Towards Hilbert-Kunz density functions in Characteristic $0$

0.23 MB·

by  Vijaylaxmi Trivedi

TOWARDS HILBERT-KUNZ DENSITY FUNCTIONS IN
CHARACTERISTIC 0
V.TRIVEDI
6
1
0 Abstract. For a pair (R,I), where R is a standard graded domain over an algebraically
2 closed field of characteristic 0 and I is a graded ideal with ℓ(R/I) < ∞, we prove that, as
Jan opthf→elicm∞opn→,vet∞hrgeeecnHocKnev(oeRfrgpfe,mpnI(pcRe)pois,fItephq)eu,iHfvoaKrleadnnetyntsfioixttyehdfeumnexc≥tisiotdeninmfcepP(orRofpjl,iRmIp.p)→T(hi∞nisLℓi(m∞Rppnl/ioeIrsp[mptmh)a]i)ts/tephqmeudiev,xafilosertneatnnctyoe
suchfixedm.
8
In particular, in char 0, to define the HK density function f(R,I) (HK multiplicity

] eHK(R,I))itisenoughtoprovetheexistenceoflimp→∞fmp(Rp,Ip)(limp→∞ℓ(Rp/Ip[pm])/pmd
C respectively),foranyfixedm≥dim ProjR.
A
.
h
t
a 1. Introduction
m
LetRbeaNoetherianringofprimecharacteristicp>0andofdimensiondandletI ⊆Rbe
[
anidealof finite colength. Then we recallthat the Hilbert-Kunz multiplicity of R with respect

1 to I is defined as
v ℓ(R/I[pn])
e (R,I)= lim ,
5 HK n→∞ pnd
7
7 whereI[pn] =n-thFrobeniuspowerofI =idealgeneratedbypn-thpowerofelementsofI. This
1 is an ideal of finite colength and ℓ(R/I[pn]) denotes the length of the R-module R/I[pn]. This
0 invariant had been introduced by E. Kunz and existence of the limit was proved by Monsky
.
1 [M]. It carries information about char p related properties of the ring, but at the same time is
0 difficulttocompute(eveninthe gradedcase)asvariousstandardtechniques,usedforstudying
6 multiplicities, are not applicable for the invariante . It is naturalto ask if the notion of this
HK
1
invariant can be extended to the ‘char 0’ case. A natural way to attempt this for a pair (R,I)
:
v (from now onwards,unless stated otherwise, by a pair (R,I), we mean R is a standard graded
i ring and I ⊂R is a graded ideal of finite colength) could be as follows: Suppose R is a finitely
X
generatedalgebraand a domain overa field k of characteristic 0 and I ⊆R is anideal of finite
r
a colength. Let (A,RA,IA) be a spread of the pair (R,I) (see Definition 3.2), where A ⊂ k is a
finitely generated algebra over Z. Then we may (tentatively) define
e∞ (R,I):= lim e (R ,I ),
HK HK s s
s→s0
where R =R ⊗ k¯(s) and I =I ⊗ k¯(s), s = the generic point of Spec A and s∈ closed
s A A s A A 0
points of Spec A. Or consider a simpler situation: R is a finitely generated Z-algebra and a
domain, I ⊂R such that R/I is an abelian group of finite rank then let
e∞HK(R,I):= lim eHK(Rp,Ip), where Rp =R⊗ZZ/pZ and Ip =I⊗ZZ/pZ.
p→∞
In case of dimension R = 1, we know that the Hilbert-Kunz multiplicity coincides with the
Hilbert-Samuel multiplicity hence it is independent of p, for large p.
For homogeneous coordinate rings of plane curves, with respect to the maximal graded
ideal (in [T1], [M]), nonsingular curves with respect to a graded ideal I (in [T2]), diagonal
hypersurfaces(in [GM]), ithas beenshownthate (R ,I )varieswith p andthe limit exists.
HK p p
2010 Mathematics Subject Classification. 13D40,14H60,14J60,13H15.
Key words and phrases. Hilbert-Kunzdensity,Hilbert-Kunzmultiplicity,characteristic0.
1
2 V.TRIVEDI
Moreoverusingtheabovementionedresultof[T2]andProposition2.17of[T4],itcanbeproved
that, for a Segre product of any finite number of projective curves, such a limit exists. Then
there are other cases, where e (R ,I ) is independent of p and therefore the limit exists.
HK p p
Since
ℓ(R /I[pn])
e∞ (R,I):= lim lim p p ,
HK p→∞n→∞ (pn)d
it seems harder to compute as such, as lim ℓ(R /I[pn])/(pn)d itself does not seem easily
n→∞ p p
computable (evenin the gradedcase). To makethis invariantmore approachablethe following
question was posed in [BLM]:
Question. Suppose e∞ (R,I) exists, is it true that for any fixed n≥1
HK
ℓ(R /I[pn])
e∞ (R,I)= lim p p ?
