) R . So e R also decompose as e R (cid:27) M ⊕ p e . Therefore, the uniqueness doesn’t hold in this case.
3. G
ENERALIZED F - SIGNATURE OF INVARIANT SUBRINGS
In this section, we show the existence of the limit and determine it.In our case, the invariant subring R has FFRT. Thus, the existence of the limit lim e → ¥ c i , e p de isguaranteed by the next proposition. So we can define this limit. Proposition 3.1. ( [SVdB, Proposition 3.3.1] , [Yao1, Theorem 3.11] ) If R has FFRT, then fori = , , · · · , n, the limit lim e → ¥ c i , e p de exists.Remark . In [SVdB], this proposition is proved under the assumption “ R is strongly F -regular and has FFRT”. After that, Y. Yao showed the condition of strongly F -regular is unnec-essary [Yao1]. Note that the existence of the limit for free direct summands ( i.e. F -signature ) is proved under more general settings as we showed before. Definition 3.3.
We call this limit the generalized F-signature of M i with respect to R and denoteit by s ( R , M i ) ≔ lim e → ¥ c i , e p de . The main theorem in this paper is the following.
Theorem 3.4 (Main theorem) . Let the notation be as above. Then for all i = , · · · , n one hass ( R , M i ) = dim k V i | G | = rank R M i | G | . Remark . The second equation follows from dim k V i = rank R M i clearly.The case that i = ( Theorem 1.4 ) . Anda similar result holds for finite subgroup scheme of SL [HS, Lemma 4.10]. Remark . From this theorem, we can see that each indecomposable MCM R -modules in thefinite set { R , M , · · · , M n } actually appear in e R as a direct summand for sufficiently large e ( seealso [TY, Proposition 2.5] ) .In order to prove this theorem, we introduce the notion of the Brauer character. In the repre-sentation theory of finite groups over C , the character gives us very effective method to distin-guish each representation. But now, we are in a positive characteristic field k , not in C . So thecharacter in the original sense doesn’t work well. Therefore we have to modify it for applying MITSUYASU HASHIMOTO AND YUSUKE NAKAJIMA to our context. For this purpose, we introduce the Brauer character ( for more details, refer tosome textbooks e.g. [CR], [Wei] ) .As we assume that m : = | G | is not divisible by p , there is a primitive m -th root of unity in k , and thus both m m ( k ) = { w ∈ k × | w m = } and m m ( C ) = { w ∈ C × | w m = } are the cyclicgroups of order m . Fix a group isomorphism F : m m ( k ) → m m ( C ) . Definition 3.7.
For a kG-module V , the Brauer character c V of V is the function c V : G → C given by c V ( g ) ≔ d (cid:229) i = F ( w i ) ∈ C ( g ∈ G ) , where w , · · · , w d are the eigenvalues of g. The following proposition is well-known for the original character over C . And this kind offormula also holds for the Brauer character. Proposition 3.8.
Let V , W be kG-modules and g ∈ G, then (1) c V ⊗ W ( g ) = c V ( g ) · c W ( g ) . (2) c V ⊕ W ( g ) = c V ( g ) + c W ( g ) . (3) c V ∗ ( g ) = c V ( g ) , where the bar denotes the conjugate of a complex number. (4) c V ( G ) = dim k V . (5) dim k V G = | G | (cid:229) g ∈ G c V ( g ) . (6) dim k Hom G ( V , W ) = | G | (cid:229) g ∈ G c V ( g ) · c W ( g ) .Proof. The statements (1)–(4) follow easily from the definition. (6) follows from (1), (3), and(5). So we only prove (5). If we show (5) for a particular choice of F , then (5) is true forarbitrary choice, say F ′ , because we can write F ′ = a ◦ F , where a is some automorphism of Q ( w ) over Q , where w is a primitive m -th root of unity in C . Let R be the ring of Witt vectorsover k . Note that R is a complete DVR (discrete valuation ring). Let t be its uniformizingparameter. We identify R / tR with k . Let ¯ w be a fixed primitive m -th root of unity in k . ByHensel’s lemma, it is easy to see that ¯ w lifts to a primitive m -th root of unity in R uniquely, sayto w . Note that V is a kG -module, and hence is an RG -module. Let V R → V be the projectivecover as an RG -module, which exists (note that RG is semiperfect). Note that V R / tV R = V , and V R is an R -free module of rank dim k V .Let R = Z [ w ] be the subring of R generated by w . Then regarding R as a subring of C ,we have that ˜ c V is a Brauer character of V , where ˜ c V ( g ) = trace V R ( g ) (the trace makes sense,since V R is a finite free R -module). Let g = | G | (cid:229) g ∈ G g ∈ RG . Then it is easy to see that g isa projector from any RG -module M to M G . In particular, the G -invariance ( ? ) G is an exactfunctor on the category of RG -modules. It follows that V G = ( V R / tV R ) G (cid:27) V GR / tV GR = k ⊗ R V GR .Let U : = ( − g ) V R . Then V R = V GR ⊕ U , and g is the identity map on V GR and zero on U . So1 | G | (cid:229) g ∈ G ˜ c V ( g ) = trace V R ( g ) = rank R V GR = dim k V G . This is what we wanted to prove. (cid:3) So we are now ready to prove the main theorem.
