Hilbert-Kunz density function for graded domains
aa r X i v : . [ m a t h . A C ] M a r HILBERT-KUNZ DENSITY FUNCTION FOR GRADED DOMAINS
VIJAYLAXMI TRIVEDI AND KEI-ICHI WATANABE
Abstract.
We prove the existence of HK density function for a pair (
R, I ), where R is a N -graded domain of finite type over a perfect field and I ⊂ R is a graded idealof finite colength. This generalizes our earlier result where one proves the existenceof such a function for a pair ( R, I ), where, in addition R is standard graded.As one of the consequences we show that if G is a finite group scheme acting linearlyon a polynomial ring R of dimension d then the HK density function f R G , m G , of thepair ( R G , m G ), is a piecewise polynomial function of degree d − R G , m G ), where G ⊂ SL ( k ) is afinite group acting linearly on the ring k [ X, Y ]. Introduction
In this paper a pair (
R, I ) is a graded pair if R is an N -graded domain of dimension d ≥ k of characteristic p >
0, and I is a gradedideal of finite colength. The main result here is to prove the existence of the HK densityfunction for such a pair.The notion of HK density function was introduced in [T] for the purpose of studyingthe Hilbert-Kunz multiplicity (or HK multiplicity) e HK ( R, I ). Recall that the notionof HK multiplicity e HK ( R, I ) was introduced by P. Monsky [M] for an arbitrary Noe-therian ring R (in characteristic p >
0) and an ideal I ⊂ R of finite colength. In thesame paper he showed that it is positive real number given by e HK ( R, I ) = lim n →∞ ℓ ( R/I [ q ] ) q d . The HK density function behaves well (when it exists) for various operations liketensor products, Segre products etc. Moreover it is a limit of a uniformly convergingsequence (which could be suitably renormalized to study a given specific property).In [T], we proved the existence of HK density function for a standard graded pair ( R, I ), where by a standard graded pair we mean a graded pair, where R is a standardgraded ring (that is, R is generated by R as a k -algebra) in addition. Theorem 1.1 [T] . Let ( R, I ) be a standard graded pair. Then for a finitely generatedgraded module M over R there is a sequence { g n ( M R , I ) : [0 , ∞ ) −→ [0 , ∞ ) } n ofcompactly supported continuous and piecewise linear functions such that (1) the sequence { g n ( M R , I ) } n is uniformly convergent. Moreover (2) the HK density function f M R ,I : [0 , ∞ ] −→ [0 , ∞ ) defined as f M R ,I ( x ) =lim n →∞ g n ( M R , I )( x ) is a compactly supported continuous function, and e HK ( M, I ) = Z ∞ f M R ,I ( x ) dx. Here, for a finitely generated graded R -module M , { g n ( M R , I ) : [0 , ∞ ) −→ [0 , ∞ ) } n denotes the sequence of functions given as follows: For x ≥
0, if x = (1 − t ) ⌊ xq ⌋ q + ( t ) ⌊ xq +1 ⌋ q , for some t ∈ [0 ,
1) then we define g n ( M R , I )( x ) = 1 q d − (cid:16) (1 − t ) ℓ ( M/I [ q ] M ) ⌊ xq ⌋ + ( t ) ℓ ( M/I [ q ] M ) ⌊ xq +1 ⌋ (cid:17) . In this paper we generalize the above result to the case of graded pair ( R, I ), where R need not be standard graded. (There are many interesting N -graded rings whichare not standard graded, for examples the ring of invariants and the positive affinesemigroup rings, in particular affine toric rings).To do this we need to generalize the notion of g n ( M R , I ) (see Definition 2.2) whichcoincides with the above notion of g n ( M R , I ) whenever gcd { n | R n = 0 } = 1.More precisely we prove the following Theorem 1.1. (Main Theorem) . If M is a finitely generated graded R -module, where ( R, I ) is a graded pair then there is a sequence { g n ( M R , I ) : [0 , ∞ ) −→ [0 , ∞ ) } n ofcompactly supported continuous and piecewise linear functions such that (1) { g n ( M R , I ) } n ∈ N is a uniformly convergent sequence of compactly supported func-tions. (2) If f M R ,I : [0 , ∞ ) −→ [0 , ∞ ) given by x → lim n →∞ g n ( M R , I )( x ) then f M R ,I is acompactly supported continuous function such that ( a ) f M R ,I = (rank M ) f R,I and ( b ) e HK ( M, I ) = Z ∞ f M R ,I ( x ) dx. We recall some key aspects of the proof in the situation of standard graded pair.If R is a standard graded domain (which need not be normal) as in [T] with I generated by homogeneous generators f , . . . , f s of degrees d , . . . , d s then there existsa very ample invertible sheaf O X ( D ) on X (associated to a Cartier divisor D ) suchthat there is a graded inclusion R −→ ⊕ m ≥ H ( X, O X ( mD )) which is an isomorphismin all graded degrees m >>
0. This gives us a short exact sequence of O X -modules(1.1) 0 −→ V −→ ⊕ i O X ((1 − d i ) D ) φ −→ O X ( D ) −→ ,φ ( P i a i ) = P i a i f i . Since O X ( D ) (in fact every O X ( mD )) is invertible the sequence(1.1) is locally split exact and hence taking its Frobenius pull backs ( F : X −→ X isthe Frobenius map induced by the map O X −→ O X given by x → x p ) gives the exactsequence (here q = p n )(1.2) 0 −→ F n ∗ V −→ ⊕ i O X (( q − qd i ) D ) φ ,q −→ O X ( qD ) −→ . Since O X ( mD ) ⊗ O X ( nD ) ≃ O X (( m + n ) D ) tensoring (1.2) by O X ( mD ) we get theexact sequence(1.3) 0 −→ F n ∗ V ⊗ O X ( mD ) −→ ⊕ i O X (( m + q − qd i ) D ) φ m,q −→ O X (( m + q ) D ) −→ . Now, for every x ≥ ⌊ xq ⌋ = m + q , for some integer m . Hence we may define stepfunctions(1.4) f n ( R, I )( x ) := f n ( R, I )( m + qq ) = 1 q d − ℓ ( R/I [ q ] ) m + q = 1 q d − (cid:2) h ( X, O X ( m + q ) D ) − ⊕ i h ( X, O X ( m + q − qd i ) D ) + h ( X, F n ∗ V ⊗ O X ( mD )) (cid:3) . ILBERT-KUNZ DENSITY FUNCTION FOR GRADED DOMAINS 3
The sequence g n ( R, I ) is obtained from f n ( R, I ) in an obvious way.In particular the computations depend on the cohomologies of the Frobenius pull-backs of the locally free sheaves V and O X ( D ) and their twists (by the line bundles O X ( mD )).On the other hand if R = ⊕ m ≥ R m is an arbitrary normal graded domain then by thetheorem of Demazure (see Theorem 3.1 below), there is a Q -divisor D such that R m = H ( X, O X ( mD )), for all m . But O X ( D ) need not be invertible and the multiplicationmap O X ( mD ) ⊗ O X ( nD ) −→ O X (( m + n ) D ) need not be an isomorphism, in general.In particular the sequence (1.1) need not be locally split exact and V may not be locallyfree. Hence a version of (1.3) cannot be derived from a single sequence like (1.2), andtherefore Ker φ m,q does not come from ‘twists of’ a single sheaf (unlike in the standardgraded situation, where Ker φ m,q = F n ∗ V ⊗ O X ( mD ), for all m and q ).However O X ( mD ), associated to such a Q -divisor, does have some special propertieswhich we exploit, for example O X ( mD ) is a reflexive sheaf of O X -modules, henceinvertible outside the singular locus of X . As a result though one does not have adirect relation between the sequences (1.3) (as m and q vary), we are able to relatetheir cohomologies by estimates | h ( X, Ker φ mp + n ,qp ) − p d − h ( X, Ker φ m,q ) | = O ( m + q ) d − , for 0 ≤ n < p. In particular the fact (Theorem 3.1) that each R m is the space of sections of thedivisor mD allows us to give a simpler proof (than in [T]) for this more general setting(a graded pair).However in [T] we prove the existence of the HK density function f M R ,I directly(and without the assumption that R is a domain). Here we prove the Main Theoremwhen R is a domain, and the proof is in three steps: We prove the theorem when( M R , I ) = ( R, I ) and where R is a normal domain such that gcd { m > | R m = 0 } = 1.This is the main part. Then we extend the result for the pair ( R, I ), where R is a generalgraded domain. Then we further extend this to graded modules over such pairs.We can extend the result to the case, when R may not be a domain, by defining f R,I := X p ∈∧ λ ( M P ) f R/P, ( I + P ) /P , where Λ = { p ∈ Spec R | dim R = dim R/P } . This is clearly an additive function andhence can be extended canonically to the notion of f M R ,I . In particular R f M R ,I ( x ) dx = e HK ( M R , I ) as e HK ( − ) is an additive function.However, if gcd { m | R m = 0 } = n , say, the equality f R,I ( x ) = lim n →∞ /q d − ℓ ( R/I [ q ] ) ⌊ xq ⌋ n may not hold any longer unless gcd { m | ( R/P ) m = 0 } = n , for all P ∈ Λ.As a consequence of our Main Theorem (Theorem 1.1) we get the following
Corollary 1.2.