HK p→∞ (pn)d
Themainresultoftheirpaperwastoangiveaffirmativeanswerinthecaseofa2dimensional
standard graded normal domain R with respect to a homogeneous ideal I. Note that the
existence of e∞ (R,I), in this case, was proved earlier in [T2].
HK
The proof goes as follows: Recall that for a vector bundle V on a smooth (projective and
polarized) variety, we have the well defined HN data, namely {r (V),µ (V)} , where r (V) =
i i i i
rank(E /E ) and µ (V)=slope of E /E and
i i−1 i i i−1
0⊂E ⊂E ⊂···⊂E ⊂V
1 2 l
is the Harder-Narasimhanfiltration of V.
Let X = Proj R , which is a nonsingular projective curve, and let I be generated by
p p p
homogeneouselements of degreesd ,...,d then we have the vector bundle V on X givenby
1 µ p p
the following canonical exact sequence of O -modules
Xp
0−→V −→⊕ O (1−d )−→O (1)−→0.
p i Xp i Xp
Then,byProposition1.16in[T2],thereisaconstantdeterminedbygenusofX andrankV ,
p p
such that for s≥1
(1.1) r (Fs∗V )µ (Fs∗V )2− r (V )µ (V )2 ≤C/p.
j p j p i p i p
(cid:12) (cid:12)
(cid:12) j i (cid:12)
(cid:12)X X (cid:12)
(cid:12) (cid:12)
(Here F is the absolu(cid:12)te Frobenius morphism, and Fs is the s-fold(cid:12)iterate.) Note that the HN
(cid:12) (cid:12)
filtration and hence the HN data of V stabilizes for p>>0 ([Mar]).
p
On the other hand, the methods used in [B] and [T1] (relating e and the HN data of
HK
Fs∗V) imply that, for s≥1,
1 R deg X deg X
ℓ p = p r (Fs∗V )µ (Fs∗V )2 − p d2 +O(1/p).
(ps)2 Ip[ps]! 2  j j p i p  2 i i!
X X
 
Hence, for any s≥1, by Equation (1.1)
1 R deg X deg X
ℓ p = p r (V )µ (V )2 − p d2 +O(1/p),
(ps)2 Ip[ps]! 2 i i p i p ! 2 i i!
X X
which implies
deg X deg X
e (R ,I )= p r (V )µ (V )2 − p d2 +O(1/p).
HK p p 2 i p i p 2 i
! !
i i
X X
TOWARDS HILBERT-KUNZ DENSITY FUNCTIONS IN CHARACTERISTIC 0 3
Also, here the ‘error term’ |O(1/p)|≤C /p , where C depends on the invariants like genus of
2 2
X and rank V . Hence
p p
1 R
p
lim e (R ,I )= lim ℓ , for any s≥1.
p→∞ HK p p p→∞(ps)2 Ip[ps]!
Thus here
(1) one relates ℓ(R /I[ps]) with the HN data of Fs∗V , for s≥1 ([B] and [T1]).
p p p
(2) The HN data of Fs∗V is related to the HN data of V ([T2]).
p p
(3) The restriction of the relative HN filtration of V on X (where V is a spread of V
A A A 0
in char 0) remains the HN filtration of V for large p ([Mar]).
p
In particular for a pair (R,I), where char R = p > 0, with the associated syzygy bundle
V (as above), the proof uses the comparison of ℓ(R/I[ps]) with the HN data of the syzygy
bundle V andthe otherwell behavedinvariantsof(R, I) (which havewelldefined notioninall
characteristics and are well behaved vis-a-vis reduction mod p).
However note that (3) is valid for dim R ≥ 2, and (2) also holds for dim R ≥ 3 (proved
relatively recently in [T3]). But (1) does not seem to hold in higher dimension, due to the
existence of cohomologies other than H0(−) and H1(−) (therefore one can not use anymore
the semistability property of a vector bundle to compute h0 of almost all its twists, by a very
ample line bundle).
Inthispaper,weapproachtheproblembycomparing 1 ℓ(R/I[pn])and 1 ℓ(R/I[pn+1])
(pn)d (pn+1)d
directly,forn≥1,takingintoaccountthatbotharegraded. Forthis wephrasetheproblemin
a more general setting: By the theory of Hilbert-Kunz density function (which was introduced
and developed in [T4]), for a pair (R,I) where R is a domain of char p > 0, there exists a
sequence of functions {fp :[0,∞)−→R} such that
n n
1 ∞ 1 ∞
ℓ(R/I[pn])= fp(x)dx and lim ℓ(R/I[pn])= fp(x)dx,
(pn)d n n→∞(pn)d
Z0 Z0
where the map fp : [0,∞) → R is given by fp(x) = lim fp(x) is called the HK density
n→∞ n
function of (R ,I ) (the existence and properties of the limit defining fp are proved in [T4]).
p p
We show here that, for all x∈[1,∞),
if lim fp(x):= lim lim fp(x) exists then lim fp(x)= lim fp(x),
n m
p→∞ p→∞n→∞ p→∞ p→∞
for any fixed m≥d−1, where d−1=dimProj R.
