Notes on the Hilbert–Kunz function

a

ingacteri-.

lly

e

Journal of Algebra 265 (2003) 619–630

www.elsevier.com/locate/jalgebr

Notes on the Hilbert–Kunz function

Douglas Hanes

School of Mathematics, University of Minnesota, Minneapolis, MN 55455, USA

Received 5 February 2002

Communicated by Craig Huneke

Abstract

We note certain properties of the Hilbert–Kunz function and Hilbert–Kunz multiplicity, includa strengthened inequality between Hilbert–Kunz and Hilbert–Samuel multiplicities and a charzation of a mixed Hilbert function incorporating both ordinary and Frobenius powers of ideals 2003 Published by Elsevier Inc.

1. Introduction

In this paper we study functions associated to anm-primary idealI of a local Noetherianring (R,m) of Krull dimensiond . The Hilbert–Samuel functionHI :N → N is given byHI(t) = (R/I t ), where(M) represents the length of an ArtinianR-moduleM. It is wellknown from the work of Hilbert and Samuel thatHI(t) is polynomial of degreed ; in otherwords, there exists a polynomialPI of degreed such thatHI(t) = PI (t) for all sufficientlylarget . We define the (Samuel) multiplicitye(I ;R) of R with respect toI to bed! timesthe leading coefficient ofPI . Alternatively, the multiplicity may be defined asymptoticaas

e(I ;R)= limt→∞

d! · (R/I t )

td.

When the ringR is understood, we will sometimes just writee(I) for the multiplicity. Themultiplicity of R with respect to its maximal idealm is sometimes denoted bye(R).

If the ring R has positive prime characteristicp, then we may similarly define thHilbert–Kunzmultiplicity c(I ;R) of anm-primary idealI . For any idealI of R and any

E-mail address:[email protected].

0021-8693/03/$ – see front matter 2003 Published by Elsevier Inc.doi:10.1016/S0021-8693(03)00233-3

620 D. Hanes / Journal of Algebra 265 (2003) 619–630

t

Kunz

ys

s

en

e. The

,rs.

ed

integere > 0, we denote byI [pe] the ideal generated by allpe th powers of elements ofI .(Because of the characteristic, the ideal is in fact generated by thepe th powers of any seof generators forI .) TheHilbert–KunzfunctionHKI :N → N of the ringR with respectto them-primary idealI is then defined by

HKI (e)= (R/I [pe]).

The fundamental properties of the Hilbert–Kunz function have been developed by[7,8] and Monsky [10]. Kunz shows that, unlike the Hilbert–Samuel function,HKI(e) isnot necessarily polynomial inpe . However, the main result of Monsky’s paper [10] sathat the Hilbert–Kunz function nonetheless induces a multiplicityc(I ;R).

Theorem 1.1(Monsky [10, Theorem 1.8]).Let (R,m) be ad-dimensional local ring ofpositive prime characteristicp, and letI be anm-primary ideal ofR. Then there exista positive real numberc(I ;R), called the Hilbert–Kunz multiplicity ofR with respect toI ,such that

HKI(e) = c(I ;R)ped + O(pe(d−1)).

In particular, the limit

lime→∞

(R/I [pe])(pe)d

exists and is equal toc(I ;R).

As above, we will usually just writec(I) when the ring is understood orc(R) for theHilbert–Kunz multiplicity with respect to the maximal ideal.

Computation of Hilbert–Kunz multiplicities has proved quite difficult; it is not evknown whetherc(R) must always be a rational number. However, if(R,m) is ad-dimen-sional local ring of positive prime characteristicp, and if I is anym-primary ideal, thenthe multiplicitiese(I) andc(I) are related by the inequalities

e(I)/d! � c(I) � e(I). (1)

SinceI [q] ⊆ Iq for any q = pe , the first inequality follows from a comparison of thleading (asymptotic) coefficients of the Hilbert–Samuel and Hilbert–Kunz functionsinequalityc(I) � e(I) is obtained by enlarging the residue field ofR so as to know thatIpossesses a minimal reductionJ . Nowe(I) = e(J ), and it follows from a result of Lech [9Corollary to Theorem 2] thate(J )= c(J ), sinceJ is generated by a system of parameteThus, sinceJ ⊆ I , we see thate(I) = c(J )� c(I).