ENERALIZED F -SIGNATURE OF INVARIANT SUBRINGS 7 Proof of Theorem 3.4.
Firstly, there is e ≥ F q G is isomorphic to thedirect product of total matrix rings over F q , where q = p e . Namely, F q G (cid:27) Mat r ( F q ) × · · · × Mat r m ( F q ) , ( r , · · · , r m ∈ N ) . Since the component of matrix representation of Frobenius twist is p − e -th power of the originalone, so if we take an appropriate basis, then any component of matrix representation is in thefinite field F q . Thus, if e = e t , then we can consider e M (cid:27) M for any G -module M .Since we know the existence of the limit, it suffices to show the subsequence { c i , e t p de t } t ∈ N converges on ( dim k V i ) / | G | . So we provelim t → ¥ c i , e t p de t = dim k V i | G | . For e = e t , we obtain e S / m e S (cid:27) e ( S / m [ q ] ) (cid:27) S / m [ q ] . And e S / m e S is also isomorphic to thefinite direct sum of irreducible representations (cf. Proposition 2.2). By Proposition 3.8 (6), themultiplicity c i , e is described as follows. c i , e = dim k Hom G ( V i , S / m [ q ] ) = | G | (cid:229) g ∈ G c V i ( g ) · c S / m [ q ] ( g ) . Set g ∈ G and suppose that the order of g is m . Then there is a basis { x , · · · , x d } of V such thateach x i is an eigenvector of g and we can write g · x i = w i x i with w i = w d i for some 0 ≤ d i < m ,where w is a primitive m -th root of unity. In this situation { x l · · · x l d d | ≤ l , . . . , l d < q } ⊂ ( q − ) d M l = Sym l V is a basis of S / m [ q ] . As each x l · · · x l d d is an eigenvector of g with the eigenvalue w l · · · w l d d ,we have c S / m [ q ] ( g ) = (cid:229) ≤ l ,..., l d < q F ( w l · · · w l d d ) = d (cid:213) i = ( + q i + · · · + q q − i ) , where q i ≔ F ( w i ) .(i) In case g =
1, by Proposition 3.8 (4), c V i ( g ) · c S / m [ q ] ( g ) q d = dim k V i · q d q d = dim k V i . (ii) In case g ,
1, we may assume q d ,
1. Then (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) c V i ( g ) · c S / m [ q ] ( g ) q d (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12) c V i ( g ) (cid:12)(cid:12) q d d − (cid:213) i = ( | | + | q i | + · · · + | q i | q − ) · (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − q qd − q d (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ dim k V i q · | − q d | t → ¥ −−−→ . MITSUYASU HASHIMOTO AND YUSUKE NAKAJIMA
The first inequation is obtained by applying the triangle inequality. Since | q i | ≤
1, we can obtainthe second inequation.From previous arguments, we may only discuss in case g =
1. Thus, we concludelim e → ¥ c i , e q d = lim e → ¥ q d · | G | (cid:229) g ∈ G c V i ( g ) · c S / m [ q ] ( g ) = dim k V i | G | . (cid:3) Next, we consider the decomposition of e M i . Since each MCM R -module M i appears in e ′ R for sufficiently large e ′ ≫ e M i (cid:27) M ⊕ d i , e ⊕ M ⊕ d i , e ⊕ · · · ⊕ M ⊕ d in , e n . In this situation, we define the limit s ( M i , M j ) ≔ lim e → ¥ d ij , e p de and call it the generalized F -signature of M j with respect to M i . The next corollary immediatelyfollows from Theorem 3.4 and [SVdB, Proposition 3.3.1, Lemma 3.3.2]. Corollary 3.9.
Let the notation be as above. Then for all i , j = , · · · , n one hass ( M i , M j ) = ( dim k V i ) · s ( R , M j ) = ( dim k V i ) · ( dim k V j ) | G | = ( rank R M i ) · ( rank R M j ) | G | . Acknowledgements.
The authors would like to thank Professor Kei-ichi Watanabe for valu-able advice and also thank Professor Peter Symonds for reading the previous version of thismanuscript and pointing out that the Brauer character is more suitable to prove the main theo-rem. R
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