The main point (Proposition 2.11) is to give a bound on the difference kfp − fp k, in
n n+1
terms of a power of p and invariants which are well behaved under reduction mod p, where
kgk := sup{g(x) | x ∈ [1,∞)} is the L∞ norm. Since the union of the support of all f
n
is contained in a compact interval, a similar bound (Corollary 2.12) holds for the difference
|ℓ(R/I[pn])/(pn)d−ℓ(R/I[pn+1])/(pn+1)d|. More precisely we prove the following
Theorem 1.1. Let R be a standard graded domain of dimension d ≥ 2, over an algebraically
closed field k of characteristic 0. Let I ⊂ R be a homogeneous ideal of finite colength. Let
(A,R ,I ) be a spread (see Definition 3.2). Then, for a closed point s ∈ Spec(A), let the
A A
function
1 R
fs(x):[1,∞)−→[0,∞) be given by fs(x)= ℓ s .
n n qd−1 I[q]
(cid:18) s (cid:19)⌊xq⌋
Let the HK density function of (R ,I ) be given by
s s
fs(x)= lim fs(x).
n
n→∞
Let s ∈SpecQ(A) denote the generic point of Spec(A). Then
0
(1) there exists a constant C (given in terms of invariants of (R ,I )) and an open dense
s0 s0
subset Spec(A′) of Spec(A) such that for every closed point s∈Spec(A′) and n≥1,
kfs−fs k<C/pn−d+2,
n n+1
4 V.TRIVEDI
where p=char k(s). In particular, for any m≥d−1,
lim kfs −fsk=0.
m
s→s0
(2) There exists a constant C (given in terms of invariants of (R ,I )) and an open
1 s0 s0
dense subset Spec(A′) of Spec(A), such that for every closed point s ∈ Spec(A′) and
n≥1, we have
1 R 1 R C
s s 1
ℓ − ℓ ≤ .
pnd I[pn] p(n+1)d I[pn+1] pn−d+2
(cid:12) (cid:18) s (cid:19) (cid:18) s (cid:19)(cid:12)
(cid:12) (cid:12)
(3) For any m≥d(cid:12)−1, (cid:12)
(cid:12) (cid:12)
1 R
s
lim ℓ −e (R ,I ) =0.
s→s0(cid:20)pmd (cid:18)Is[pm](cid:19) HK s s (cid:21)
As a result we have
Corollary 1.2. Let R be a standard graded domain and a finitely generated Z-algebra, let
I ⊂ R be a homogeneous ideal of finite colength. such that for almost all p, the fiber over p,
Rp :=R⊗ZZ/pZ is a standard graded ring of dimension d, which is geometrically integral, and
I ⊂R is a homogenous ideal of finite colength. Then
p p
(1) there exists a constant C given in terms of invariants of R and I such that, for n≥1,
1
we have
1 R 1 R C
p p 1
ℓ − ℓ ≤ .
(cid:12)(cid:12)pnd Ip[pn]! p(n+1)d Ip[pn+1]!(cid:12)(cid:12) pn−d+2
(cid:12) (cid:12)
(2) For any fixed m(cid:12) ≥d−1, (cid:12)
(cid:12) (cid:12)
1 R
p
lim e (R ,I )− ℓ =0.
p→∞" HK p p pmd Ip[pm]!#
In particular, for any fixed m≥d−1,
1 R
e∞ (R,I):= lim e (R ,I ) exists ⇐⇒ lim ℓ p exists.
HK p→∞ HK p p p→∞pmd Ip[pm]!
In particular the last assertionof the abovecorollaryanswers the above mentioned question
of [BLM] affirmatively, for all (R,I), where R is a standard graded domain and I ⊂ R is a
graded ideal of finite colength.
Moreovertheproof,eveninthecaseofdimension2(unliketheproofin[BLM])doesnotrely
on earlier results of [B], [T1] and [T2]. In particular, since we do not use Harder-Narasimhan
filtrations, we do not need a normality hypothesis on the ring R.