It has been shown that in any Noetherian local ring(R,m) there existm-primary idealsI for which the ratio ofe(I) to d! ·c(I) is arbitrarily close to one. In fact, it has been prov

D. Hanes / Journal of Algebra 265 (2003) 619–630 621

dt suchfs

tionsas the

oviden

nce in

een

independently by the author [4] and by Watanabe and Yoshida [13] that for anym-primaryidealI ,

limt→∞

(e(I t )

d! · c(I t ))

= 1.

However, the arguments do not show that equality betweene(I) andd!c(I) can be achievefor any idealI , and it has been proposed by Watanabe and Yoshida [12] that in facequality is not possible whend � 2. In the next section we will give two different prooof this statement, as well as some applications.

In the final section we introduce amixedHilbert function

HI(e, t) =

(R

(I [pe])t

)

associated to anm-primary idealI of a local ring (R,m) of prime characteristic. Weexplore the relation of this function to certain Hilbert–Samuel and Hilbert–Kunz funcand prove that the mixed function is, in a certain sense, at least as well-behavedHilbert–Kunz function (Theorem 3.2).

2. Relations between Hilbert–Samuel and Hilbert–Kunz multiplicities

We begin this section with a proof thatd! · c(I) > e(I) for anym-primary idealI ofa d-dimensional local ring(R,m) of positive prime characteristic, provided thatd � 2.Although the proof of Theorem 2.2 is quite general and satisfying, it does not prany effective lower bound for the differencec(I) − e(I)/d!. We have therefore taken aalternative approach in Theorem 2.4 which produces a lower bound for the differeterms of the dimensiond and the minimal number of generatorsν(I) of the idealI .

For the proof of Theorem 2.2 we require the following result of [4], which has bproved independently by Watanabe and Yoshida [12].

Proposition 2.1 [4, Proposition II.1].Let J ⊆ I be anyn-primary ideals of a local ring(S,n) of prime characteristicp > 0. Then for anyq = pe ,

(S/J [q]) �

(S/I [q]) +

(S/n[q]) · (I/J ).

Consequently,c(J )� c(I)+ c(n) · (I/J ).In particular, c(I) � (S/I) · c(n) for anyn-primary idealI .

We proceed to the main theorem.

Theorem 2.2.Let I be anym-primary ideal of a local ring(R,m) of prime characteris-tic p, and assume thatd = dim(R) > 1. Thene(I) < d! · c(I).


ts

]

s

Proof. We may reduce to the case that(R,m) is a complete local ring with infinitecoefficient field, by passing to the completion at the maximal ideal ofR[z]mR[z] andthe expansion ofI . This is a faithfully flat extension of the original ringR, of the samedimension, and it is easy to see that lengths mod the expansions ofm-primary ideals remainunchanged.

We may now letJ = (x1, . . . , xd) be a minimal reduction ofI . We note that there exisa fixedk > 0 so thatIk+t ⊆ J t for all t > 0. SinceI [q] ⊆ Iq for all q = pe , and sinceν(I [q]) � ν(I) f or all q = pe , it follows that

(I [q] + J q

J q

)� ν(I) · (R/Ik

),

for all q . We setM = ν(I) · (R/Ik), a fixed constant.For anyq = pe ,

c(I) · qd = c(I [q]) � c

(I [q] + J q

). (2)

Moreover, since the length of(I [q] + J q)/J q is bounded byM, it follows from Proposi-tion 2.1 that

c(I [q] + J q

)� c

(J q

) −M · c(m), (3)

for anyq = pe . (2) and (3) together imply that the differencec(J q)− c(I) · qd is boundedabove by a fixed constantM · c(m).

Finally, letK be a coefficient field forR, and letAJ = K❏x1, . . . , xd❑ be the completeregular subring ofR generated by thex ’s. Notice that

rankAJ R = e(J ;R)= e(I),

sinceI is contained in the integral closure ofJ . Consequently,

c(J q;R) = rankAJ R · c(J q;AJ

) = e(I) · c(J q;AJ

), (4)

and sinceAJ is a power series ring with maximal idealJ , we have by the work of Kunz [7that

c(J q;AJ

) =

(AJ

J q

)=

(q + d − 1

d

). (5)

