Generalized Hilbert-Kunz function of the Rees algebra of the face ring of a simplicial complex
aa r X i v : . [ m a t h . A C ] M a r GENERALIZED HILBERT-KUNZ FUNCTION OF THE REES ALGEBRA OFTHE FACE RING OF A SIMPLICIAL COMPLEX
ARINDAM BANERJEE, KRITI GOEL, AND J. K. VERMA
Abstract.
Let R be the face ring of a simplicial complex of dimension d − R ( n ) be theRees algebra of the maximal homogeneous ideal n of R. We show that the generalized Hilbert-Kunzfunction HK ( s ) = ℓ ( R ( n ) / ( n , n t ) [ s ] ) is given by a polynomial for all large s. We calculate it in manyexamples and also provide a Macaulay2 code for computing HK ( s ) . Dedicated to Roger Wiegand and Silvia Wiegand on the occasion of their th birthday Introduction
The objective of this paper is to find the generalized Hilbert-Kunz function of the maximal homo-geneous ideal of the Rees algebra of the maximal homogeneous ideal of the face ring of a simplicialcomplex. The Hilbert-Kunz functions of the Rees algebra, associated graded ring and the extendedRees algebra have been studied by K. Eto and K.-i. Yoshida in [3] and by K. Goel, M. Koley andJ. K. Verma in [5].In order to recall one of the main results of Eto and Yoshida, we set up some notation first. Let( R, m ) be a Noetherian local ring of dimension d and of prime characteristic p. Let q = p e where e isa non-negative integer. The q th Frobenius power of an ideal I is defined to be I [ q ] = ( a q | a ∈ I ) . Let I be an m -primary ideal. The Hilbert-Kunz function of I is the function HK I ( q ) = ℓ ( R/I [ q ] ) . Thisfunction, for I = m , was introduced by E. Kunz in [8] who used it to characterize regular local rings.The Hilbert-Kunz multiplicity of an m -primary ideal I is defined as e HK ( I ) = lim q →∞ ℓ ( R/I [ q ] ) /q d . It was introduced by P. Monsky in [10]. We refer the reader to an excellent survey paper of C.Huneke [7] for further details. Let e ( I ) denote the Hilbert-Samuel multiplicity of R with respectto I. We write e ( m ) = e ( R ) for a local ring ( R, m ) and e HK ( R ) = e HK ( n ) where n is the uniquemaximal homogeneous ideal of a graded ring R. Eto and Yoshida calculated the Hilbert-Kunz multiplicity of various blowup algebras of an idealunder certain conditions. Put c ( d ) = ( d/
2) + d/ ( d + 1)! . They proved the following.
Theorem 1.1.
Let ( R, m ) be a Noetherian local ring of prime characteristic p > with d = dim R ≥ . Then for any m -primary ideal I , we have e HK ( R ( I )) ≤ c ( d ) · e ( I ) . Primary 13A30, 13D40, 13F55.
Key words and phrases : Generalized Hilbert-Kunz function, Generalized Hilbert-Kunz multiplicity, Stanley-Reisnerring. ARINDAM BANERJEE, KRITI GOEL, AND J. K. VERMA
Moreover, equality holds if and only if e HK ( R ) = e ( I ) . When this is the case, e HK ( R ) = e ( R ) and e HK ( I ) = e ( I ) . Here e HK ( R ) = e HK ( m ) and e ( R ) = e ( m ) . It is natural to ask if there is a formula for the Hilbert-Kunz function and the Hilbert-Kunzmultiplicity of the maximal homogeneous ideal ( m , It ) of the Rees algebra R ( I ) = ⊕ ∞ n =0 I n t n where I is an m -primary ideal, in terms of invariants of the ideals m and I. In this paper we answer thisquestion for the Rees algebra of the maximal homogeneous ideal of the face ring of a simplicialcomplex. In fact, we find its generalized Hilbert-Kunz function. The generalized Hilbert-Kunzfunction was introduced by Aldo Conca in [2]. Let ( R, m ) be a d -dimensional Noetherian local(resp. standard graded) ring with maximal (resp. maximal homogeneous) ideal m and I be an m -primary (resp. a graded m -primary) ideal. Fix a set of generators of I , say I = ( a , a , . . . , a g ) . Wechoose these as homogeneous elements in case R is a graded ring. Define the s th Frobenius powerof I to be the ideal I [ s ] = ( a s , a s , . . . , a sg ) . The generalized Hilbert-Kunz function of I is definedas HK I ( s ) = ℓ ( R/I [ s ] ) . The generalized Hilbert-Kunz multiplicity is defined as lim s →∞ HK I ( s ) /s d ,whenever the limit exists.We now describe the contents of the paper. Let ∆ be a simplicial complex of dimension d − . Let k be any field, k [∆] denote the face ring of ∆ and n be its maximal homogeneous ideal. Let R ( n ) = ⊕ ∞ n =0 n n t n be the Rees algebra of n . In section 2, we collect some preliminaries requiredfor estimation of the asymptotic reduction number in terms of the a -invariants of local cohomol-ogy modules and Hilbert-Samuel polynomial of the maximal homogeneous ideal of the face ringof a simplicial complex. Section 3 is devoted to the computation of the generalized Hilbert-Kunzfunction HK ( n , n t ) ( s ), where ( n , n t ) is the maximal homogeneous ideal of the Rees algebra R ( n ) . Wealso estimate an upper bound on the postulation number of HK ( n , n t ) ( s ) in terms of a -invariants ofthe local cohomology modules. This enables us to explicitly calculate the generalized Hilbert-Kunzfunction for the Rees algebra in several examples such as the edge ideal of a complete bipartitegraph, the real projective plane and a few other examples of simplicial complexes. We have im-plemented the formula for the Hilbert-Kunz function in an algorithm written in the language ofMacaulay2. Acknowledgement:
We thank the anonymous referee for his valuable suggestions.2.