Let ( S, I ) be a graded pair of dimension d > . Suppose there isa graded ring R with a degree preserving map S ⊂ R such that R is S -finite, andproj dim R ( R/IR ) < ∞ .Then the HK density function f S,I is a piecewise polynomial function of degree d − ,explicitly given in terms of the graded Betti numbers of the resolution of IR .In particular, if R = k [ X , . . . , X d ] is a polynomial ring and G is a finite group(scheme) acting linearly on R then for any graded pair ( R G , I ) , where R G is the ringof invariants, the function f R G ,I is a piecewise polynomial of degree d − . VIJAYLAXMI TRIVEDI AND KEI-ICHI WATANABE
We explicitly write down (in the tame case) the HK density function f R G ,I , where R = k [ X , X ] and G ⊂ SL ( k ) a finite group and I is the graded maximal ideal of R G .Similar to the case of standard graded pairs, the HK density function is multiplicative(Theorem 6.1) for the graded pairs too. In particular the HK density function of theSegre product of two graded pairs can be written in terms of the HK density functionsof those pairs. 2. preliminaries Notations 2.1.
By a graded pair ( R, I ) we mean that R = ⊕ m ≥ R m is a Noetheriangraded domain of dimension d ≥
2, and of finite type over a perfect field k = R ofcharacteristic p >
0, and I ⊂ R is a graded ideal such that ℓ ( R/I ) < ∞ .Let ( R, I ) be a graded pair, and let M be a finitely generated graded R -module. Weextend the definition of g n ( M R , I ) (given in the introduction) as follows. Definition 2.2.
Let n = gcd { n | R n = 0 } . Then f n ( M R , I ) : [0 , ∞ ) −→ [0 , ∞ ) is thestep function given by f n ( M R , I )( x ) = 1 q d − (cid:16) ℓ ( M/I [ q ] M ) ⌊ xq ⌋ n + · · · + ℓ ( M/I [ q ] M ) ⌊ xq ⌋ n + n − (cid:17) . If x = (1 − t ) ⌊ xq ⌋ q + ( t ) ⌊ xq +1 ⌋ q , for some t ∈ [0 ,
1) then the function g n ( M R , I ) :[0 . ∞ ) −→ [0 , ∞ ) is given by g n ( M R , I )( x ) = (1 − t ) f n ( M R , I )( x ) + ( t ) f n ( M R , I )( x + 1 q ) . In particular, each g n ( R, I ) is continuous, and the uniform convergence of the se-quence { g n ( M R , I ) } n is equivalent to the uniform convergence of the sequence { f n ( M R , I ) } n .We also make the following observation that the functions g n ( M R , I ) and f n ( M R , I )are compactly supported with a bound on the support which is independent of n . Lemma 2.3.
Each g n ( M R , I ) is a compactly supported continuous function. Moreover,for a given pair ( M R , I ) there is a constant e m (independent of n ) such that supp g n ( M R , I ) ⊆ [0 , e m ] , for all n ≥ . In particular supp f n ( M R , I ) ⊆ [0 , e m ] , for all n ≥ .Proof. We choose the integers s , l , m µ and n ν as follows: Let µ ( I ) = s . Let J = ⊕ m> R m with a set of homogeneous generators h , . . . , h µ of degrees, say, m ≤ · · · ≤ m µ respectively. Let l be an integer such that J l ⊆ I . Let M be generated by homo-geneous elements g , . . . , g ν of degrees n ≤ · · · ≤ n ν .Since R m = h R m − m + · · · + h µ R m − m µ and M m = g R m − n + · · · + g ν R m − n µ , m − n ν ≥ ( m µ ) lsq = ⇒ M m ⊆ J lsq M ⊆ I sq M ⊆ I [ q ] M. Hence (
M/I [ q ] M ) m = 0, for all m ≥ n s + ( m µ ) lsq . (cid:3) The following is a well known result.
Lemma 2.4.
Let R = ⊕ n ≥ R n be a Noetherian graded domain such that R is a field.Then the following three conditions are equivalent: (1) gcd { n > | R n = 0 } = n . (2) n > is the least integer with the property: there is m > such that R mn = 0 ,for all m ≥ m . ILBERT-KUNZ DENSITY FUNCTION FOR GRADED DOMAINS 5 (3) n > is the least integer such that the quotient field of R has an homogeneouselement of degree n .Proof. Left as an exercise for the reader. (cid:3) The HK density functions for normal graded domains
In this section we prove the existence of the HK density function (in Proposition 3.8)for a graded pair (
R, I ), where, in addition, R is a normal domain. We will make useof a technical lemma (Lemma 3.7), which we will prove in Section 5.For such a ring R we will be use the following result of Demazure [D]. Theorem 3.1. (Demazure) . Let R = ⊕ n ≥ R n be a normal graded domain of finitetype over a field k . Suppose there is an homogeneous element T of degree in thequotient field of R . Then for X = Proj R , there exists a unique Weil Q -divisor D in W div ( X, Q ) such that R n = H ( X, O X ( nD )) .T n , for every n ≥ . We recall some general facts about Q -divisors. Notations 3.2.
Let X be a normal projective variety over a perfect field k (in ourcase X = Proj R , where R is a normal graded domain).The set W div( X ) is the set of Weil divisors, where a Weil divisor is a formal sum ofcodimension 1 integral subschemes (prime divisors) of X . The setDiv( X, Q ) = W div( X, Q ) = W div( X ) ⊗ Z Q , is the set of formal linear combinations of codimension one integral subschemes of X with coefficients in Q (called Q -divisors). Let K ( X ) denote the function field of X .For D ∈ W div( X, Q ) the O X -sheaf O X ( D ) is the sheaf whose space of sections on anopen set U ⊂ X is given by H ( U, O X ( D )) = { f ∈ K ( X ) | div( f ) | U + D | U ≥ } , where div( f ) = P i v D i ( f ) D i and v D i : K ( X ) −→ Z ∪ {∞} is the discrete valuation of K ( X ) corresponding to the prime divisor D i .In particular, if D = P i a i D i ∈ W div( X, Q ) is a formal sum of prime divisors D i ,where a i ∈ Q then O X ( D ) = O X ( ⌊ D ⌋ ), where ⌊ D ⌋ = ⊕ i ⌊ a i ⌋ D i .For the following basic theory of reflexive sheaves we refer to [H1] (one can also lookup the notes by [S] on his homepage). Definition 3.3.
A coherent sheaf F on X is reflexive if the natural map of O X -modules α : F −→ ( F ∧ ) ∧ is an isomorphism, where F ∧ = H om O X ( F , O X ).A rank one reflexive sheaf is invertible on the regular locus of X . In fact {O X ( D ) | D ∈ W div( X ) } = { rank 1 reflexive subsheaves of K ( X ) } and (even if R is not normal) { the Cartier divisors of X } = { invertible (hence reflexive) subsheaves of K ( X ) } . Hence if D is a Cartier divisor then D = P a i D i , where a i ∈ Z and hence D = ⌊ D ⌋ As we discussed earlier, in case R is standard graded there is a Cartier divisor D suchthat for m >> R m = H ( X, O X ( mD )). On the other hand if R is graded normaldomain then (by the above theorem of Demazure) there exists a Q -divisor D (whichneed not be Cartier, but some positive integer multiple of D is a Cartier divisor) suchthat R m = H ( X, O X ( mD )), for all m ≥ VIJAYLAXMI TRIVEDI AND KEI-ICHI WATANABE
We recall (in Lemma 3.4) some relevant properties of R and O X ( nD ) (see [D]).By Lemma 2.4 the existence of an homogeneous element T of degree 1 in the quotientfield of R is equivalent to the condition that R m = 0 for all m >> { m > | R m = 0 } = 1. Lemma 3.4.