Putting the results of Eqs. (2)–(5) together, we get that

c(I) · qd � e(I) ·(q + d + 1

d

)−M · c(m), (6)

for all q = pe . We now have only to note that the right side of (6) is a degreed polynomialin q with leading term(e(I)/d!)qd , and that the coefficient onqd−1 is positive as long a


tly

s. Ourr

n.eeng

d � 2. Thus, provided thatd � 2, the expression on the right side of (6) will be stricgreater than its leading term(e(I)/d!)qd for all sufficiently largeq . Dividing through byqd in (6) therefore shows thatc(I) > e(I)/d!, as claimed. ✷Corollary 2.3. Let I be anym-primary ideal of a local ring(R,m) of prime character-istic p, and assume thatd = dim(R) > 1. ThenHKI(e) >

e(I)d ! (pe)d for all sufficiently

large e.

Theorem 2.2 shows thatc(I) > e(I)/d! wheneverd � 2, but it is difficult to applythis result without some lower bound on the difference between the two constantnext result will give such a bound in terms of the dimensiond and the minimal numbeof generatorν(I) of the idealI . The idea here is to note thatI [q] ⊆ Iq for anyq = pe .It follows immediately from this thatHKI(e) � HI (q), as discussed in the introductioHowever, if d � 2, then the number of generators ofIq tends to infinity, whereas thnumber of generators ofI [q] is at mostν(I) for all q . This leads us to believe that thcontribution ofIq/I [q] toHKI(e) should not be insignificant, as is proved in the followitheorem.

Theorem 2.4.Let I be anym-primary ideal of a local ring(R,m) of prime characteris-tic p, and assume thatd = dim(R) > 1. Then

c(I) � e(I)

d! · ν(I)

(d−1√ν(I)− 1)d−1

.

Proof. We note thatI [q] ⊆ Iq for all q = pe , and thatν(I [q]) � ν(I) for all q . It followsthat, for anyq = pe and anys ∈ N,

(I [q] + Iq+s

I q+s

)� ν(I) · (R/Is

).

Therefore, for anyq = pe and anys ∈ N, we have

(R

I [q]

)�

(R

I [q] + Iq+s

)�

(R

Iq+s

)− ν(I) · (R/Is

). (7)

Moreover, using the theory of Samuel functions, we know that

(R/I t

) = e(I)

d! td + O(td−1). (8)

The issue is to wisely choose the parameters.For the approximation of (7) to yield new results on multiplicities, we needs to grow

proportionately toq . Supposing that this is the case, and settings = qh, we may applyEqs. (7) and (8) in order to obtain that

(R

[q]

)� e(I) [

(q + qh)d − ν(I)(qh)d] − O

(qd−1).

I d!


the

,

g therees

.

sionies., hasfrom

l

Now, if we ignore the O(qd−1) term and derivate in order to maximize the function onright in terms ofh, we obtainh = 1/(d−1√ν(I) − 1). The greatest lower bound forc(I)given by Eq. (7) is therefore achieved by setting

s = q

d−1√ν(I) − 1

.

We do have to deal with the fact thats must be an integer. We therefore seth =1/(d−1√ν(I) − 1) and apply Eq. (7) withs = �qh�. Note thath > 0, sinceν(I) � d � 2,and thats therefore grows proportionately toq . Also note that we may writes = q(h− ε),whereε < 1/q .

Applying Eqs. (7) and (8) with this value ofs gives us that

(R/I [q]) � e(I)

d![(q + s)d − ν(I)sd

] − O(qd−1)

� e(I)

d! qd[(1+ h− ε)d − ν(I)(h − ε)d

] − O(qd−1). (9)

Dividing through byqd in (9), and lettingq tend toward infinity (so thatε tends toward 0)we obtain the estimate

c(I) � e(I)

d![(1+ h)d − ν(I)hd

]. (10)

It is easily calculated that

(1+ h)d − ν(I)hd = ν(I)

(d−1√ν(I) − 1)d−1

,

which completes the proof of the theorem.✷Remark 2.5.In general, the approximation(I [q]/(I [q] + Iq+s )) � ν(I) · (R/I s ) is quitecrude; these numbers will often differ significantly. In fact, the question of determininmaximal Hilbert function of an ideal with a given number of generators of given degis interesting in itself (see, e.g., [1,6]). On the other hand, since we know thatc(I) can beas great ase(I), the fact that our estimate can be way off should come as no surprise

Of course, for a fixed dimensiond the constantν(I)/(d−1√ν(I) − 1)d−1 tends towardone for large values ofν(I). However, if we make some assumptions on the dimenor on the number of generators ofI , Theorem 2.4 does yield some interesting corollarCorollary 2.6, which gives the statement of the theorem for two-dimensional ringsalready appeared in work of Watanabe and Yoshida [13], but also follows quite easilyTheorem 2.4.