Remark 1.3. If e∞ (R,I) exists for a pair (R,I), whenever R is a standard graded domain,
HK
defined overanalgebraicallyclosedfield of characteristic0, then one cancheck that e∞ (R,I)
HK
exists for any pair (R,I) where R is a standard graded ring over a field k of characteristic 0:
Let R¯ = R⊗ k¯. Let {q ,...,q } = {q ∈ Ass(R¯) | dim R¯/q = dim R} then we have a spread
k 1 r
(A,R¯ ,I¯ ) of (R¯,I¯) such that {q ,...,q }={q ∈Ass(R¯ )|dim R¯ /q =dim R¯ } and, for
A A 1s rs s s s s s
each i, ℓ((R¯ )q )=l , a constant independent of s. This implies that
s is i
r R¯ I¯ +q
e (R¯ ,I¯)= l e s, s is ,
HK s s i HK
q q
i=1 (cid:18) is is (cid:19)
X
which implies
r R¯ I¯ +q ) r R¯ I¯+q
lim e (R¯ ,I¯)= l lim e s, s is = l e∞ , i .
s→s0 HK s s i=1 is→s0 HK(cid:18)qis qis (cid:19) i=1 i HK(cid:18)qi qi (cid:19)
X X
TOWARDS HILBERT-KUNZ DENSITY FUNCTIONS IN CHARACTERISTIC 0 5
Hence, in this situation, one can define
r R¯ I¯+q
e∞ (R,I):=e∞ (R¯,I¯)= l e∞ , i .
HK HK i HK q q
i=1 (cid:18) i i (cid:19)
X
2. A key proposition
Throughout this section, R is a Noetherian standard graded integral domain of dimension
d over an algebraically closed field k of char p > 0, I is a homogeneous ideal of R such that
ℓ(R/I) < ∞. Let h ,··· ,h be a set of homogeneous generators of I of degrees d ,...,d
1 µ 1 µ
respectively.
Let X = Proj R; then we have an associated canonical short exact sequence of locally free
sheaves of O -modules (moreover the sequence is locally split exact)
X
(2.1) 0−→V −→⊕ O (1−d )−→O (1)−→0,
i X i X
where O (1−d )−→O (1) is given by the multiplication by the element h .
X i X i
For a coherent sheaf Q of O -modules, the sequence of O -modules
X X
0−→Fn∗V ⊗Q(m)−→⊕ Q(q−qd +m)−→Q(q+m)−→0
i i
is exact as the short exact sequence (2.1) is locally split as O -modules (as usual, q =pn and
X
Fn is the nth iterate of the absolute Frobenius morphism). Therefore we have a long exact
sequence of cohomologies
(2.2) 0−→H0(X,Fn∗V ⊗Q(m))−→⊕ H0(X,Q(q−qd +m))φm→,q(Q)H0(X,Q(q+m))
i i
−→H1(X,Fn∗V ⊗Q(m))−→··· ,
for m≥0 and q =pn.
Definition 2.1. Let Q be a coherentsheafofO -modulesandlet O (1)be a veryample line
X X
bundle on X. We say that Q is m-regular (or m is a regularity number of Q) with respect to
O (1), if for all m≥m
X
(1) the canonicalmultiplicatieon map H0(Xe,Q(m))⊗H0(X,O (1))−→H0(X,Q(m+1))
X
is surjective aned
(2) Hi(X,Q(m−i))=0, for i≥1.
Notations 2.2. Let
m+d−1 m+d−2
P(R,m)(m)=e0 d −e1 d−1 +···+(−1)ded
(cid:18) (cid:19) (cid:18) (cid:19)
be the Hilbert-SamuelpolynomialofR withrespectto the gradedmaximalideal m. Therefore
e e e
m+d−1 m+d−2
χ(X,O (m))=e −e +···+(−1)d−1e .
X 0 1 d−1
d−1 d−2
(cid:18) (cid:19) (cid:18) (cid:19)
Let m¯ be a positive integer such that
e e e
(1) m¯ is a regularity number for (X,O (1)), and
X
(2) Rm = h0(X,OX(m)), for all m ≥ m¯. In particular ℓ(R/mm) = P(R,m)(m), for all
m≥m¯.
Let l =h0(X,O (1)) and let n ≥1 be an integer such that R ⊆I.
1 X 0 n0
We also denote dim Coker φ (Q) by coker φ (Q).
k m,q m,q
Remark 2.3. (1) The canonical map ⊕ R −→ ⊕ H0(X,O (m)) is injective, as R is
m m m X
an integral domain.
6 V.TRIVEDI
(2) For m+q ≥ m (q) = m¯ +n ( d )q, we have coker φ (O ) = ℓ(R/I[q]) = 0:
R 0 i i m,q X m+q
Becausem (q)=m¯+n µq+n ( (d −1))q =⇒ q−qd +m≥m¯,foralli. Hencethe
R 0 0P i i i
mapφ (O )isthemap⊕ R −→R ,wherethemapR →R is
givenbmy,qmuXltiplication by thieqe−leqPmdi+enmt hq. Thme+reqfore,cokerφ (Oq−)qd=i+ℓm(R/I[q]m)+q .
i m,q X m+q
Moreover,by Lemma 2.10, we have ℓ(R/I[q]) =0, as m+q ≥m¯ +n µq.