For R and D as in Theorem 3.1, let h , . . . , h µ denote a set of homo-geneous generators of R as an R -algebra, of degrees m , . . . , m µ respectively, and let l = lcm ( m , . . . , m µ ) . Then (a) for n ∈ l N , the sheaf O X ( nD ) is a line bundle on X . In particular the canonicalmultiplication map O X ( nD ) ⊗ O X ( iD ) −→ O X (( n + i ) D ) is an isomorphism, for all i. (b) For r = l µ the line bundle O X ( rD ) is very ample on X .Proof. (a): The variety X has the affine open cover { D + ( h i ) } i , where O X ( nD ) | D + ( h i ) = { f /h mi | deg( f ) − m deg( h i ) = n, f ∈ R deg( f ) } = h n/m i i O X | D + ( h i ) . is generated by the element h n/m i i ∈ H ( D + ( h i ) , O X | D + ( h i ) ), for all i .Since O X ( nD ) is a Cartier divisor O X ( nD + ⌊ iD ⌋ ) = O X ( ⌊ ( n + i ) D ⌋ ).(b): If R ( r ) := ⊕ m ≥ R rm and R ( r ) m := R rm . Then R ( r ) is a standard graded ring, as for m ≥ R ( r ) m = R mr ⊆ R µ ( m − l R l µ ⊆ R m − l µ R l µ = ( R ( r )1 ) m . Since X = Proj R ( r ) , the sections of O X ( rD ) give a closed immersion of X into P hk where h = h ( X, O X ( rD )) − (cid:3) In the rest of the section we have the following notations.
Notations 3.5.
The pair (
R, I ) is a fixed graded pair, where R is a normal gradeddomain and gcd { n | R n = 0 } = 1.We fix a homogeneous element T of degree 1 in the quotient field of R .Let D ∈ Div ( X, Q ) be the divisor, as in Theorem 3.1 so that R = R ( X, D ) = ⊕ n ≥ H ( X, O X ( nD )) .T n .We fix r ∈ N , so that O X ( rD ) is a very ample divisor on X and R m = 0, for all m ≥ r .For I , we fix a set of homogeneous generators f , · · · , f s of degrees d , . . . , d s respec-tively.For the sake of abbreviation we adopt the following Notations.Let O n = O X ( nD ).Let L = O r be the very ample line bundle on X .Let m q,d i = m + q − qd i where q = p n , for some n ≥ m q,d i ) = ⌊ m + qr ⌋ r − qd i .Since gcd { m > | R m = 0 } = 1, the definition of the sequences { f n ( R, I ) } n and { g n ( R, I ) } n is same as in the case of standard graded pair (given in [T]). Definition 3.6.
For the pair (
R, I ), the function { f n ( R, I ) : [0 , ∞ ) → [0 , ∞ ) } n ∈ N isgiven by f n ( R, I )( x ) = 1 /q d − ℓ ( R/I [ q ] ) ⌊ xq ⌋ , where q = p n ILBERT-KUNZ DENSITY FUNCTION FOR GRADED DOMAINS 7 and, for x = (1 − t ) m/q + t ( m + 1) /q , where t ∈ [0 , g n ( R, I )( x ) = (1 − t ) f n ( R, I )( m/q ) + ( t ) f n ( R, I )( m + 1 /q ) . For given m ∈ N and q = p n , we consider the following short exact sequence of O X -modules(3.1) 0 −→ F m,q −→ ⊕ si =1 O m q,di ϕ m,q −→ O m + q −→ , where ϕ m,q ( a , . . . , a s ) = P i a i f qi .Then f n ( R, I )( m + qq ) = q d − ℓ ( R/I [ q ] ) m + q = q d − [ ℓ ( R m + q ) − P si =1 ℓ ( f qi R m + q − qd i )]= q d − h h ( X, O m + q ) − ⊕ i h ( X, O m q,di ) + h ( X, F m,q ) i . To compare f n and f n +1 , we use the following crucial technical result, which will beproved below in Section 5. Lemma 3.7. (Main Lemma) . For F m,q as in (3.1), there exists a constant C suchthat, for all m ≥ and q = p n and ≤ n < p , | h ( X, F mp + n ,qp ) − p d − h ( X, F m,q ) | ≤ C ( mp + qp ) d − , and | h ( X, O m q p + n ) − p d − h ( X, O m q ) | ≤ C ( mp + qp ) d − , where m q = m q,d i , m q = ( m q,d i ) , for ≤ i ≤ s , or m q = m + q . The following proposition proves the existence of the HK density function for normalgraded domains.
Proposition 3.8. If R = ⊕ n ≥ R n is a normal graded domain and gcd { n | R n = 0 } = 1 then for a graded pair ( R, I ) the sequence { f n ( R, I ) } n is uniformly convergent.Proof. For brevity, in the rest of the proof, we denote f n ( R, I ) by f n .Let x ≥
1. For q = p n and m + q ≤ xq < m + q + 1, f n ( x ) = 1 /q d − ℓ ( R/I [ q ] ) ⌊ xq ⌋ = 1 /q d − ℓ ( R/I [ q ] ) m + q . Therefore there is n such that 0 ≤ n < p and f n +1 ( x ) = 1 / ( qp ) d − ℓ ( R/I [ qp ] ) mp + qp + n . For O m = O X ( mD ) consider the short exact sequences of O X -modules (as in (3.1)0 −→ F m,q −→ ⊕ si =1 O m + q − qd i ϕ m,q −→ O m + q −→ , −→ F mp + n ,qp −→ ⊕ si =1 O mp + qp + n − qpd i ϕ mp + n ,qp −→ O mp + qp + n −→ . Therefore f n ( x ) = p d − [ h ( X, O m + q ) − P i h ( X, O m + q − qd i ) + h ( X, F m,q )]( qp ) d − ,f n +1 ( x ) = [ h ( X, O mp + qp + n ) − P i h ( X, O mp + qp + n − qpd i ) + h ( X, F mp + n ,qp )]( qp ) d − . VIJAYLAXMI TRIVEDI AND KEI-ICHI WATANABE
By the Main Lemma 3.7, there is a constant C > | f n ( x ) − f n +1 ( x ) | ≤ C ( mp + qp ) d − / ( qp ) d − . Since supp f n ⊆ [0 , e m ], where e m is as in Lemma 2.3, we can assume ( m + q ) /q ≤ e m and hence there is a constant C such that | f n ( x ) − f n +1 ( x ) | ≤ C /qp, for all x ≥ . We can further choose C such that the above inequality also holds for all 0 ≤ x ≤ ≤ x < f n ( x ) = ℓ ( R ) ⌊ xq ⌋ /q d − = P ( ⌊ xq ⌋ ) /q d − for all q = p n >> , where P ( x ) ∈ Q [ X ] is the Hilbert polynomial of R hence of degree d −
1. This provesthe proposition. (cid:3) The Main theorem
In this section we will prove that the Main Theorem holds for the general gradedpairs (
R, I ). (Here we still assume the Main Lemma 3.7 which will be proved in thenext section).Throughout this section (
R, I ) is a graded pair and gcd { m | R m = 0 } = n .For a finitely generated graded R -module, the functions f n ( M R , I ) and g n ( M R , I )are as in Definition 2.2. Remark 4.1. If S −→ R is a degree preserving module-finite map of graded domains,where ( S, I ) is a graded pair and n = gcd { n | R n = 0 } and m = gcd { n | S n = 0 } ,then the HK density function of ( R, IR ) as a module over itself is ( q = p n ) f R,IR ( x ) := lim n →∞ f n ( R, IR )( x ) = 1 q d − (cid:16) ℓ ( R/I [ q ] R ) ⌊ xq ⌋ n (cid:17) . Whereas the HK density function of (
R, IR ) as a module over S is f R S ,I ( x ) := lim n →∞ f n ( R S , I )( x ) = 1 q d − (cid:16) ℓ ( R/I [ q ] R ) ⌊ xq ⌋ m + · · · + ℓ ( R/I [ q ] R ) ⌊ xq ⌋ m + m − (cid:17) . The existence of both the limits is shown in the following Theorem 1.1.We use the following lemma to reduce the problem of convergence of { f n ( M R , I ) } tothe problem of convegence of { f n ( S, IS ) } , where S is the normalization of R in Q ( R ). Lemma 4.2.
If gcd { m > | R m = 0 } = 1 and N , N ′ are finitely generated graded R -modules with the exact sequence of graded R -linear maps (4.1) 0 −→ N φ −→ N ′ −→ Q ′′ −→ such that the supp dim Q ′′ < d and the map φ is of degree .Then the sequence { f n ( N R , I ) } n is uniformly convergent if and only if { f n ( N ′ R , I ) } n is so. Moreover in that case lim n →∞ f n ( N R , I ) = lim n →∞ f n ( N ′ R , I ) . Proof.