Preliminaries
In this section we gather some results which we shall use in the later sections.Let R be a ring and I be an R -ideal. Let G ( I ) = ⊕ n ≥ I n /I n +1 be the associated graded ring of I .An ideal J ⊆ I is called a reduction of I if J I n = I n +1 , for all large n. A minimal reduction of I isa reduction of I minimal with respect to inclusion. For a minimal reduction J of I , we set r J ( I ) = min { n | I m +1 = J I m for all m ≥ n } . The reduction number of I is defined as r ( I ) = min { r J ( I ) | J is a minimal reduction of I } . ENERALIZED HILBERT-KUNZ FUNCTION OF THE REES ALGEBRA 3
Let ( R, m ) be a d -dimensional local ring and I be an m -primary ideal. It is known that H I ( n ) := ℓ ( R/I n ) is a polynomial function of n of degree d , for large n . In particular, there exists a polynomial P I ( x ) ∈ Q [ x ] such that H I ( n ) = P I ( n ) for all large n . The postulation number of I is defined as n ( I ) = max { n | H I ( n ) = P I ( n ) } . Let M be a finitely generated R -module. We define a i ( M ) to be max { u ∈ Z | [ H i m ( M )] u = 0 } if H i m ( M ) = 0, and −∞ otherwise. We shall use the following results to estimate the reductionnumber of powers of an ideal. Theorem 2.1 ([9, Corollary 2.21]) . Let ( R, m ) be a d -dimensional Cohen-Macaulay local ring withinfinite residue field and I be an m -primary ideal such that grade( G ( I ) + ) ≥ d − . Then for k ≥ , r ( I k ) = (cid:22) n ( I ) k (cid:23) + d. Theorem 2.2 ([4, Theorem 2.1]) . Let ( R, m ) be a Noetherian local ring and let I ⊆ m be an R -ideal. Then r J ( I n ) is independent of J and stable if n is large. In particular, for all n > max {| a i ( G ( I )) | : a i ( G ( I )) = −∞} , we get r J ( I n ) = s if a s ( G ( I )) ≥ ,s − if a s ( G ( I )) < , where J is any minimal reduction of I n and s is the analytic spread of I. Let S be a d -dimensional Cohen-Macaulay local ring and let I be a parameter ideal. Fix s ∈ N . For a fixed set of generators of I , define functions F ( n ) := H I ( I [ s ] , n ) = ℓ S I [ s ] I [ s ] I n ! and H ( n ) := H I ( S, n ) = ℓ S (cid:18) SI n (cid:19) = e ( I ) (cid:18) n + d − d (cid:19) for all n. Note that if S is 1-dimensional, then F ( n ) = H ( n ) for all n and for all s . In [5], theauthors prove that the function F ( n ) is a piecewise polynomial in n. Theorem 2.3 ([5, Theorem 3.2]) . Let S be a d -dimensional Cohen-Macaulay local ring and I be aparameter ideal. Let d ≥ . For a fixed s ∈ N , F ( n ) = d H ( n ) if ≤ n ≤ s, d − X i =1 ( − i +1 (cid:18) di (cid:19) H ( n − ( i − s ) if s + 1 ≤ n ≤ ( d − s − ,H ( n + s ) − s d e ( I ) if n ≥ ( d − s. Let ∆ be a ( d − n ] = { , , . . . , n } . Let k bea field and S = k [ x , x , . . . , x n ] be the polynomial ring over k. For F ⊂ [ n ] , we put x F = Y i ∈ F x i . The ideal of ∆ is I ∆ = ( x F | F / ∈ ∆) S. The face ring or the Stanley-Reisner ring of ∆ is the ring k [∆] := S/I ∆ . Let n denote the unique maximal homogeneous ideal of k [∆] . The f -vector of ∆ is f (∆) = ( f − , f , . . . , f d − ) , where f − = 1 and f i is the number of i -dimensional faces of ∆, for ARINDAM BANERJEE, KRITI GOEL, AND J. K. VERMA i = 0 , , . . . , d − . The Hilbert series of k [∆] is the formal power series H ( k [∆] , t ) = ∞ X i =0 dim k k [∆] i t i where k [∆] i is the graded component of k [∆] consisting of homogeneous elements of degree i in k [∆] . Stanley showed the following:
Theorem 2.4 ([11, Theorem 1.4]) . H ( k [∆] , t ) = d − X i = − f i t i +1 (1 − t ) i +1 By taking the lcm of the denomenators we write the Hilbert series of k [∆] as the rational function H ( k [∆] , t ) = ( h + h t + · · · + h d t d ) / (1 − t ) d . The vector ( h , h , . . . , h d ) is called the h -vector of ∆.Let h ( i ) ( t ) denote the i th derivative of h ( t ) = h + h t + · · · + h d t d . Theorem 2.5 ([6, Theorem 6.2]) . The Hilbert-Samuel function ℓ ( k [∆] / n s ) , for all s ≥ , is givenby ℓ (cid:18) k [∆] n s (cid:19) = d X i =0 ( − i h ( i ) (1) i ! (cid:18) s − d − id − i (cid:19) . The following result of Conca computes the generalized Hilbert-Kunz function of k [∆]. Theorem 2.6 ([2, Remark 2.2]) . For s ≥ , the generalized Hilbert-Kunz function of k [∆] is givenby the equation ℓ (cid:18) k [∆] n [ s ] (cid:19) = d X i =0 f i − ( s − i . The generalized Hilbert-Kunz function of ( n , n t )Let S = k [ x , . . . , x r ] be a polynomial ring in r variables over a field k and let m = ( x , . . . , x r )denote the maximal homogeneous ideal of S. Let P j , for j = 1 , . . . , α and α ≥
2, be distinct S -ideals generated by subsets of { x , . . . , x r } . Let I = ∩ αj =1 P j and R = S/I.