Corollary 2.6 (cf. [13, Lemma 2.1]).For any m-primary ideal I of a two-dimensionalocal ring (R,m) of positive prime characteristic,c(I) � e(I)ν(I)/(2ν(I) − 2).


at

tic,

mation

rrimerin the

et

.1,

onto

orems

Corollary 2.7. If I is anm-primary ideal of ad-dimensional local ring(R,m) of positiveprime characteristic, and ifν(I) � 2d−1, thenc(I) � e(I)2d−1/d!.

Proof. The statement is trivial whend = 1. Otherwise, we apply Theorem 2.4, noting ththe function

F(x) = x

(d−1√x − 1)d−1

=(

1+ 1

(d−1√x − 1)

)d−1

is decreasing forx > 1, and thatF(2d−1) = 2d−1. ✷Corollary 2.8. Let (R,m) be ad-dimensional hypersurface ring of prime characteriswhered � 3. Thenc(m)� e(m)2d−1/d!.

Proof. Apply the previous corollary, noting thatν(m) = d + 1 � 2d−1 for d � 3. ✷Theorem 2.4 and its corollaries should help to strengthen several extant approxi

results. For example, it is a theorem of Lech [9] that

e(I) � d! (R/I)e(m),

for anym-primary idealI of ad-dimensional local ring(R,m). It has been noted in earliework [4] that this inequality can be recovered from Proposition 2.1 for rings of pcharacteristic, sincee(I) � d! c(I) and c(m) � e(m). But we now see that the formeinequality can be replaced by the one given by Theorem 2.4. We therefore obtaanother corollary.

Corollary 2.9. Let (R,m) be ad-dimensional local ring of prime characteristic, and lI ⊆ R be anym-primary ideal. Then

e(I) � d! (d−1√ν(I) − 1)d−1

ν(I)(R/I)c(m).

Proof. We simply apply Theorem 2.4 to the idealI , and then note that by Proposition 2c(I) � (R/I)c(m). ✷

As before, Corollary 2.9 is most relevant when the number of generators ofI isrelatively small. If, for example,ν(I) � 2d−1, we obtain the inequalities

e(I) � d! (R/I)c(m)

2d−1 � d! (R/I)e(m)

2d−1 . (11)

In fact, all of the bounds one(I) obtained in [4] are achieved by combining a boundc(I) with the inequalitye(I) � d! · c(I). We may therefore apply Theorem 2.4 in orderstrengthen the inequalities. The resulting bounds are given by the following two the(which correspond to Propositions III.5 and III.8 of [4]).


-

onualn in [4]

unz

gs of

f

ere

Theorem 2.10.Let (R,m) be ad-dimensional local ring of prime characteristicp, andlet I be anm-primary ideal. Then for any(minimal) reductionJ of m,

e(I) � d! · (d−1√ν(I)− 1)d−1

ν(I)·[e(m)+

(J + I

I

)· c(m)

].

Theorem 2.11.Let (R,m) be a local ring of dimensiond and positive prime characteristic p, and letI be anm-primary ideal withI ⊆ mt . Then

e(I) � d! · (d−1√ν(I) − 1)d−1

ν(I)·[e(m) ·

(t + d − 1

d

)+ c(m) · (mt/I

)].

Remark 2.12. As in [4], we may replace the termc(m) by e(m) in each of thethree preceding results, sincec(m) � e(m), and then we may show by a reductito characteristicp that the resulting inequalities are equally valid over rings of eqcharacteristic zero. Since these reductions will be essentially the same as those give(all that is required is that we also preserve the number of generators ofI while makingthe reduction), we do not carry them out here.

As a simple example, we offer the following.

Example 2.13.Let R be the hypersurface ring

R = Z/3Z❏X1, . . . ,X5❑

(X21 + · · · +X2

5).