Here if M is a graded R -module then the function f n ( M R , I ) : [0 , ∞ ) −→ [0 , ∞ )is given by x → ℓ ( M/I [ q ] M ) ⌊ xq ⌋ /q d − , where q = p n .Let I have homogeneous generators f , . . . , f s of degree d , . . . , d s respectively. Thenfor any graded R -module M , we defineΦ M : ⊕ si M ( − qd i ) −→ M given by ( m , . . . , m s ) → X i f qi m i ILBERT-KUNZ DENSITY FUNCTION FOR GRADED DOMAINS 9
This gives graded degree 0 maps of graded R -modules, functorial in M ,0 −→ Ker Φ M −→ ⊕ s M ( − qd i ) Φ M −→ M −→ Coker Φ M −→ . Now the snake lemma applied to (4.1) gives the following exact sequence of graded R -modules −→ Ker Φ Q ′′ −→ Coker Φ N −→ Coker Φ N ′ −→ Coker Φ Q ′′ −→ , where f n ( N ′ R , I )( m + qq ) = ℓ (Coker Φ N ′ ) m + q and f n ( N R , I )( m + qq ) = ℓ (Coker Φ N ) m + q .Let C Q ′′ be constant such that, for all m > ℓ ( Q ′′ m ) ≤ C Q ′′ m d − (such a constantexists by the hypothesis on support dimensions). | ℓ (Coker Φ N ′ ) m + q − ℓ (Coker Φ N ) m + q | ≤ C Q ′′ ( m + q ) d − . Now, for x ≥ m + q ≤ xq < m + q + 1 for some m ≥ | f n ( N R , I )( x ) − f n ( N ′ R , I )( x ) | ≤ C Q ′′ x d − /q, where by Lemma 2.3, we may fix an x such that supp f n ( N R , I ) and supp f n ( N ′ R , I )are subsets of [0 , x ], for all n ≥ ≤ x < m ≤ xq < m + 1, for some m < q . It is easy to check that in thiscase | f n ( N R , I )( x ) − f n ( N ′ R , I )( x ) | = 2 C Q ′′ m d − /q d − ≤ C Q ′′ /q. This proves the lemma. (cid:3)
Now we are ready to prove the Main Theorem.Proof of the Main Theorem 1.1: Let gcd { n | R n = 0 } = n . Let S = R ( n ) , wherethe n th degree component of R ( n ) is R nn . Then S is a graded domain, where gcd { n | S n = 0 } = 1. (Note that S = R as rings, but the grading is changed.)Note that assertion (2) (b) follows from assertion (1).It is easy to prove that the uniform convergence of { g n ( M R , I ) } n is equivalent to theuniform convergence of { f n ( M R , I ) } n .We first prove the theorem for M = R , where it is sufficient to prove the uniformconvergence of { f n ( R, I ) } n . By definition, f n ( R, I ) = f n ( S, I ), for all n .Let e S = ⊕ n e S n denote the normalization of S in its quotient field then the inclusionmap S −→ e S is a module finite graded map of degree 0, and we have the short exactsequence of graded S -modules0 −→ S −→ e S −→ Q ′′ −→ , where support dim Q ′′ ≤ d − { f n ( ¯ S, I ¯ S ) is uniformly convergent. But f n ( ¯ S S , I ) = f n ( ¯ S, I ¯ S ). Hence the uniform convergence of { f n ( R, I ) } n follows by Lemma 4.2.We now consider the general case of a finite graded module M .Let ¯ M = ⊕ n ¯ M n , where ¯ M n = M nn + · · · + M nn + n − denotes the degree n compo-nent of ¯ M . If M is generated by homogeneous elements g , . . . , g ν as an R -module then M is generated by g , . . . , g ν as an S -module. Hence M is a finitely generated graded S -module. Also, for all n ≥ f n ( M R , I ) = f n ( M S , IS ) and rank R M = rank S M . Claim . There exists an exact sequence of graded S -modules0 −→ ⊕ n S ( − a ) φ −→ M −→ Q ′′ −→ , where φ is a graded map of degree 0 and dim ( Q ′′ ) ≤ d − T = S \ { } , the T − S -module T − M is free of finite rank, say n and is generated by a finite set of homogeneouselements. Hence we can choose homogeneous elements m , . . . , m n in M of degrees d , . . . , d n respectively such that the m ′ i s give a basis for T − M .Since gcd { n | S n = 0 } = 1, we have m such that S m = 0, for all m ≥ m . Let a > a ≥ max { m + d i , m } i and let s i ∈ S a − d i \ { } . Then s m , . . . , s n m n ∈ M are homogeneous elements (each of degree a ) and generate T − M as T − S -module.Hence we have a generically isomorphic map ⊕ n S ( − a ) −→ M of graded S -modules ofdegree 0. The map is injective as S is a domain. This proves the claim.Now the theorem (1) and (2) (a) follows from Lemma 4.2. (cid:3) Note that assertion (2) (b) follows from assertion (1).As we remarked earlier (Remark 4.1), for a finite map S −→ R as in Notations 4.1,the two HK density functions, for the pair ( R, IR ), namely f R,IR and f R S ,I need notbe the same functions but can be recovered from each other as follows. Lemma 4.3.
Let S −→ R be the module-finite map as in Notations 4.1. Let m = gcd { n > | S n = 0 } and n = gcd { n > | R n = 0 } then ( l ) f R,IR ( xl ) = f R S ,I ( x ) = (rank S R ) f S,I ( x ) , for all x ∈ R ≥ , where l = m /n is an integer. Hence f R,IR ≡ f R S ,I if m = n .Proof. We first prove that n divides m . Otherwise m = n l + n , where 0 < n
ILBERT-KUNZ DENSITY FUNCTION FOR GRADED DOMAINS 11 Proof of the Main Lemma
Here we prove that the Main technical Lemma 3.7 which will complete the proof ofthe Main Theorem.Throughout this section we follow the Notations 3.5.As we mentioned earlier, the sequence (3.1) need not be locally split exact as thesheaf O m + q is not invertible in general. Hence we consider the following locally split(as O ⌊ m + q/r ⌋ r ≃ L ⌊ m + q/r ⌋ is invertible) exact sequence of O X -modules(5.1) 0 −→ G m,q −→ ⊕ si =1 O ( m q,di ) ¯ ϕ m,q −→ L ⌊ m + q/r ⌋ −→ , where ¯ ϕ m,q ( a , . . . , a s ) = P i a i f qi and where G m,q = F ⌊ m + q/r ⌋ r − q,q (note however that G m,q may not be locally free). In case m + q is divisible by r , the sequence (5.1) is sameas the sequence 0 −→ F m,q −→ ⊕ si =1 O m q,di ϕ m,q −→ O m + q −→ , as in (3.1).In Lemma 5.3, we show that the length of the cohomology of F m,q (or of O m + q )differs from the length of the cohomology of G m,q ( L ⌊ m + q/r ⌋ respectively) by a functionof order O ( m + q ) d − , where dim X = d −
1. Hence it will be sufficient to prove theMain Lemma 3.7 for G m,q instead of F m,q .In the rest of the section we use the following Terminology
We fix a pair (
R, I ), the line bundle L , the integer r along with a choiceof generators f , . . . , f s of I as in Notations 3.5.Given L , where L might be a number, a set, a map or a coherent sheaf, C L denotesa constant which depends only on L (with the fixed data ( R, I ), L etc. as above). Bysupp dim F , we mean the dimension of the support of F .Here supp dim O n = supp dim X = d − ≥ Lemma 5.1.