Let n = m /I denote themaximal homogeneous ideal of R. In this section, we show that the generalized Hilbert-Kunz function of the maximal homogeneousideal ( n , n t ) of the Rees algebra R ( n ) of R is a polynomial for large s. We begin by proving thatfor s, n ∈ N , ℓ S ( S/I + m [ s ] m n ) is a piecewise polynomial in s and n. First we prove the followingresult which is a consequence of Theorem 2.3.
Corollary 3.1.
Let S = k [ x , . . . , x d ] be a polynomial ring in d variables over a field k. Let m = ( x , . . . , x d ) be its maximal homogeneous ideal. Let s, n ∈ N . (1) If d = 1 , then ℓ (cid:18) S m [ s ] m n (cid:19) = s + n for all s, n ≥ . (2) If d = 2 , then ℓ (cid:18) S m [ s ] m n (cid:19) = s + n + n if ≤ n ≤ s, (cid:18) n + s + 12 (cid:19) if n ≥ s. ENERALIZED HILBERT-KUNZ FUNCTION OF THE REES ALGEBRA 5 (3) If d ≥ , then ℓ (cid:18) S m [ s ] m n (cid:19) = s d + d (cid:18) n + d − d (cid:19) if ≤ n ≤ s,s d + d − X i =1 ( − i +1 (cid:18) di (cid:19)(cid:18) n − ( i − s + d − d (cid:19) if s + 1 ≤ n ≤ ( d − s − , (cid:18) n + s + d − d (cid:19) if n ≥ ( d − s. Proof.
Let s, n ∈ N . If d = 1, then S = k [ x ] and m = ( x ) implying that ℓ (cid:18) S m [ s ] m n (cid:19) = ℓ (cid:18) k [ x ]( x s + n ) (cid:19) = s + n. Let d ≥ . Since ℓ (cid:18) S m [ s ] m n (cid:19) = ℓ (cid:18) S m [ s ] (cid:19) + ℓ m [ s ] m [ s ] m n ! and ℓ ( S/ m [ s ] ) = s d , the result follows from Theorem 2.3. (cid:3) Let T = ⊕ n ≥ T n be a Noetherian graded ring, where T is an Artinian ring. Let M = ⊕ n ≥ M n be a finitely generated graded T -module. Then ℓ T ( M n ) < ∞ . The Hilbert series H ( M, λ ) of M isdefined by H ( M, λ ) = X n ≥ ℓ T ( M n ) λ n . Theorem 3.2.
Let T be a standard graded Artinian ring and let I , . . . , I α , for α ≥ , be homoge-neous T -ideals. Let I = ∩ αi =1 I i . Then H (cid:18) TI , λ (cid:19) = α X i =1 H (cid:18) TI i , λ (cid:19) − α X i,j =1 i Apply induction on α. Let α = 2 . Consider the following short exact sequence0 −→ TI ∩ I −→ TI M TI −→ TI + I −→ . Then H (cid:18) TI , λ (cid:19) = H (cid:18) TI ∩ I , λ (cid:19) = H (cid:18) TI , λ (cid:19) + H (cid:18) TI , λ (cid:19) − H (cid:18) TI + I , λ (cid:19) . Let α > −→ T ∩ αi =1 I i −→ T ∩ α − i =1 I i M TI α −→ T ∩ α − i =1 I i + I α −→ . ARINDAM BANERJEE, KRITI GOEL, AND J. K. VERMA Using induction hypothesis, it follows that H (cid:18) T ∩ αi =1 I i , λ (cid:19) = H (cid:18) T ∩ α − i =1 I i , λ (cid:19) + H (cid:18) TI α , λ (cid:19) − H (cid:18) T ∩ α − i =1 I i + I α , λ (cid:19) = α − X i =1 H (cid:18) TI i , λ (cid:19) − α − X i,j =1 i Corollary 3.3. Let S = k [ x , . . . , x r ] be a polynomial ring in r variables over a field k and let m = ( x , . . . , x r ) be the maximal homogeneous ideal of S. Let P , . . . , P α , for α ≥ , be distinct S -ideals generated by subsets of { x , . . . , x r } . Let I = ∩ αi =1 P i . Then for s, n ∈ N , ℓ (cid:18) SI + m [ s ] m n (cid:19) = α X i =1 ℓ (cid:18) SP i + m [ s ] m n (cid:19) − X ≤ i Since S/ m [ s ] m n is a standard graded Artinian ring, using Theorem 3.2 it follows that H (cid:18) SI + m [ s ] m n , λ (cid:19) = α X i =1 H (cid:18) SP i + m [ s ] m n , λ (cid:19) − α X i,j =1 i ENERALIZED HILBERT-KUNZ FUNCTION OF THE REES ALGEBRA 7 The generalized Hilbert-Kunz function of ( n , n t ) .Theorem 3.4. Let S = k [ x , . . . , x r ] be a polynomial ring in r variables over a field k and let m be the maximal homogeneous ideal of S. Let P , . . . , P α , for α ≥ , be distinct S -ideals generatedby subsets of { x , . . . , x r } . Let I = ∩ αi =1 P i and R = S/I. Suppose n = m /I denotes the maximalhomogeneous ideal of R and dim( R ) = d. Set δ = max {| a i ( R ) | : a i ( R ) = −∞} . Then for s > δ , ℓ (cid:18) R ( n )( n , n t ) [ s ] (cid:19) is a polynomial in s. Proof. Since R is a standard graded ring, it follows that R ≃ G ( n ) . Let s > δ. Using Theorem 2.2,it follows that r ( n s ) = d − a d ( R ) < ,d if a d ( R ) = 0 . In other words, r ( n s ) = d − j , where j is either 0 or 1 as per the above observation. As n [ s ] isa minimal reduction of n s , we get, n [ s ] n ( d − j ) s = n ( d − j +1) s . In other words, n [ s ] n n − s = n n , for all n ≥ ( d − j + 1) s. This implies that( n , n t ) [ s ] = ( n [ s ] , n [ s ] t s ) = s − M n =0 n [ s ] n n t n ! + M n ≥ s n [ s ] n n − s t n = s − M n =0 n [ s ] n n t n ! + ( d − j +1) s − M n = s n [ s ] n n − s t n + M n ≥ ( d − j +1) s n n t n . Therefore, for s > δ , ℓ (cid:18) R ( n )( n , n t ) [ s ] (cid:19) = s − X n =0 ℓ (cid:18) n n n [ s ] n n (cid:19) + ( d − j +1) s − X n = s ℓ (cid:18) n n n [ s ] n n − s (cid:19) = s − X n =0 ℓ (cid:18) R n [ s ] n n (cid:19) + ( d − j +1) s − X n = s ℓ (cid:18) R n [ s ] n n − s (cid:19) − ( d − j +1) s − X n =0 ℓ (cid:18) R n n (cid:19) = s − X n =1 ℓ (cid:18) SI + m [ s ] m n (cid:19) + ( d − j +1) s − X n = s +1 ℓ (cid:18) SI + m [ s ] m n − s (cid:19) − ( d − j +1) s − X n =1 ℓ (cid:18) R n n (cid:19) + 2 ℓ (cid:18) R n [ s ] (cid:19) = 2 s − X n =1 ℓ (cid:18) SI + m [ s ] m n (cid:19) + ( d − j ) s − X n = s ℓ (cid:18) SI + m [ s ] m n (cid:19) − ( d − j +1) s − X n =1 ℓ (cid:18) R n n (cid:19) + 2 ℓ (cid:18) R n [ s ] (cid:19) . The result now follows from Corollary 3.3, Theorem 2.5 and Theorem 2.6. (cid:3) The generalized Hilbert-Kunz function of ( n , n t ) for Cohen-Macaulay k [∆] .Theorem 3.5. Let S = k [ x , . . . , x r ] be a polynomial ring in r variables over a field k and let m be the maximal homogeneous ideal of S. Let P , . . . , P α , for α ≥ , be distinct S -ideals generated ARINDAM BANERJEE, KRITI GOEL, AND J. K. VERMA by subsets of { x , . . . , x r } . Let I = ∩ αi =1 P i and R = S/I. Suppose n = m /I denotes the maximalhomogeneous ideal of R and dim( R ) = d. Suppose that R is Cohen-Macaulay. Then ℓ (cid:18) R ( n )( n , n t ) [ s ] (cid:19) is given by a polynomial for s ≥ . Proof. Since R is a standard graded ring, it follows that R ≃ G ( n ) . Let h (∆) = ( h , . . . , h d ) denotethe h -vector of R. Note that − d < n ( n ) ≤ . If n ( n ) = − d , then h = 1 and h i = 0 for all i = 0,implying that 0 = h = r − d. It follows that I is a height zero ideal, which is not true. Hence, − d < n ( n ) ≤ . Suppose n ( n ) = 0 . Using Theorem 2.1, it follows that r ( I s ) = d , for all s ≥ . Using the samearguments as in the proof of Theorem 3.4, it follows that for s ≥ ,ℓ (cid:18) R ( n )( n , n t ) [ s ] (cid:19) = 2 s − X n =1 ℓ (cid:18) SI + m [ s ] m n (cid:19) + ds − X n = s ℓ (cid:18) SI + m [ s ] m n (cid:19) − ( d +1) s − X n =1 ℓ (cid:18) R n n (cid:19) + 2 ℓ (cid:18) R n [ s ] (cid:19) . The result now follows from Corollary 3.3, Theorem 2.5 and Theorem 2.6.Now suppose that n ( n ) < . If s < | n ( n ) | , write | n ( n ) | = k s + k , where k ∈ { , , . . . , s − } . Using Theorem 2.1, it follows that r ( n s ) = d − k if s < | n ( n ) | , k = 0 ,d − k − s < | n ( n ) | , k = 0 ,d − s ≥ | n ( n ) | . In other words, r ( n s ) = d − j, where j ∈ { , k , k + 1 } as per the above observation. Using thesame arguments as in the proof of Theorem 3.4, we are done. (cid:3) Examples In this section, we illustrate the above results using some examples. Example 4.1. Let ∆ be the simplicial complex x x x x Then R = k [ x , x , x , x ] / (( x , x ) ∩ ( x , x )) is the face ring of ∆ . Observe that R is a 2-dimensionalring with f -vector f (∆) = (1 , , 2) and h -vector h (∆) = (1 , , − . Set S = k [ x , x , x , x ] , P =( x , x ), P = ( x , x ) . Since depth( R ) = 1, it follows that a ( R ) = −∞ . In order to find a ( R ) and a ( R ), we consider the following short exact sequence.0 −→ SP ∩ P −→ SP M SP −→ SP + P −→ n ( R ) ≃ H m ( S/ ( P + P )) and H n ( R ) ≃ H x ,x ) ( k [ x , x ]) ⊕ H x ,x ) ( k [ x , x ]) . ENERALIZED HILBERT-KUNZ FUNCTION OF THE REES ALGEBRA 9 This implies that a ( R ) = 0 and a ( R ) = − . Hence, δ = max {| a i ( R ) | : a i ( R ) = −∞} = 2 . Since a ( R ) < 0, using Theorem 3.4 it follows that for all s > ,ℓ (cid:18) R ( n )( n , n t ) [ s ] (cid:19) = 2 s − X n =1 ℓ (cid:18) SI + m [ s ] m n (cid:19) − s − X n =1 ℓ (cid:18) R n n (cid:19) + 2 ℓ (cid:18) R n [ s ] (cid:19) . From Corollary 3.3, Theorem 2.5 and Theorem 2.6, we obtain ℓ (cid:18) R ( n )( n , n t ) [ s ] (cid:19) = 2 s − X n =1 (cid:20) ℓ (cid:18) SP + m [ s ] m n (cid:19) + ℓ (cid:18) SP + m [ s ] m n (cid:19) − ℓ (cid:18) SP + P + m [ s ] m n (cid:19)(cid:21) − s − X i =1 " X i =0 ( − i h ( i ) (1) i ! (cid:18) n + 1 − i − i (cid:19) + 2 X i =0 f i − ( s − i . Substituting the values and using Corollary 3.1, we get ℓ (cid:18) R ( n )( n , n t ) [ s ] (cid:19) = 2 s − X n =1 (cid:20) s + n + n ) − (cid:21) − s − X n =1 (cid:20) (cid:18) n + 12 (cid:19) − (cid:21) + 2 (cid:20) s − 1) + 2( s − (cid:21) . Simplifying the above expression, we obtain that for all s > ℓ (cid:18) R ( n )( n , n t ) [ s ] (cid:19) = 83 s − s − 1= 16 (cid:18) s + 23 (cid:19) − (cid:18) s + 12 (cid:19) + 2 s − . Example 4.2. Let ∆ be the simplicial complex x x x x Then R = k [ x , x , x , x ] / (( x ) ∩ ( x , x )) is the face ring of ∆ . Observe that R is a 3-dimensionalring with f -vector f (∆) = (1 , , , 1) and h -vector h (∆) = (1 , , − , . Set S = k [ x , x , x , x ] ,P = ( x , x ), P = ( x ) . Since depth( R ) = 2, it follows that a ( R ) = a ( R ) = −∞ . In order tofind a ( R ) and a ( R ), we consider the following short exact sequence.0 −→ SP ∩ P −→ SP M SP −→ SP + P −→ n ( R ) ≃ H x ,x ,x ) ( k [ x , x , x ]) and 0 → H x ) ( k [ x ]) → H n ( R ) → H x ,x ) ( k [ x , x ]) → . This implies that a ( R ) = − a ( R ) = − . Hence, δ = max {| a i ( R ) | : a i ( R ) = −∞} = 3 . Since a ( R ) < 0, using Theorem 3.4 it follows that for all s > ,ℓ (cid:18) R ( n )( n , n t ) [ s ] (cid:19) = 2 s − X n =1 ℓ (cid:18) SI + m [ s ] m n (cid:19) + s − X n = s ℓ (cid:18) SI + m [ s ] m n (cid:19) − s − X n =1 ℓ (cid:18) R n n (cid:19) + 2 ℓ (cid:18) R n [ s ] (cid:19) . From Corollary 3.3, Theorem 2.5 and Theorem 2.6, we obtain ℓ (cid:18) R ( n )( n , n t ) [ s ] (cid:19) = 2 s − X n =1 (cid:20) ℓ (cid:18) SP + m [ s ] m n (cid:19) + ℓ (cid:18) SP + m [ s ] m n (cid:19) − ℓ (cid:18) SP + P + m [ s ] m n (cid:19)(cid:21) + s − X n = s (cid:20) ℓ (cid:18) SP + m [ s ] m n (cid:19) + ℓ (cid:18) SP + m [ s ] m n (cid:19) − ℓ (cid:18) SP + P + m [ s ] m n (cid:19)(cid:21) − s − X i =1 " X i =0 ( − i h ( i ) (1) i ! (cid:18) n + d − i − d − i (cid:19) + 2 X i =0 f i − ( s − i . Substituting the values and using Corollary 3.1, we get ℓ (cid:18) R ( n )( n , n t ) [ s ] (cid:19) = 2 s − X n =1 (cid:20) ( s + n + n ) + s + 3 (cid:18) n + 23 (cid:19) − ( s + n ) (cid:21) + s − X n = s (cid:20)(cid:18) n + s + 12 (cid:19) + s + 3 (cid:18) n + 23 (cid:19) − (cid:18) n − s + 23 (cid:19) − ( s + n ) (cid:21) − s − X n =1 (cid:20)(cid:18) n + 23 (cid:19) + (cid:18) n + 12 (cid:19) − n (cid:21) + 2 (cid:20) s − 1) + 4( s − + ( s − (cid:21) . Simplifying the above expression, we obtain that for all s > ℓ (cid:18) R ( n )( n , n t ) [ s ] (cid:19) = 138 s + 1312 s − s − s = 39 (cid:18) s + 34 (cid:19) − (cid:18) s + 23 (cid:19) + 14 (cid:18) s + 12 (cid:19) . Example 4.3. Let ∆ be the simplicial complex x x x x Then R = k [ x , x , x , x ] / (( x , x ) ∩ ( x , x ) ∩ ( x , x ) ∩ ( x , x )) is the face ring of ∆ . Observe that R is a 2-dimensional Cohen-Macaulay ring with f -vector f (∆) = (1 , , 4) and h -vector h (∆) = (1 , , . This implies that n ( n ) = 0 . Using Theorem 3.5, it follows that for s ≥ ,ℓ (cid:18) R ( n )( n , n t ) [ s ] (cid:19) = 2 s − X n =1 ℓ (cid:18) SI + m [ s ] m n (cid:19) + s − X n = s ℓ (cid:18) SI + m [ s ] m n (cid:19) − s − X n =1 ℓ (cid:18) R n n (cid:19) + 2 ℓ (cid:18) R n [ s ] (cid:19) . Substituting, we get ℓ (cid:18) R ( n )( n , n t ) [ s ] (cid:19) = 2 s − X n =1 (cid:20) s + n + n ) − s + n ) + 1 (cid:21) + s − X n = s (cid:20) (cid:18) n + s + 12 (cid:19) − s + n ) + 1 (cid:21) − s − X n =1 (cid:20) (cid:18) n + 12 (cid:19) − n + 1 (cid:21) + 2 (cid:20) s − 1) + 4( s − (cid:21) . ENERALIZED HILBERT-KUNZ FUNCTION OF THE REES ALGEBRA 11 Simplifying the above expression, we obtain that for all s ≥ ℓ (cid:18) R ( n )( n , n t ) [ s ] (cid:19) = 163 s − s − s + 1= 32 (cid:18) s + 23 (cid:19) − (cid:18) s + 12 (cid:19) + 8 s + 1 . Example 4.4. Let ∆ be a 1-dimensional simplicial complex on r vertices, for some r ≥ x x x x r − x r For i = 1 , . . . , r − 1, set P i = (cid:0) { x , . . . , x r } \ { x i , x i +1 } (cid:1) . Then R = k [ x , . . . , x r ] / ∩ r − i =1 P i is theface ring of ∆ . It is a two-dimensional Cohen-Macaulay ring with f -vector f (∆) = (1 , r, r − 1) and h -vector h (∆) = (1 , r − , . Since the a -invariant a ( R ) = − 1, using Theorem 3.5, it follows thatfor s ≥ ,ℓ (cid:18) R ( n )( n , n t ) [ s ] (cid:19) = 2 s − X n =1 ℓ (cid:18) SI + m [ s ] m n (cid:19) − s − X n =1 ℓ (cid:18) R n n (cid:19) + 2 ℓ (cid:18) R n [ s ] (cid:19) = 2 s − X n =1 r − X i =1 ℓ (cid:18) SP i + m [ s ] m n (cid:19) − r − X i,j =1 i 1, if { x i , x i +1 } ∩ { x j , x