m+q 0
(3) For C =(µ)h0(X,O (m¯)), we have
R X
(2.3) |coker φ (O )−ℓ(R/I[q]) |≤C ,
m,q X m+q R
for all n,m≥0 and q =pn: Because
if m+q <m¯, then
|coker φ (O )−ℓ(R/I[q]) |≤h0(X,O (m+q))≤h0(X,O (m¯)).
m,q X m+q X X
If m+q ≥m¯, then h0(X,O (m+q))=ℓ(R ) and therefore
X m+q
µ
|coker φ (O )−ℓ(R/I[q]) |≤ |h0(X,O (q−qd +m))−ℓ(R )|.
m,q X m+q X i q−qdi+m
i
X
Now, ifq−qd +m<m¯ then ℓ(R )≤h0(X,O (q−qd +m))≤h0(X,O (m¯),
i q−qdi+m X i X
and if q−qd +m≥m¯ then R =H0(X,O (q−qd +m)).
i q−qdi+m X i
Hence
|coker φ (O )−ℓ(R/I[q]) |≤µh0(X,O (m¯)).
m,q X m+q X
Definition 2.4. Let Q be a coherent sheaf of O -modules of dimension d¯and let m ≥ 1 be
X
the least integer which is a regularity number for Q with respect to O (1). Then we define
X
C0(Q) and D0(Q) as follows: Let a1,...,ad¯ ∈ H0(X,OX(1)) be such that we havee a short
exact sequence of O -modules
X
0→Q (−1)→ai Q →Q →0, for 0<i≤d¯,
i i i−1
where Qd =Q and Qi =Q/(ad¯,...,ai+1)Q, for 0≤i<d¯, with dim Qi =i. We define
d¯
C0(Q)=min{ h0(X,Qi)|a1,...,ad¯is a Q−sequence as above},
i=0
X
D0(Q)=h0(X,Q(m))+2(d¯+1)(max{q0,q1,...,qd¯}),
where
χ(X,Q(m))=q0 md+¯ed¯ −q1 md+¯−d¯1−1 +···+(−1)d¯qd¯
(cid:18) (cid:19) (cid:18) (cid:19)
is the Hilbert polynomial of Q.
Lemma 2.5. Let Q be a coherent sheaf of O -modules of dimension d¯. Let P be a locally-free
X
sheafofO -modules whichfitsintoashortexactsequenceoflocally-free sheaves ofO -modules
X X
(2.4) 0−→P −→⊕ O (−b )−→P′′ −→0, where b ≥0.
i X i i
Then, for µ=rk(P)+rk(P′′) and, for all n,m≥0, we have
h0(X,Q(m+q))≤D (Q)(m+q)d¯ and h0(Fn∗P ⊗Q(m))≤(µ)C (Q)(md¯+1).
0 0
e
Proof. Let m be a regularity number for Q, then by Definition 2.4, we have
e
h0(X,Q(m+q))≤D (Q)(m+q)d¯, for all n,m≥0.
0
e
Let Qd¯=Q. Let ad¯,...,a1 ∈H0(X,OX(1)) with the exact sequence of OX-modules
0−→Q (−1)−a→i Q −→Q −→0,
i i i−1
where Qi = Qd¯/(ad¯,...,ai+1)Qd¯, for 0 ≤i ≤ d¯, and realizing the minimal value C0(Q). Now,
by the exact sequence (2.4), we have the following short exact sequence of O -sheaves
X
0−→Fn∗P ⊗Q −→⊕ Q (−qb )−→Fn∗P′′⊗Q −→0.
i j i j i
TOWARDS HILBERT-KUNZ DENSITY FUNCTIONS IN CHARACTERISTIC 0 7
This implies H0(X,Fn∗P ⊗Q )֒→⊕ H0(X,Q (−qb )). Therefore
i j i j
(2.5) h0(X,Fn∗P ⊗Q )≤ h0(X,Q (−qb ))≤(µ)h0(X,Q ),
i i j i
j
X
as −b ≤0. Since Fn∗P is a locally-free sheaf of O -modules, wee have
j X
0−→Fn∗P ⊗Q (m−1)−a→i Fn∗P ⊗Q (m)−→Fn∗P ⊗Q (m)−→0,
i i i−1
which is a short exact sequence of O -sheaves. Now by induction on i, we prove that, for
X
m≥1,
h0(X,Fn∗P ⊗Q (m))≤(µ) h0(X,Q )+···+h0(X,Q ) (mi).
i i 0
For i=0, the inequality holds as h0(X,Fn∗P ⊗Q (m))≤(µ)h0(X,Q ) (as dim Q =0).