For a given coherent sheaf N of O X -modules with supp dim N < d − ,there is a constant C N such that (1) h j ( X, N ⊗ L m ) ≤ C N ( | m | ) d − , for every j ≥ and m ∈ Z . (2) h j ( X, O m ⊗ N )) ≤ C N ( | m | ) d − , for all j ≥ and for all m ∈ Z .In particular, for m ≥ and ≤ j ≤ d − , (3) there exists C such that h j ( X, O m q ⊗ N ) ≤ C N ( m + q ) d − , where, for ≤ i ≤ s , m q = m q,d i , m q = ( m q,d i ) or m q = m + q (as in Notations 3.5), and (4) h j ( X, G m,q ⊗ N )) ≤ C N ( m + q ) d − , for all m, j ≥ and q .Proof. (1) By the Serre vanishing theorem ([H]) h j ( X, N ⊗ L m ) = 0, for j > m >>
0. Also, for m >> h ( X, N ⊗ L m ) is a polynomial of degree equal todim N < d −
1. Hence the assertion (1) follows by induction on dim N and the Serre’sduality ([H]).(2) By Lemma 3.4, we have O m = L ⌊ m/r ⌋ ⊗ O r , where r = m − ⌊ m/r ⌋ r < r . Sincethe support of O m ⊗ N = the support of N , the assertion (2) follow from the fact that O r ⊗ N belongs to the finite set {O ⊗ N , O ⊗ N , . . . , O r − ⊗ N } of coherent sheavesof O X -modules.The assertion (3) follows from (2) as | m + q − d i q | ≤ d i ( m + q ).Since the sequence (5.1) is locally split exact, the induced sequence0 −→ G m,q ⊗ N −→ ⊕ si =1 O ( m q,di ) ⊗ N ¯ ϕ m,q ⊗N −→ L ⌊ m + q/r ⌋ ⊗ N −→ is exact. Now the assertion (4) follows from (2) and (3). (cid:3) Lemma 5.2. (1) Let S = { ψ j : E j −→ F j | ≤ j ≤ s } be a finite set of O X -linearmaps, where E j and F j are coherent sheaves of O X -modules. For m ∈ Z , let ψ j ( m ) := Id L m ⊗ ψ j : L m ⊗ E j −→ L m ⊗ F j be the canonically induced maps. Assume that supp dim (ker ψ j ) and supp dim (coker ψ j ) are each < d − . Then there exists a constant C S such that h i ( X, ker ψ j ( m )) ≤ C S m d − and h i ( X, coker ψ mj ) ≤ C S m d − , for all i ≥ . (2) Moreover if { −→ N ′ m −→ M ′ m φ m −→ M m −→ N m −→ } m ∈ Z denote a familyof exact sequences of O X -modules and C and C are constants such that h i ( X, N ′ m ) ≤ C ( n m ) d − and h i ( X, N m ) ≤ C ( n m ) d − , for all i ≥ , then | h ( X, M ′ m ) − h ( X, M m ) | ≤ ( C + C )( n m ) d − . Proof.
We note that, for any m ∈ Z ,ker ψ j ( m ) ≃ L m ⊗ ker ψ j and coker ψ j ( m ) ≃ L m ⊗ coker ψ j , where ker ψ j and coker ψ j are in a fixed family of finite number of coherent sheaves of O X -modules. Hence the first assertion follows by Lemma 5.1.The second assertion follows by splitting the exact sequence into two canonical twoshort exact sequences 0 −→ N ′ m −→ M ′ m −→ Im( φ m ) −→ , −→ Im( φ m ) −→ M m −→ N m −→ . (cid:3) Lemma 5.3.
For all m and q = p n , (1) there is a constant C such that (cid:12)(cid:12) h ( X, G m,q ) − h ( X, F m,q ) (cid:12)(cid:12) ≤ C ( m + q ) d − . (2) For given integer l , there exists a constant C l such that for every ≤ l ≤ l , (cid:12)(cid:12) h ( X, G m,q ) − h ( X, G m + l,q ) (cid:12)(cid:12) ≤ C l ( m + q ) d − , (cid:12)(cid:12) h ( X, O m q ) − h ( X, O m q + l ) (cid:12)(cid:12) ≤ C l ( m + q ) d − , where, for ≤ j ≤ s , m q = m q,d j or m q = ( m q,d j ) , or m q = m + q .Proof. Claim (A). For a given e r ∈ Z , if H ( X, O ˜ r ) = { } then there exists a constant C ˜ r such that, for i ≥ m ≥ | h ( X, F m,q ) − h ( X, F m + ˜ r,q ) | ≤ C ˜ r ( m + q ) d − . Proof of the claim: An element h ∈ H ( X, O ˜ r ) \ { } gives an injective map Φ h : O m −→ O m + ˜ r , for all m . In particular we have the following canonical diagram ofsheaves of O X -modules: ILBERT-KUNZ DENSITY FUNCTION FOR GRADED DOMAINS 13 −→ F m + ˜ r,q −→ ⊕ si =1 O m + q + ˜ r − qd i ϕ m +˜ r,q −→ O m + ˜ r + q −→ ↑ Φ ′ h ↑ ⊕ i Φ h ↑ Φ h −→ F m,q −→ ⊕ si =1 O m + q − qd i ϕ m,q −→ O m + q −→ ↑ ↑ ↑ ⊕ i Φ h : ⊕ i O m + q − qd i −→ ⊕ i O m + ˜ r + q − qd i is same as ⊕ i (Id L ⊗ φ i ) : ⊕ i ( L ⌊ m q,di /r ⌋ ⊗ E i ) −→ ⊕ i ( L ⌊ m q,di /r ⌋ ⊗ F i )and where E i = O m + q − qd i −⌊ m q,di /r ⌋ r and F i = O m + ˜ r + q − qd i −⌊ m q,di /r ⌋ r and the map φ i : E i −→ F i is the multiplication map by h .Also the map Φ h : O m + q −→ O m + ˜ r + q isId L ⊗ φ : L ⌊ m + q/r ⌋ ⊗ E −→ L ⌊ m + q/r ⌋ ⊗ F , where E = O m + q −⌊ m + q/r ⌋ r and F = O m + ˜ r + q −⌊ m + q/r ⌋ r and the map φ : E −→ F is given by the multiplication by h . Note that E i ∈{O , . . . , O r − } and F i ∈ {O ˜ r , . . . , O ˜ r + r − } and supp dim (coker φ i ) < d −
1. Hencethe claim follows by Lemmas 5.1 and 5.2 and the short exact sequence0 −→ coker Φ ′ h −→ coker ( ⊕ si Φ h ) −→ coker Φ h −→ . Assertion (2). It is enough to prove the Assertion (2) for F m,q instead of G m,q . Sincethere exists x ∈ H ( X, O r ) \ { } , the above claim implies that we have a constant C r such that | h ( X, F m,q ) − h ( X, F m +2 r,q ) | ≤ C r ( m + q ) d − , for i ≥ . Case 1. If l ≤ r then there exists x ∈ H ( X, O r − l ) \ { } , and therefore we have aconstant C r − l such that, for i ≥ | h ( X, F m + l,q ) − h ( X, F m +2 r,q ) | ≤ C r − l ( m + q ) d − . Case 2. If l ≥ r then we can choose x ∈ H ( X, O l ) \ { } and therefore get a constant C l such that | h ( X, F m,q ) − h ( X, F m + l,q ) | ≤ C l ( m + q ) d − , for i ≥
0. Since, for given 0 ≤ l ≤ l , there are finitely many choices of such C l , we getAssertion (2) of the lemma.Similarly we prove the lemma for O m q .Assertion (1). It follows from the proof of Assertion (2). (cid:3) The Main Lemma for G m,q . Here we compare h ( X, G mp,qp ) ( h ( X, ( O m q p ))with h ( X, ⊕ p d − G m,q ) ( h ( X, ⊕ p d − O m q ) respectively) in Lemma 5.4 and in Lemma 5.5.Since the sequence (5.1) is locally split exact, it remains exact for the functor ( − ) ⊗M , for any sheaf of O X -modules M . In particular we construct below a genericallyisomorphic map F ∗ G m,q −→ G m ′ ,qp , provided | mp − m ′ | bounded by constant for all m and m ′ . Lemma 5.4.