j +1 } 6 = ∅ ,then S/ ( P i + P j ) ≃ k [ x ] and there are r − S/ ( P i + P j ) ≃ k. It is alsoeasy to observe that S/ ( P i + · · · + P i u ) ≃ k, for all u ≥ i , . . . , i u ∈ { , . . . , r − } . Therefore, ℓ (cid:18) R ( n )( n , n t ) [ s ] (cid:19) = 2 s − X n =1 (cid:20) ( r − s + n + n ) − h ( r − s + n ) + (cid:18) r − (cid:19) − ( r − i + (cid:18) r − (cid:19) + · · · + ( − r − (cid:21) − s − X n =1 (cid:20) ( r − (cid:18) n + 12 (cid:19) − ( r − n (cid:21) + 2 (cid:2) r ( s − 1) + ( r − s − (cid:3) . Since, r − X i =2 ( − i (cid:18) r − i (cid:19) = r − , simplifying the above expression we get ℓ (cid:18) R ( n )( n , n t ) [ s ] (cid:19) = 43 ( r − s − ( r − s − ( r − s = 8( r − (cid:18) s + 23 (cid:19) − r − (cid:18) s + 12 (cid:19) + (2 r − s. We need some terminologies for the next example. Definition 4.5. Let G be a finite simple graph with vertices V = V ( G ) = { x , . . . , x n } and theedges E = E ( G ) . The edge ideal of I ( G ) of G is defined to be the ideal in K [ x , . . . , x n ] generatedby the square free quadratic monomials representing the edges of G , i.e., I ( G ) = h x i x j | x i x j ∈ E i . A vertex cover of a graph is a set of vertices such that every edge has at least one vertex belongingto that set. A minimal vertex cover is a vertex cover such that none of its subsets is a vertex cover.For any graph G with the set of all minimal vertex covers C , the edge ideal I ( G ) has the primarydecomposition: I ( G ) = \ { x i ,...,x iu }∈ C ( x i , . . . , x i u ) . For example, when G is a five cycle, the primary decomposition of the edge ideal I ( G ) is I ( G ) = ( x x , x x , x x , x x , x x ) = ( x , x , x ) ∩ ( x , x , x ) ∩ ( x , x , x ) ∩ ( x , x , x ) ∩ ( x , x , x ) . Example 4.6 ( Complete Bipartite Graphs). A complete bipartite graph K α,β is a graph whoseset of vertices is decomposed into two disjoint sets such that no two vertices within the same setare adjacent and that every pair of vertices in the two sets are adjacent. x x x α y y y β Figure 1. K α,β Let S = k [ x , . . . , x α , y , . . . , y β ] , where 3 ≤ α ≤ β. The edge ideal of K α,β is the ideal I = (cid:0) x i y j | ≤ i ≤ α, ≤ j ≤ β (cid:1) . Observe that R = S/I is a β -dimensional ring. Let P = ( x , . . . , x α ), P = ( y , . . . , y β ) . Then I = P ∩ P . Note that I is the Stanley-Reisner ideal of the union of an α -simplex and a β -simplex.In order to find the a -invariants we consider the following short exact sequence.0 −→ SP ∩ P −→ SP M SP −→ SP + P −→ ENERALIZED HILBERT-KUNZ FUNCTION OF THE REES ALGEBRA 13 Using the corresponding long exact sequence of local cohomology modules, it follows thatH n ( R ) ≃ H m (cid:18) SP + P (cid:19) , H α n ( R ) ≃ H α ( x ,...x α ) ( k [ x , . . . , x α ]) and H β n ( R ) ≃ H β ( y ,...,y β ) ( k [ y , . . . , y β ]) . Therefore, a ( R ) = 0, a α ( R ) = − α and a β ( R ) = − β. Hence, δ = max {| a i ( R ) | : a i ( R ) = −∞} = β. Since a β ( R ) < 0, using Theorem 3.4 it follows that for all s > β,ℓ (cid:18) R ( n )( n , n t ) [ s ] (cid:19) = 2 s − X n =1 ℓ (cid:18) SI + m [ s ] m n (cid:19) + ( β − s − X n = s ℓ (cid:18) SI + m [ s ] m n (cid:19) − βs − X n =1 ℓ (cid:18) R n n (cid:19) + 2 ℓ (cid:18) R n [ s ] (cid:19) . From Corollary 3.3, Theorem 2.5 and Theorem 2.6, we obtain ℓ (cid:18) R ( n )( n , n t ) [ s ] (cid:19) = 2 s − X n =1 (cid:20) ℓ (cid:18) SP + m [ s ] m n (cid:19) + ℓ (cid:18) SP + m [ s ] m n (cid:19) − ℓ (cid:18) SP + P + m [ s ] m n (cid:19)(cid:21) + ( β − s − X n = s (cid:20) ℓ (cid:18) SP + m [ s ] m n (cid:19) + ℓ (cid:18) SP + m [ s ] m n (cid:19) − ℓ (cid:18) SP + P + m [ s ] m n (cid:19)(cid:21) − βs − X n =1 " β X i =0 ( − i h ( i ) (1) i ! (cid:18) n + β − i − β − i (cid:19) + 2 β X i =0 f i − ( s − i . As the f -vector is f (∆) = (cid:18) , α + β, (cid:18) α (cid:19) + (cid:18) β (cid:19) , . . . , (cid:18) αα (cid:19) + (cid:18) βα (cid:19) , (cid:18) βα + 1 (cid:19) , . . . , (cid:18) ββ (cid:19)(cid:19) and the h -vector can be computed using [1, Lemma 5.1.8], substituting the values and using Corollary 3.1,it follows that for all s > β,ℓ (cid:18) R ( n )( n , n t ) [ s ] (cid:19) = 2 s − X n =1 (cid:20) s β + β (cid:18) n + β − β (cid:19) + s α + α (cid:18) n + α − α (cid:19) − (cid:21) + ( β − s − X n = s (cid:20) s β + β − X i =1 ( − i +1 (cid:18) βi (cid:19)(cid:18) n − ( i − s + β − β (cid:19) − (cid:21) + ( α − s − X n = s (cid:20) s α + α − X i =1 ( − i +1 (cid:18) αi (cid:19)(cid:18) n − ( i − s + α − α (cid:19)(cid:21) + ( β − s − X n =( α − s (cid:18) n + s + α − α (cid:19) − βs − X n =1 (cid:20) β X i =0 ( − i h ( i ) (1) i ! (cid:18) n + β − i − β − i (cid:19)(cid:21) + 2 (cid:20) β X i =1 (cid:18) βi (cid:19) ( s − i + α X i =1 (cid:18) αi (cid:19) ( s − i (cid:21) . In particular, when α = 3 and β = 4 , we obtain that for all s > ,ℓ (cid:18) R ( n )( n , n t ) [ s ] (cid:19) = 6130 s + 1924 s − s − s − s − . Sometimes, certain invariants of the Stanley-Reisner ring may depend on the characteristic of thering. Triangulation of the real projective plane is one such example where the Cohen-Macaulayproperty of the ring is characteristic dependent. We prove that in this example, the Hilbert-Kunzfunction is characteristic independent. ba e cd fc ab Example 4.7 ( Triangulation of real projective plane). Let ∆ be the triangulation of the realprojective plane.Let k be a field and R be the corresponding Stanley-Reisner ring of ∆ . It is known that R is Cohen-Macaulay if and only if char k = 2 . The f -vector of R is f (∆) = (1 , , , 10) and h -vector of R is h (∆) = (1 , , , . Let char k = 2 . Then R is Cohen-Macaulay and n ( n ) = − . Using Macaulay2 code, we obtain that for s ≥ ℓ (cid:18) R ( n )( n , n t ) [ s ] (cid:19) = 390 (cid:18) s + 34 (cid:19) − (cid:18) s + 23 (cid:19) + 372 (cid:18) s + 12 (cid:19) − s. (4.1)We save the code in a file named as HKPolySC.m2 and make the following session in Macaulay2. i1 : S = QQ[a..f];i2 : I = ideal"abe, ade, acd, bcd, bdf, abf, acf, cef, bce, def";i3 : needsPackage"Depth"i4 : needsPackage"SimplicialComplexes"i5 : needsPackage"SimplicialDecomposability"i6 : load"HKPolySC.m2"i7 : HKPolySC(I)The postulation number is: -1Enter a number bigger than or equal to the absolute value of the postulation number: 2The value of the Hilbert-Kunz function at the point 2 is: 104Do you wish to enter one more point? (true/false): trueEnter a number bigger than or equal to the absolute value of the postulation number: 3The value of the Hilbert-Kunz function at the point 3 is: 759Do you wish to enter one more point? (true/false): trueEnter a number bigger than or equal to the absolute value of the postulation number: 4The value of the Hilbert-Kunz function at the point 4 is: 2806Do you wish to enter one more point? (true/false): trueEnter a number bigger than or equal to the absolute value of the postulation number: 5The value of the Hilbert-Kunz function at the point 5 is: 7475Do you wish to enter one more point? (true/false): trueEnter a number bigger than or equal to the absolute value of the postulation number: 6The value of the Hilbert-Kunz function at the point 6 is: 16386Do you wish to enter one more point? (true/false): false ENERALIZED HILBERT-KUNZ FUNCTION OF THE REES ALGEBRA 15 One may check that if char k = 2, then the a -invariant of R is negative and depth( R ) = 2 . UsingTheorem 3.4 it follows that ℓ ( R ( n ) / ( n , n t ) [ s ] ) has the same formula as in (4.1) for s > δ , where δ = max {| a ( R ) | , | a ( R ) |} . This proves that the Hilbert-Kunz function is characteristic independentin this example.5. Macaulay2 code for Cohen-Macaulay Stanley-Reisner rings In this section we present a Macaulay2 code which uses the idea of Theorem 3.5 to calculate thevalue of the generalized Hilbert-Kunz function at a point. The code requires Macaulay2 packagesSimplicialComplexes, SimplicialDecomposability and Depth. The code accepts the Stanley-Reisnerideal as an input. It then calculates the postulation number, after ensuring that the correspondingring is Cohen-Macaulay, and prompts the user to enter a point according to the postulation numbercalculated. The value of the generalized Hilbert-Kunz function at the point is produced as an outputand the user is given a choice to enter more points. HKPolySC = (SCIdeal) -> (polyRing := ring SCIdeal; Step 1 : Check if the Stanley-Reisner ring is Cohen-Macaulay if isCM(polyRing/SCIdeal) == false then error "Stanley-Reisner ring is not Cohen-Macaulay";dimSC := dim (polyRing/SCIdeal);SComplex := simplicialComplex monomialIdeal SCIdeal;fvect := fVector(SComplex);hvect := hVector(SComplex); Step 2 : Calculate the derivatives of the polynomial corresponding to the h -vector at 1 Diffh = (i) -> (TT := QQ[tt];hPoly := sum(0..dimSC, j -> (hvect Find the list of minimal primes MinPrimeList := primaryDecomposition SCIdeal;numPrime := Ring required for the output polynomial OutputRing = QQ[s]; Redefining the binomial function binom = (aa, bb) -> (if aa > 0 then return binomial(aa,bb)else if (aa == 0 and bb == 0) then return 1else return 0); Step 3 : Calculate and print the postulation number PostNum := -position(toList apply(0..dimSC, i-> dimSC-i), i-> hvect Step 4 : Obtain the point from the user as an input and calculate the Hilbert-Kunz polynomial atthat point pointer := true;while pointer == true do(point = read "Enter a number bigger than or equal to the absolute value of thepostulation number: ";point = value point; Step 5 : The function FunctionF calculates length as in Corollary 3.1. FunctionF = (QtI, n) -> (dimQt = dim (polyRing/QtI);use OutputRing;if dimQt == 0 then return 1else if dimQt == 1 then return point + nelse if dimQt == 2 then (if n <= point then return point^2 + n^2 + n ENERALIZED HILBERT-KUNZ FUNCTION OF THE REES ALGEBRA 17 else return (n + point + 1)*(n + point)/2)else (if n <= point then return point^dimQt + dimQt*binom(n+dimQt-1,dimQt)else if (point+1 <= n and n <= (dimQt-1)*point-1) thenreturn point^dimQt + sum(1..(dimQt-1), i ->((-1)^(i+1))*binom(dimQt,i)*binom(n-(i-1)*point+dimQt-1,dimQt))else return binom(n+point+dimQt-1,dimQt))); Step 6 : The function AltSumLength calculates length as in Corollary 3.3 AltSumLength = (n) -> (polySum = 0;for i from 1 to numPrime do (CL := subsets(numPrime, i);midSum = 0;for j from 0 to Step 7 : Calculate the Hilbert-Kunz polynomial at the point use OutputRing;if PostNum == 0 thenpolyPoint = 2*sum(1..(point-1), n -> AltSumLength(n))+ sum(point..(dimSC*point-1), n -> AltSumLength(n))- sum(1..((dimSC+1)*point-1), n ->sum(0..dimSC,i->(-1)^i*Diffh(i)*(1/i!)*binom(n+dimSC-i-1,dimSC-i)))+ 2*sum(0..dimSC, n -> (fvect polyPoint3 = sum(1..(dimSC*point-1), n ->sum(0..dimSC, i -> (-1)^i*Diffh(i)*(1/i!)*binom(n+dimSC-i-1,dimSC-i)));polyPoint4 = 2*sum(0..dimSC, n -> (fvect If the a -invariant of the ring is known, the above code can also be used for the non Cohen-Macaulaycase with minor modifications. References 1. Winfried Bruns and J¨urgen Herzog. Cohen-Macaulay rings , volume 39 of Cambridge Studiesin Advanced Mathematics . Cambridge University Press, Cambridge, 1993. 132. Aldo Conca. Hilbert-Kunz function of monomial ideals and binomial hypersurfaces. Manuscripta Math. , 90(3):287–300, 1996. 2, 43. Kazufumi Eto and Ken-ichi Yoshida. Notes on Hilbert-Kunz multiplicity of Rees algebras. Comm. Algebra , 31(12):5943–5976, 2003. 14. Le Tuan Hoa. Reduction numbers and Rees algebras of powers of an ideal. Proc. Amer. Math.Soc. , 119(2):415–422, 1993. 35. Kriti Goel, Mitra Koley, and J. K. Verma. Hilbert-Kunz function and Hilbert-Kunz multiplicityof some ideals of the Rees algebra. arXiv preprint arXiv:1911.03889 . 1, 36. Kriti Goel, Vivek Mukundan and J. K. Verma. Tight closure of powers of ideals and tightHilbert polynomials. Mathematical Proceedings of the Cambridge Philosophical Society , 1-21. 47. Craig Huneke. Hilbert-Kunz multiplicity and the F-signature. Commutative algebra , 485–525,Springer, New York, 2013. 18. Ernst Kunz. On Noetherian rings of characteristic p . Amer. J. Math. , 98(4):999–1013, 1976. 19. Thomas John Marley. Hilbert functions of ideals in Cohen-Macaulay rings . ProQuest LLC,Ann Arbor, MI, 1989. Thesis (Ph.D.)–Purdue University. 310. Paul Monsky. The Hilbert-Kunz function. Math. Ann. , 263(1):43–49, 1983. 111. Richard P. Stanley. Combinatorics and Commutative Algebra. Second edition, Progress inMathematics Birkh¨auser 412. Richard P. Stanley. Enumerative combinatorics. Vol. 1 , volume 49 of Cambridge Studies inAdvanced Mathematics . Cambridge University Press, Cambridge, 1997. With a foreword byGian-Carlo Rota, Corrected reprint of the 1986 original. ENERALIZED HILBERT-KUNZ FUNCTION OF THE REES ALGEBRA 19 Ramakrishna Mission Vivekananda Educational and Research Institute, Belur, India Email address : Indian Institute of Technology Bombay, Mumbai, India 400076 Email address : [email protected] Indian Institute of Technology Bombay, Mumbai, India 400076 Email address ::