(cid:2) 0 0 (cid:3) 0
Now, for m≥1, by the inequality 2.5 aend by induction on i, we have
h0(X,Fn∗P⊗Q (m))≤h0(X,Fn∗P⊗Q )+h0(X,Fn∗P⊗Q e(1))+···+h0(X,Fn∗P⊗Q (m))
i i i−1 i−1
≤(µ)h0(X,Q )+µ h0(X,Q )+···+h0(X,Q ) (1+2i−1+···+mi−1)
i i−1 0
≤(µ) h0(X,Qi(cid:2))+···+h0(X,Q0) mi. (cid:3)
e e
This implies
(cid:2) (cid:3)
h0(X,Fn∗eP ⊗Q(m))=h0(X,Fn∗P ⊗Qd¯(m))≤µC0(Q)md¯,
for all m ≥ 1. Therefore, for all 0 ≤ i ≤ d¯, h0(X,Fn∗P ⊗Q(m)) ≤ µC (Q)(md¯+1), for all
0
m≥0. This proves the lemma. e (cid:3)
e
Lemma 2.6. There exists a short exact sequence of coherent sheaves of O -modules
X
0−→⊕pd−1O (−d)−→F O −→Q−→0,
X ∗ X
where Q is a coherent sheaf of O -modules such that dim supp(Q)<d−1.
X
Proof. Note that X =Proj R, where R=⊕ R , is a standard graded domain such that R
n≥0 n 0
is an algebraically closed field. Therefore there exists a Noether normalization
k[X ,...,X ]−→R,
0 d−1
which is an injective, finite separable graded map of degree 0 (as k is an algebraically closed
field). This induces a finite separable affine map π :X −→Pd−1 =S.
k
Note that there is also an isomorphism
η :OS⊕n0 ⊕OS(−1)⊕n1 ⊕···⊕OS(−d+1)⊕nd−1 −→F∗OS
of O -modules, where n =pd−1.
S i
Now the isomorphismPof η implies that the map π∗(η) : ⊕di=−01OX(−i)⊕ni −→π∗F∗Os is an
isomorphism of O -sheaves. Since 0 ≤ i ≤ d−1, we also have an injective and generically
X
isomorphic map of O -sheaves
X
⊕pd−1O (−d)−→⊕d−1O (−i)⊕ni.
X i=0 X
Composingthismapwithπ∗(η)givesaninjectiveandgenericallyisomorphicmapofO -sheaves
X
α:⊕pd−1O (−d)−→π∗F O .
X ∗ S
Since π is separable, there is a canonical map β : π∗F O −→ F O , of sheaves of O -
∗ S ∗ X X
modules, which is generically isomorphic.
Now we have the composite map
β◦α:⊕pd−1O (−d)−→π∗F O →F O
X ∗ S ∗ X
which is generically an isomorphism. Hence dimCoker(β◦α) < dim X = d−1 and the map
β◦α : ⊕pd−1O (−d) −→ F O is injective, as X is an integral scheme. This proves the
X ∗ X
lemma. (cid:3)
8 V.TRIVEDI
Lemma 2.7. Let
0−→⊕pd−1O (−d)−→F O −→Q−→0
X ∗ X
as in the Lemma 2.6. Then
(1) Q is m-regular, where m = max{m¯ +d,l −1}, where m¯ and l are as given in Nota-
1 1
tions 2.2.
(2) For aegiven d, there exeists a universal polynomial function P¯1d(X0,...,Xd−1,Y) with
rational coefficients (and hence independent of p) such that
2C (Q)+D (Q)≤pd−1P¯d(e ,e ,...,e ,m¯).
0 0 1 0 1 d−1
Proof. (1) The above short exact sequence of O -sheaves gives a long exact sequence of coho-
X
e e e
mologies
⊕pd−1Hi(X,O (m−d))−→Hi(X,O (mp))−→Hi(X,Q(m))−→⊕pd−1Hi+1(X,O (m−d)).
X X X
But hi(X,O (m−d−i))=0, for all m ≥m¯ +d and i≥1, which implies that if m ≥m¯ +d
X
then hi(X,Q(m−i))=0, for i≥1, and the map
f :H0(X,(F O )(m))−→H0(X,Q(m))
1,m ∗ X
is surjective. Moreover the canonical map
H0(X,(F O )(m))⊗H0(X,O (1))−→H0(X,(F O )(m+1))
∗ X X ∗ X
is
f :H0(X,O (mp))⊗H0(X,O (1))[p] −→H0(X,O (mp+p)),
2,m X X X
is surjective for m≥m because it fits into the following canonical diagram
R ⊗R[p] −→ R
mp 1 mp+p
e
↓ ↓
H0(X,O (mp))⊗H0(X,O (1))[p] f−2→,m H0(X,O (mp+p))
X X X
where the top horizontal map is surjective for m ≥ l −1. Now the following commutative
1
diagram of canonical maps
H0(X,(F O )(m))⊗H0(X,O (1)) −→ H0(X,Q(m))⊗H0(X,O (1))
∗ X X X
↓f ↓
2,m
H0(X,(F O )(m+1)) f1−,m→+1 H0(X,Q(m+1))
∗ X
implies that the second vertical map is surjective, for m ≥ m, as the maps f and f
2,m 1,m+1
surjective. This proves that Q is m-regular. Hence the assertion (1).