Han and Monsky, who have a general algorithm for computing the Hilbert–Kmultiplicities of diagonal hypersurfaces [3], computec(R) = 23/19. Since dim(R) = 4ande(m) = 2, the general inequalitye(I) � d! · (R/I)e(m) yields thate(I) � 48(R/I)

for any idealI of finite colength inR.Applying Corollary 2.9 certainly gives a better bound. In particular, ifI is an ideal with

ν(I) � 8, we see that

e(I) � d!8(R/I)c(m) = 69

19(R/I).

We may give one last application of Theorem 2.4, or even of Theorem 2.2, for rindimension two. Note that if(R,m) is a complete local ring of characteristicp, and if theresidue fieldR/m is perfect, then the extension ringsR1/q are finiteR-modules for allq = pe , e(m;R1/q) = qde(m;R) for all q , and c(R) = lime→∞(ν(R1/q)/qd). Thestatement thatc(m) > e(m)/d! is therefore equivalent to the statement that(d!)ν(R1/q) >

e(R1/q) for all sufficiently largeq . This is of interest in light of the following result oUlrich [11].

Theorem 2.14(Ulrich). Let R be a local Cohen–Macaulay ring, and suppose that thexists a finitely generatedR-moduleM with positive rank such that


n of

result

em

hs ofuiredbenius.

t andionl

t

(i) 2ν(M) > e(R) rank(M), and(ii) ExtiR(M,R) = 0 for 1 � i � dim(R).

ThenR is Gorenstein.

When the dimension is two, we obtain the following corollary.

Corollary 2.15. If R is a two-dimensional complete local Cohen–Macaulay domaipositive prime characteristicp, with perfect residue fieldk, thenR is Gorenstein if andonly if Ext1R(R

1/q,R) = Ext2R(R1/q,R) = 0 for all q = pe .

This result bears close resemblance to work of Goto [2], who proved a similarunder weaker conditions in all dimensions, but with the assumption thatR1/q be self-dualas anR-module for someq = pe . Other applications and extensions of Ulrich’s theorhave recently been studied in [5].

3. Characterization of a mixed Hilbert function

In this section we seek to characterize a “mixed” Hilbert function given by colengtFrobenius powers of ordinary powers of an Artinian ideal. For this, of course, it is reqthat the ring have nonzero prime characteristic. Considering both ordinary and Fropowers of ideals simultaneously, we are quite naturally led to the following definition

Definition 3.1.Let (R,m) be ad-dimensional local ring of prime characteristicp, and letI ⊆ R be anm-primary ideal. We define themixed Hilbert functionHI :N × N → N by

HI(e, t) =

(R

(I [pe])t

).

One should note that, for anyq = pe and anyt ,

(I [q])t = (

I t)[q]

.

The order in which we take ordinary and Frobenius powers is therefore irrelevanwe may writeI [q]t , or I t [q], without causing any confusion. The mixed Hilbert functrecovers various Hilbert–Kunz and Hilbert–Samuel functions associated to the ideaI . Inparticular,

(1) For any fixede0 andq0 = pe0, HI (e0, t) = HI [q0](t).(2) For any fixedt0, HI (e, t0) = HKIt0 (e).

Sincee(I [q]) = qde(I) for anyq = pe , we see that ife0 is fixed, then the mixed Hilberfunction must take the form:

HI (e0, t) = HI [pe0](t) = e(I)(pe0

)dtd + O

(td−1). (12)

d!


n

Likewise, for fixedt0, we have by Theorem 1.1 that

HI(e, t0) = HKIt0(e) = c(I t0

) · (pe)d + O

((pe

)d−1). (13)

We do not in general have a very good grasp of the leading coefficientc(I t0); however, itis known in general that

c(I t

)� e(I t )

d! = e(I)

d! td ,

as well as that

limt→∞

(e(I t )

d! · c(I t ))

= 1 (14)

(as discussed in the introduction). This leads one to surmise that the functionHI (e, t)

should be asymptotic to a polynomial inq = pe andt , of degreed in each variable, withleading term(e(I)/d!)qdtd . The following theorem makes this more precise.

Theorem 3.2.Let (R,m) andI be as in Definition3.1. Then

HI (e, t) = e(I)

d!(pe

)dtd + (

pe)d · O

(td−1).

More specifically, there exist positive real numbersa andb such that

e(I)

d!(pe

)dtd − a

(pe

)d−1td−1 � HI(e, t) � e(I)

d!(pe

)dtd + b

(pe

)dtd−1

for all positive integerse andt .