There is a constant C such that (cid:12)(cid:12) h ( X, ( F ∗ G m,q )) − h ( X, G mp,qp ) (cid:12)(cid:12) ≤ C ( mp + qp ) d − , (cid:12)(cid:12) h ( X, ( F ∗ O m q )) − h ( X, O m q p ) (cid:12)(cid:12) ≤ C ( mp + qp ) d − , where m q = m q,d j , m q = ( m q,d j ) , for ≤ j ≤ s , or m q = m + q and where m ≥ and q = p n .Proof. Claim . For given n there is a generically isomorphic map ψ n : F ∗ O n −→ O np .Proof of the claim: By notation O n = O X ( nD ), where D is a Q -Weil divisor. Let D = P a i D i , where a i ∈ Q and D i are prime divisors. Then ⌊ npD ⌋ = X i ⌊ a i n ⌋ pD + X i m i D i = p ⌊ nD ⌋ + X i m i D i , where 0 ≤ m i ≤ p are integers. Let M = O X ( p ⌊ nD ⌋ ) then M ¯ f n −→ O np is an inclusionsuch that supp dim coker ¯ f n < d − φ n : F ∗ O n −→ M as follows: For theFrobenius map F : X −→ X let F ∗ O n = F − O n ⊗ F − O X O X and let { D + ( f ) } f denote the affine open cover of X , where f ∈ R is an homogeneous element of R . Thenthe map φ n | D + ( f ) is given by v/f j ⊗ u/f i → ( v/f j ) p · u/f i , if v/f j ∈ F − O n and u/f i ∈ O X . The map φ n is isomorphism on the regular locus X reg of X as O n | X reg is invertible.In particular ψ n = ¯ f n · φ n is generically an isomorphism. This proves the claim.Now the map ψ = ⊕ i ψ ( m q,di ) : ⊕ i F ∗ O ( m q,di ) −→ ⊕ i O ( m q,di ) p is generically an iso-morphism, and φ is an isomorphism such that φ ◦ F ∗ φ m,q = ¯ φ m ′ ,qp ◦ φ . This gives us amap F ∗ G m,q −→ G m ′ ,qp such that the following diagram commutes0 −→ G m ′ ,qp −→ ⊕ si =1 O ( m q,di ) p ¯ ϕ m ′ ,qp −→ L ⌊ m + q/r ⌋ p −→ ↑ f m,q ↑ ψ ↑ φ −→ F ∗ G m,q −→ ⊕ si =1 F ∗ O ( m q,di ) F ∗ ϕ m,q −→ F ∗ L ⌊ m + q/r ⌋ −→ , where m ′ = ⌊ ( m + q ) /r ⌋ rp − qp . Therefore m ′ = mp − r p , for some 0 ≤ r < r .Note that the map ψ ( m q,di ) : F ∗ O ( m q ,d i ) −→ O ( m q ,d i ) p is the same as the map ψ ⌊ m q,di /r ⌋ r ⊗ ψ i j : F ∗ L ⌊ m q,di /r ⌋ ⊗ F ∗ O i j −→ L ⌊ m q,di /r ⌋ p ⊗ O i j p , where the map ψ ⌊ m q,di /r ⌋ r : F ∗ L ⌊ m q,di /r ⌋ −→ L ⌊ m q,di /r ⌋ p is an isomorphism and thegenerically isomorphic map ψ i j ∈ { ψ j : F ∗ O j −→ O jp | − r ≤ j ≤ r } .Similarly for any m ∈ Z , the map ψ m : F ∗ O m −→ O mp is same as the map ψ ⌊ m/r ⌋ ⊗ ψ i : F ∗ L ⌊ m/r ⌋ ⊗ F ∗ O i −→ L ⌊ m/r ⌋ p ⊗ O ( ip ) , where 0 ≤ i < r and where ψ ⌊ m/r ⌋ is an isomorphism. Since the map φ is an isomorphism, we haveker f m,q = ker ψ and coker f m,q = coker ψ and each have supp dim < d −
1. Hencethe lemma follows by Lemma 5.2. (cid:3)
ILBERT-KUNZ DENSITY FUNCTION FOR GRADED DOMAINS 15
Lemma 5.5.
There is a constant C such that | p d − h ( X, G m,q ) − h ( X, F ∗ G m,q ) | ≤ C ( mp + qp ) d − | p d − h ( X, O m q ) − h ( X, F ∗ O m q ) | ≤ C ( mp + qp ) d − , where m q = m q,d j , m q = ( m q,d j ) , for ≤ j ≤ s , or m q = m + q , and where m ≥ and q = p n .Proof. Recall X = Proj R = Proj R r , where R r is a standard graded domain. Thereforeby Lemma 2.9 in [T], there is an integer m ∈ N (it will be a multiple of r ) such thatwe have a short exact sequence of sheaves of O -modules(5.2) 0 −→ ⊕ p d − O X ( − m D ) η −→ F ∗ O X −→ Q ′′ −→ , where support dimension Q ′′ is < d − M = ⊕ p d − O − m and M = F ∗ O X . Then the short exact sequences 0 −→ M η −→ M −→ Q ′′ −→ ↑ ↑ ↑ −→ G m,q ⊗ Q ′′ −→ ⊕ si =1 O ( m q,di ) ⊗ Q ′′ −→ L ⌊ m + q/r ⌋ ⊗ Q ′′ −→ ↑ h G m,q ↑ h L m,q ↑ −→ G m,q ⊗ M −→ ⊕ si =1 O ( m q,di ) ⊗ M ¯ ϕ m,q −→ L ⌊ m + q/r ⌋ ⊗ M −→ ↑ f G m,q ↑ f L m,q ↑ −→ G m,q ⊗ M −→ ⊕ si =1 O ( m q,di ) ⊗ M ϕ m,q −→ L ⌊ m + q/r ⌋ ⊗ M −→ ↑ ↑ ↑ ker( f G m,q ) = ker( f L m,q ) 0 , ↑ ↑ O ( m q,di ) = L ⌊ m q,di /r ⌋ ⊗ E i , where E i = O ( m q,di ) −⌊ m q,di /r ⌋ r ∈ {O , O , . . . , O r } the map f L m,q is same as the map ⊕ si =1 Id L ⊗ ψ i : ⊕ si =1 L ⌊ m q,di /r ⌋ ⊗ E i ⊗ M −→ ⊕ si =1 L ⌊ m q,di /r ⌋ ⊗ E i ⊗ M, where the map ψ i = Id E i ⊗ η : E i ⊗ M −→ E i ⊗ M is generically isomorphic. Thereforesupp dim of (cid:16) ker( f G m,q ) = ker( f L m,q ) = ⊕ si =1 ( L ⌊ m q,di /r ⌋ ⊗ ker ψ i ) (cid:17) < d − . Now the long exact sequence0 −→ ker( f G m,q ) −→ G m,q ⊗ M −→ G m,q ⊗ M −→ G m,q ⊗ Q ′′ −→ | h ( X, G m,q ⊗ M ) − h ( G m,q ⊗ F ∗ O X ) | = C η ( m + q ) d − , for some constant C η . On the other hand, as F is a finite map, for any coherent sheaf M of O X -modules the projection formula F ∗ ( F ∗ G m,q ⊗ M ) = G m,q ⊗ F ∗ M holds.This implies(5.4) h ( X, G m,q ⊗ F ∗ O X ) = h ( X, F ∗ ( F ∗ G m,q )) = h i ( X, F ∗ G m,q )Now the lemma follows by (5.3) and (5.4). The second assertion follows by the same line of arguments. (cid:3)
Proof of Main Lemma 3.7 It follows from Lemma 5.4, Lemma 5.5 and Lemma 5.3 (1). (cid:3) HK density functions for segre products of graded rings
Here we show that the HK density function is multiplicative. Let (
R, I ) and (
S, J )be two pairs, where R = ⊕ n ≥ R n and S = ⊕ n ≥ S n are graded domains of dimension d ≥ d ≥ k , and I ⊂ R and J ⊂ S are gradedideals of finite colengths.Moreover let F R : [0 , ∞ ) −→ [0 , ∞ ) and F S : [0 , ∞ ) −→ [0 , ∞ ) be the Hilbert-Samueldensity functions given by F R ( x ) = e ( R ) x d − / ( d − , and F S ( x ) = e ( S ) x d − / ( d − , where, for a graded ring R , e ( R ) is the Hilbert-Samuel multiplicity of R with respectto its graded maximal ideal.In [T], we had proved that the HK density function is multiplicative for Segre prod-ucts of standard graded rings. In Theorem 6.1 and Corollary 6.3, we show that thisproperty extends to graded domains. Theorem 6.1.
For the pairs ( R, I ) and ( S, J ) as above if gcd { m | R m = 0 } = 1 andgcd { m | S m = 0 } = 1 . Then the HK density function of the pair ( R S, I J ) , where R S = ⊕ n ≥ R n ⊗ k S n is the Segre product of R and S , is given by F R S − f R S,I J = [ F R − f R,I ] [ F S − f S,J ] and also e HK ( R S, I J ) = e ( R )( d − R ∞ x d − f S,J ( x ) dx + e ( S )( d − R ∞ x d − f R,I ( x ) dx − R ∞ f R,I ( x ) f S,J ( x ) dx. Proof.