(2) If e
e
m+d−2 m+d−3
(2.6) χ(X,Q(m))=q −q +···+(−1)d−2q ,
0 1 d−2
d−2 d−3
(cid:18) (cid:19) (cid:18) (cid:19)
then by Lemma 4.1, (in the Appendix, below)
|q |≤pd−1Pd(e ,...,e ),
i i 0 i+1
where Pd(X ,...,X ) is a universal polynomial function with rational coefficients.
i 0 i+1
Now, Q is m-regular implies that, for 0 ≤ i <ed, Qie:= Q/(ad¯,...,ai+1)Q is m-regular, for
any Q-sequence a1,...,ad¯∈H0(X,OX(1)). Therefore
h0(X,Q )e ≤ h0(X,Q (m))≤h0(X,Q (m))≤···≤h0(X,Q(m))=χ(Xe,Q(m))
i i i+1
m+d−2 m+d−3
≤ |q | +|q | +···+|q |.
0 1 d−2
d−e2 d−e3 e e
(cid:18) (cid:19) (cid:18) (cid:19)
This implies h0(X,Q ) ≤epd−1Pd(e ,...,e e,m), where Pd(X ,...,X ,Y) is a universal
i 0 d−1 0 d−1
polynomialfunctionwithrationalcoefficients. ThereforeC (Q)≤(d−1)pd−1Pd(e ,...,e ,m).
0 0 d−1
The inequality for D0(Q) followsesimilarely. Tehis proves the assertion (2) and hence the
lemma. (cid:3)
e e e
TOWARDS HILBERT-KUNZ DENSITY FUNCTIONS IN CHARACTERISTIC 0 9
Lemma 2.8. Let m ≥ 0 and n ≥ 0 be two integers. Consider the short exact sequences of
0 2
O -modules
X
0−→O (−m )−→O −→Y −→0 and 0−→O −→O (n )−→Y −→0.
X 0 X 1 X X 2 2
Then, for given d, there exist universal polynomial functions P¯d(X ,...,X ,Y) and
2 0 d−1
P¯d(X ,...,X ,Y), with rational coefficients such that
3 0 d−1
2C (Y )+D (Y )≤md−1P¯d(e ,...,e ,m¯),
0 1 0 1 0 2 0 d−1
2C (Y )+D (Y )≤nd−1P¯d(e ,...,e ,m¯).
0 2 0 2 2 3 0 d−1
e e
Proof. Without loss of generality one can assume m ≥1 and n ≥1. Since O is m¯-regular,
0 0 X
e e
the sheaf Y is m¯ +m -regularsheaf of O -modules ofdimension d−2. Therefore,for any Y -
1 0 X 1
sequencea ,...,a ,thesheafY :=Y /(a ,...,a )Y ism¯+m -regular,asO -modules.
1 d−2 1i 1 d−2 i+1 1 0 X
This implies
h0(X,Y ) ≤ h0(X,Y (m¯ +m ))≤h0(X,Y (m¯ +m ))≤h0(X,O (m¯ +m ))
1i 1i 0 1 0 X 0
m¯ +1+d−2 m+1+d−2
≤ md−1 |e | +|e | +···+|e |
0 0 d−1 1 d−2 d−1
(cid:20) (cid:18) (cid:19) (cid:18) (cid:19) (cid:21)
= m d−1Pd(e ,...,e ,m¯),
0 0 d−1
e e e
where Pd(X ,...,X ,Y) is a universal polynomial function with rational coefficients.
Let e (Y)0denoted−th1e ith ecoeefficienteof the Hilbert polynomial of (Y ,O (1)). Then by
i 1 X
Lemmae4.1, we have e (Y )≤mi+1Pd(e ,...,e ), where Pd(X ,...,X ) is a universal polyno-
i 1 0 i 0 i i 0 i
mial with rational coefficients.
Now the bound for 2C (Y )+D (Y ) follows. The identical proof follows for Y . (cid:3)
0 1 0 1e e 2
Notations 2.9. For a pair (R,I), where R is a standard graded ring of char p>0, we define
(similartothesequencewehaddefinedin[T4]),asequenceoffunctions{f :[1,∞)→[0,∞)} ,
n n
as follows: Fix n ∈ N and denote q = pn. Let x ∈ R then there exists a unique nonnegative
integer m such that (m+q)/q ≤x<(m+q+1)/q. We define
f (x)=1/qd−1ℓ(R/I[q]) .
n m+q
Lemma 2.10. Each f : [1,∞) −→ [0,∞), defined as in Notations 2.9, is a compactly sup-
n
ported function such that ∪ Supp f ⊆[1,n µ], where R ⊆I and µ=µ(I).
n≥1 n 0 n0
Proof. Since R is standard graded ring, for m ≥ n µq, we have R ⊆ (R )µq ⊆ Iµq ⊆ I[q].