We make a couple of remarks before proceeding to the proof of the theorem.

Remark 3.3.The earlier result that

limt→∞

e(I t )

d! · c(I t ) = 1

(cf. [4,13]) easily follows from the statement of Theorem 3.2.

Remark 3.4. We remark that, according to Theorem 3.2, the degreed coefficient onHI(e, t), as a function oft , is (e(I)/d!)(pe)d . In other words,

limt→∞

HI (e, t)

td= e(I)

d!(pe

)d,

for any fixede. But we already knew from Eq. (12) that, ifHI (e, t) could be expressed ithe way given by Theorem 3.2, then this must be the case.


r. In

uired

lingarious

g

ce

Likewise, the coefficient on(pe)d must bec(I t ) for any fixedt , which is always greatethan or equal to(e(I)/d!) · td , with the inequality generally being strict (see Section 2)particular, ford � 2 it is not truethat

HI (e, t) = e(I)

d!(pe

)dtd + O

((pet

)d−1).

We even know that for any fixedt , there exists someεt > 0 such thatHI(e, t)/qd �

(e(I)/d!)td + εt for all sufficiently largeq = pe (by Theorem 2.2 or 2.4).

Proof of Theorem 3.2. For anyq = pe and anyt , one easily sees that

HI(e, t) =

(R

(I [q])t

)�

(R

Iqt

)= HI (tq),

and it is well known that

HI (tq) = e(I)

d! (tq)d + O((tq)d−1).

This shows that the mixed Hilbert function is bounded below by a formula of the reqtype.

To prove the other inequality, we may reduce to the case thatR is a complete locaring with infinite coefficient fieldK, since completing at the maximal ideal and extendthe residue field does not change any of the finite lengths used in determining the vHilbert-type functions. In this case, we letJ = (x1, . . . , xd) ⊆ I be a minimal reductionof the idealI , and we setA = K❏x1, . . . , xd❑ ⊆ R, the complete regular local subringenerated by thex ’s.

If r is the torsion-free rank ofR overA, then we may consider a short exact sequen

0 → Ar → R → C → 0

of A-modules, whereC is a torsion module. It follows that, for allt and allq = pe ,

HJ (e, t) � r · (

A

(J t )[q]

)+

(C

(J t )[q]C

). (15)

We now remark that:

(1) HI (e, t) � HJ (e, t) for all e andt , sinceJ ⊆ I .(2) r = e(J ;R)= e(I ;R).(3) SinceA is regular, for anyt andq = pe ,

(A

(J t )[q]

)= qd ·

(A

(J t )

)= qd ·

(t + d − 1

d

).

(4) (C/(J t )[q]C) = O((tq)d−1).


en-

notes

roof

ociate. Inertn theuse theoveeviouss notvary

–

2.and

ath.

Note that dim(C) < d , and that one may assume point (4) by an induction on the dimsion. Alternatively, one may simply note that

(C/

(J t

)[q]C

)�

(C/

(J dqt

)C

)

and use well-known results on the Hilbert–Samuel function. Combining Eq. (15) and(1)–(4) yields the inequality:

HI (e, t) � e(I) · qd ·(t + d − 1

d

)+ O

((tq)d−1)

� e(I)

d! qdtd + qdO(td−1) + O

((qt)d−1). (16)

From this it is clear that we may chooseb as in the statement of the theorem, and the pis complete. ✷

The theorem is, theoretically, quite satisfying, in that it shows that we may assto the mixed Hilbert function a well-defined leading coefficient, i.e., a multiplicityfact, unlike the Hilbert–Kunz multiplicity, the leading coefficient of the mixed Hilbfunction is a familiar constant that is rational and can quite often be calculated. Oother hand, the results presented here do suggest some limits upon attempts toHilbert–Kunz multiplicity as a fundamentally “new” invariant in examples, or to imprexisting approximations/inequalities by reference thereto (as, for example, in our prwork [4] or in Section 2). Such methods may still give nice results, but only if one doeapply them to high powers of ideals, where the new invariant does not significantlyfrom a familiar one.

References

[1] D. Anick, Thin algebras of embedding dimension three, J. Algebra 100 (1986) 235–259.[2] S. Goto, A problem on Noetherian local rings of characteristicp, Proc. Amer. Math. Soc. 64 (1977) 199

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Documents

Notes on the Hilbert–Kunz function