Note that R S is a graded integral domain with gcd { n | ( R S ) n = 0 } = 1.Therefore ( q d − q d − ) f n ( R S, I J )( m/q ) = ℓ ( R S/ ( I J ) [ q ] ) m = ℓ ( R m ) ℓ ( S m ) − h ℓ ( R m ) − ℓ ( R/I [ q ] ) m i h ℓ ( S m ) − ℓ ( S/J [ q ] ) m i . Hence f n ( R S, I J )( x ) = f n ( S, J ) ℓ ( R ) ⌊ xq ⌋ q d − + f n ( R, I ) ℓ ( S ) ⌊ xq ⌋ q d − − f n ( R, I ) f n ( S, J ) . Since { f n ( R S, I J ) } n , { f n ( R, I ) } n and { f n ( S, J ) } n are uniformly convergent se-quences with bounded supports, taking limit as n → ∞ we get, f R S,I S ( x ) = F R ( x ) f S,J ( x ) + F S ( x ) f R,I ( x ) − f R,I ( x ) f S,J ( x ) for all x ≥ . The rest of the proof follows as F R S ( x ) = F R ( x ) F S ( x ). (cid:3) Notations 6.2.
For a graded domain R = ⊕ n ≥ R n with a graded ideal I = ⊕ n ≥ I n ,we denote R ( m ) = ⊕ n R nm with degree n component = R nm and I ( m ) = I ∩ R ( m ) = ⊕ n I nm . ILBERT-KUNZ DENSITY FUNCTION FOR GRADED DOMAINS 17
Corollary 6.3.
If gcd { m | R m = 0 } = n and gcd { m | S m = 0 } = n . Then F R S − f R S,I J = h F R ( l ) − f R ( l ) ,I ( l ) i h F S ( l ) − f S ( l ) ,J ( l ) i and e HK ( R S, I J ) = e ( R ( l ) )( d − R ∞ x d − f S ( l ) ,J ( l ) ( x ) dx + e ( S ( l ) )( d − R ∞ x d − f R ( l ) ,I ( l ) ( x ) dx − R ∞ f R ( l ) ,I ( l ) ( x ) f S ( l ) ,J ( l ) ( x ) dx, where l = lcm ( n , n ) .Proof. Since R S = R ( l ) S ( l ) and I J = I ( l ) J ( l ) and gcd { m | R ( l ) m = 0 } = gcd { m | S ( l ) m = 0 } = 1, the corollary follows from the above theorem. (cid:3) Applications and examples
The coefficients of the HK function for a pair (
R, I ) (given as HK ( q ) = 1 /q d ℓ ( R/I [ q ] ))have a nice geometric description (see [K]) provided proj dim R ( R/I ) < ∞ . Here weprove that the HK density function too has a nice description in the case of such gradedpairs. Proposition 7.1.
Let ( R, I ) be a graded pair such that proj dim R ( R/I ) < ∞ then theHK density function f R,I is a piecewise polynomial function of degree d − , where f R,I (and hence e HK ( R, I ) ) is given in terms of the graded Betti numbers of the minimalgraded R -resolution of R/I .Proof.
Consider the minimal graded resolution of
R/I over the graded ring R −→ ⊕ j ∈ Z R ( − j ) β d,j −→ ⊕ j ∈ Z R ( − j ) β d − ,j −→ · · · −→ ⊕ j ∈ Z R ( − j ) β ,j −→ R −→ R/I −→ . Since the functor of Frobenius is exact on the category of modules of finite type andfinite projective dimension (a corollary of the acyclicity lemma by Peskine-Szpiro [PS]),we have a long exact sequence0 −→ ⊕ j ∈ Z R ( − qj ) β d,j −→ ⊕ j ∈ Z R ( − qj ) β d − ,j −→ · · · −→ ⊕ j ∈ Z R ( − qj ) β ,j −→ R −→ R/I −→ . Let e e = e ( R, m ) / ( d − B ( j ) = β j − β j + β j + · · · + ( − d β dj . Note that β = 1 and β ,j = 0 for j = 0. For j < B ( j ) = 0. If l be the largestinteger such that β il = 0 for some i . Then ℓ ( R/I [ q ] ) m = ℓ ( R m ) + B (1) ℓ ( R m − q ) + B (2) ℓ ( R m − q ) + · · · + B ( l ) ℓ ( R m − lq )and therefore f R,I ( x ) = e e (cid:2) x d − (cid:3) ≤ x ≤ e e (cid:2) x d − + B (1)( x − d − (cid:3) ≤ x ≤ e e (cid:2) x d − + B (1)( x − d − + · · · + B ( i )( x − i ) d − (cid:3) i ≤ x ≤ ( i + 1)= e e (cid:2) x d − + B (1)( x − d − + · · · + B ( l − x − l + 1) d − (cid:3) l − ≤ x ≤ l = 0 l ≤ x. Note that f R,I is a compactly supported function which implies that the polynomial x d − + B (1)( x − d − + · · · + B ( l )( x − l ) d − = 0 . Hence supp ( f R,I ) ⊆ [0 , l ].Moreover e HK ( R, I ) = e d ! h B (0) l d + B (1)( l − d + · · · + B ( i )( l − i ) d + · · · + B ( l − i (cid:3) Proof of Corollary 1.2: Let m = gcd { n > | S n = 0 } and n = gcd { n > | R n = 0 } and l = m /n . Then, by Theorem 1.1 and Lemma 4.3, for x ≥ f S,I ( x ) = 1rank S R f R S ,I ( x ) = l rank S R f
R,IR ( xl ) and e HK ( S, I ) = e HK ( R, IR )rank S R .
Hence the corollary follows from Proposition 7.1. (cid:3)
Remark 7.2.
If (
R, I ) is a graded pair such that
R/IR has the finite pure resolution0 −→ ⊕ β d R ( − j d ) −→ · · · −→ ⊕ β R ( − j ) −→ ⊕ β R ( − j ) −→ R −→ R/I −→ j < j < · · · < j d and B (1) = · · · = B ( j −
1) = 0 and B ( j ) = − β B ( j n − + 1) = · · · = B ( j n −
1) = 0 and B ( j n ) = ( − n β n . Hence f R,I ( x ) = e e (cid:2) x d − (cid:3) ≤ x ≤ j = e e (cid:2) x d − − β ( x − j ) d − (cid:3) j ≤ x ≤ j = e e (cid:2) x d − − β ( x − j ) d − + · · · + ( − d − β d − ( x − j d − ) d − (cid:3) j d − ≤ x ≤ j d = 0 j d ≤ x. Here the maximum support of f R,I = α ( R, I ) = j d , as β d = 0.7.1. Some concrete examples.
We recall the Hilbert-Burch theorem (see [BH]).
Theorem 7.3.
Let ψ : R n −→ R n +1 be a R -linear map, where R is a Noetherian ring.Let I = I n ( ψ ) be the ideal generated by n + 1 elements consisting of n × n minors ofthe matrix given by ψ .Then grade I n ( ψ ) ≥ implies the ideal I has the resolution of the form −→ R n ψ −→ R n +1 −→ I −→ . In the following examples we compute the HK density function f S,I , where S = R G is the ring of invariants in R = k [ x , x ] with G ∈ { A n , D n , E , E , E } and I ⊂ S isits graded maximal ideal. This also recovers the computations of e H ( R G , I ) given inTheorem 5.1 of [WY]. For this it is enough to construct the minimal graded resolutionof IR as a R -module.Note that in all the following cases R G = k [ h , h , h ] ⊂ k [ x , x ], where h , h , h areexplicit homogeneous polynomials in x , x (see Chap X, page 225 of [MBD]). Usingthe Hilbert-Burch theorem, we will construct a R -resolution for IR = ( h , h , h ) R ,which is of the folllowing type:(7.1)0 −→ R ( − l ) ⊕ R ( − l ) ψ −→ R ( − deg h ) ⊕ R ( − deg h ) ⊕ R ( − deg h ) φ −→ IR −→ , ILBERT-KUNZ DENSITY FUNCTION FOR GRADED DOMAINS 19 where φ is given by the matrix [ h , h , h ]. In the forthcoming set of examples we definethe map ψ by giving a 3 × R (this will also determine the values l , l ). Sincegrade IR = 2, to prove that (7.1) is exact, it only remains to check that I ( ψ ) = IR which can be done easily. Example 7.4.