0 m n0
This implies ℓ(R/I[q]) = 0, if m ≥ n µq. Therefore support f ⊆ [1,n µ], for every n ≥ 0.
m 0 n 0
This proves the lemma. (cid:3)
Proposition 2.11. For f as given in Notations 2.9, we have
n
(1) |f (x)−f (x)|≤C/pn−d+2, for every x∈[1,∞), and for all n≥0.
n n+1
(2) In particular, ||f −f ||≤C/pn−d+2 and ||f −f ||≤C/p,
n n+1 d−1 d
where
µ d−2
(2.7) C =C +µ m¯ +n ( d )+1 (P¯d+dd−1P¯d+P¯d)
R 0 i 1 2 3
!
i=1
X
and the integers m¯ and n are given as in Notations 2.2, and d ,...,d are degrees of a chosen
0 1 µ
generators of I. Moreover C = µh0(X,O (m¯)), for X = Proj R, and P¯d, P¯d and P¯d are
R X 1 2 3
given as in Lemma 2.7 and Lemma 2.8 respectively.
Proof. Fix x ∈ [1,∞). Therefore, for given q = pn, there exists a unique integer m ≥ 0, such
that (m+q)/q ≤x<(m+q+1)/q and
(m+q)p+n (m+q)p+n +1
2 2
≤x< , for some 0≤n <p.
2
qp qp
10 V.TRIVEDI
Hence
1 R 1 R
f (x)= ℓ and f (x)= ℓ .
n qd−1 I[q] n+1 (qp)d−1 I[qp]
(cid:18) (cid:19)m+q (cid:18) (cid:19)mp+qp+n2
Now, by Equation (2.3) in Remark 2.3, we have
coker φ (O ) C coker φ (O ) C
(2.8) f (x)− m,q X < R and f (x)− mp+n2,qp X < R .
n qd−1 qd−1 n+1 (qp)d−1 (qp)d−1
(cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
Conside(cid:12)r the short exact sequenc(cid:12)e of OX-modu(cid:12)les (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
0−→O (−d)−→O −→Y −→0.
X X 1
Then, for any locally free sheaf P of O -modules and for m ≥ 0, we have the following short
X
exact sequence of O -modules
X
0−→Fn∗P ⊗O (−d+m)−→Fn∗P ⊗O (m)−→Fn∗P ⊗Y (m)−→0.
X X 1
Since
coker φ (O )=h0(X,O (m+q))− h0(X,O (q−qd ))+h0(X,(Fn∗V)(m))
m,q X X X i
i
X
we have (by taking P =V and = O (1−d )) respectively),
X i
|coker φm,q(OXP)−coker φm−d,q(OX)|
≤h0(X,Y (m+q))+h0(X, O (q−qd )⊗Y (m))+h0(X,Fn∗V ⊗Y (m))
1 X i 1 1
i
X
which, by Lemma 2.5 is
≤(µ)D (Y )(m+q)d−2+2(µ)C (Y )(md−2+1)≤(µ)[2C (Y )+D (Y )](m+q)d−2.
0 1 0 1 0 1 0 1
Therefore
(2.9) |pd−1cokerφ (O )−pd−1cokerφ (O )|≤(µ)[2C (Y )+D (Y )](m+q)d−2pd−1.
m,q X m−d,q X 0 1 0 1
Since, for a locally free sheaf P, we have
h0(X,Fn∗P ⊗(F O )(m))=h0(X,F(n+1)∗P ⊗O (mp)),
∗ X X
the short exact sequence in the statement of Lemma 2.7 gives a canonical long exact sequence
0−→H0(X,(Fn∗P)(m−d))−→H0(X,(F(n+1)∗P)(mp))−→H0(X,Q(m))−→··· ,
which implies
(2.10) |pd−1coker φ (O )−coker φ (O )|≤(µ)[2C (Q)+D (Q)](m+q)d−2.
(m−d),q X mp,qp X 0 0
The short exact sequence of O -modules
X
0−→O −→O (n )−→Y −→0
X X 2 2
gives
0−→H0(X,(F(n+1)∗P)(mp))−→H0(X,(F(n+1)∗P)(mp+n ))−→H0(X,(F(n+1)∗P)⊗Y (mp)),
2 2
which gives
|coker φ (O )−coker φ (O )|
mp,qp X mp+n2,qp X
≤h0(X,F(n+1)∗V ⊗Y (mp))+h0(X, O (qp−qpd )⊗Y (mp))+h0(X,Y (mp+qp))
2 i X i 2 2
≤2(µ)C (Y )((mp)d−2+1)+(µ)D (YP)(mp+qp)d−2.
0 2 0 2
Therefore
(2.11) |coker φ (O )−coker φ (O )|≤(µ)[2C (Y )+D (Y )](mp+qp)d−2.
mp,qp X mp+n2,qp X 0 2 0 2