Let G = A n then | G | = n ≥ k = p ≥ p, n ) = 1. R G = k [ h , h , h ] ∼ = k [ x , x , x ]( x n + x x ) , where h = x x , h = x n and h = x n . The map ψ is given by the matrix (cid:20) x n − − x x n − − x (cid:21) . Then the sequence0 −→ R ( − n − ⊕ R ( − n − ψ −→ R ( − ⊕ R ( − n ) ⊕ R ( − n ) −→ IR φ −→ IR as I ( ψ ) = I . Here B (2) = − B ( n ) = − B ( n + 1) = 2. If n is even then the HK density function f S,I is given by f S,I ( x ) = 4 x/ ( n + 1) if 0 ≤ x ≤
1= 4 / ( n + 1) if 1 ≤ x ≤ n/
2= 2(2 − x + 2 n ) / ( n + 1) if n/ ≤ x ≤ ( n + 1) / n is odd then the HK density function f S,I is given by f S,I ( x ) = x/ ( n + 1) if 0 ≤ x ≤
2= 2 / ( n + 1) if 2 ≤ x ≤ n = (2 − x + 2 n ) / ( n + 1) if n ≤ x ≤ n + 1 Example 7.5.
Let G = D n the dihedral group then | G | = 4 n and char k = p ≥ p, n ) = 1. R G = k [ h , h , h ] = k [ x , x , x ]( x + x x + x n +11 ) , where h = − x x , h = x n + ( − n x n , h = x x ( x n − ( − n x n . We assume n is even. Let map ψ is given by the matrix (cid:20) − x n − x x x − x n − − x x x (cid:21) Then the sequence0 −→ R ( − n − ⊕ R ( − n − ψ −→ R ( − ⊕ R ( − n ) ⊕ R ( − n − φ −→ IR −→ IR as I ( ψ ) = IR .Here B (4) = − B (2 n ) = − B (2 n + 2) = −
1. If n is even then the HK densityfunction f S,I is given by f S,I ( x ) = x/n − ≤ x ≤
2= 2 / ( n −
2) if 2 ≤ x ≤ n = ( n + 2 − x ) / ( n −
2) if n ≤ x ≤ n + 1= (2 n + 3 − x ) / ( n −
2) if n + 1 ≤ x ≤ n + 3 / e HK ( R G , I ) = 2 − / n . Example 7.6.
Let G = E the tetrahedral group then | G | = 24 and char k = p ≥ R G = k [ h , h , h ] = k [ x , x , x ](6 ax − x + x ) , where a = 2 √− h = x x − x x , h = x + ax x + x , h = x − ax x + x . Let ψ be given by the matrix (cid:20) x − ( a/ x x − x ( a/ x x − x x x + ( a/ x x x − ( a/ x x (cid:21) Then I ( ψ ) = ( ah , h , h ) R . If a = 0 in k then the canonical sequence0 −→ R ( − ⊕ R ( − ψ −→ R ( − ⊕ R ( − ⊕ R ( − φ −→ IR −→ R -resolution of IR .Here B (4) = − B (6) = − B (7) = 2 and the HK density function f S,I is givenby f S,I ( x ) = x/ ≤ x ≤
2= (4 − x ) / ≤ x ≤
3= (7 − x ) / ≤ x ≤ /
2= 0 otherwise
Example 7.7.
Let G = E octahedral group then | G | = 24 and char k ≥ R G = k [ h , h , h ] = k [ x , x , x ](108 x − x + x ) , where h = x x − x x , h = x + 14 x x + x , h = x − x x ) − x x ) + x . Let ψ be given by the matrix (cid:20) − x x − x x x x x + x x x (cid:21) If char k > h , h , h ) R = I ( ψ ) and hence the minimal R -resolution for IR is given by0 −→ R ( − ⊕ R ( − ψ −→ R ( − ⊕ R ( − ⊕ R ( − φ −→ IR −→ . Here B (6) = − B (8) = − B (12) = − B (13) = 2. Hence the HK density of( S, I ) is given by f S,I ( x ) = x/
48 if 0 ≤ x ≤
6= 6 /
48 if 6 ≤ x ≤
8= (14 − x ) /
48 if 8 ≤ x ≤
12= (26 − x ) /
48 if 12 ≤ x ≤
13= 0 otherwise . and hence e HK ( R G , I ) = 2 − (1 / Example 7.8.
Let G = E the icosahedral group then | G | = 120 and char k ≥
7. Now R G = k [ h , h , h ] = k [ x , x , x ]( x + x − x ) , where ILBERT-KUNZ DENSITY FUNCTION FOR GRADED DOMAINS 21 h = x x ( x + 11 x x − x ) h = x + x + 522( x x − x x ) − x x + x x ) h = − x − x + 228( x x − x x ) − x x ) . Let ψ be given by the matrix (cid:20) x f f x g g (cid:21) where f = − x − (11 / x x . f = x + ax x + ( b/ x x g = − x + (11 / x x g = − x + ax x − ( b/ x x and where a = 228 and b = 494.In particular ( h , h , h ) R = I ( ψ )Hence the minimal R -resolution for IR is given by0 −→ R ( − ⊕ R ( − ψ −→ R ( − ⊕ R ( − ⊕ R ( − −→ IR −→ . B (12) = − B (20) = − B (30) = − B (31) = 2.Hence the HK density of ( S, I ) is given by f S,I ( x ) = x/
30 if 0 ≤ x ≤
6= 6 /
30 if 6 ≤ x ≤
10= (16 − x ) /
30 if 10 ≤ x ≤
15= (31 − x ) /
30 if 15 ≤ x ≤ /
2= 0 otherwiseand hence e HK ( R G , I ) = 2 − (1 / Remark 7.9.
If (
R, I ) is a two dimensional graded pair then its HK density function f R,I is an explicit piecewise linear polynomial with rational coefficients and rationalbreak points (the proof follows from the same arguments as in [TW]):Let f , . . . , f s be a set of homogeneous generators of I of degrees d , . . . , d s . Let e S denote the normalization of R ( n ) = ⊕ n ≥ R nn , where gcd { m | R m = 0 } = n . Let X = Proj( e S ), Then for the Q -Weil divisor D (which is Cartier in this case)corresponding to the normal ring e S (as in Theorem 3.1) the sheaf O n = O X ( D ) isinvertible. Hence the sequence (3.1) is0 −→ F n ∗ V ⊗ O m −→ ⊕ i O m + q − qd i φ m,q −→ O m + q −→ , where 0 −→ V −→ ⊕ i O − d i φ −→ O −→ , where φ ( x , . . . , x s ) = P x i f i . This gives f R,I ( x ) = f V, O ( x ) − f ⊕ i O − di , O ( x ) , for x ≥ , where, for a vector bundle E on X with strong HN data ( { a , . . . , a l +1 } , { r , . . . , r l +1 } )and d = deg O , the function f E, O denotes the HK density function of E with respectto O and is given by x < − a /d = ⇒ f E, O ( x ) = − hP l +1 i =1 a i r i + d ( x − r i i − a i /d ≤ x < − a i +1 /d = ⇒ f E, O ( x ) = − hP l +1 k = i +1 a k r k + d ( x − r k i . References [BH] W. Bruns and J. Herzog,
Cohen-Macaulay rings , Cambridge Stud. Adv. Math., Vol 39, Cam-bridge University Press, Cambridge, 1993.[D] M. Demazure,
Anneaux gradu´es normaux , in Seminaire Demazure-Giraud-Teissier, Singularitiesdes surfaces, Ecole Polytechnique, 1979.[H1] R. Hartshorne,,
Algebraic geometry , Springer 1977.[H1] R. Hartshorne,,
Stable reflexive sheaves , Math. Ann. (1980), no. 2, 121-176.[K] K. Kurano,
The singular Riemann-Roch theorem and Hilbert-Kunz functions , J. Algebra, 304,487-499.[MBD] G.A. Miller, H.F. Blichfeldt and L.E. Dickson,
Theory and applications of finite groups ,Reprint of 1916 Edition with corrections, New York, G.F Stechert & Co., 1938.[M] P. Monsky,
The Hilbert-Kunz function , Math. Ann. 263 (1983) 43-49.[PS] C. Peskine and L. Szpiro,
Dimension projective finie et cohomologie locale , Publ. Math. I.H.E.S.42 (1972), 47-119.[S] K. Schwede,
Generalised divisors and reflexive sheaves , Homepage.[T] V. Trivedi,
Hilbert-Kunz Density Function and Hilbert-Kunz Multiplicity , Trans. Amer. Math. Soc.370 (2018), no. 12, 8403-8428.[TW] V. Trivedi and K.I. Watanabe,
Hilbert-Kunz Density Functions and F -thresholds [WY] K.I. Watanabe and K.I. Yoshida Hilbert-Kunz multiplicity and an inequality between multiplicityand colength , J. Algebra, (2000), 295-317
School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road,Mumbai-40005, India
E-mail address : [email protected] Department of Mathematics, College of Humanities and Sciences, Nihon University,Setagaya-Ku, Tokyo 156-0045, Japan
E-mail address ::