TYPE SEQUENCES OF ONE-DIMENSIONAL LOCAL ANALYTICALLY IRREDUCIBLE RINGS

TYPE SEQUENCES OF ONE-DIMENSIONAL LOCAL ANALYTICALLY IRREDUCIBLE RINGS

VALENTINA BARUCCI and IOANA CRISTINA ¸SERBAN

Abstract

We extend the notion of type sequence to rings that are not necessarily residually rational. Using this invariant we characterize different types of rings as almost Gorenstein rings and rings of maximal length.

1. Introduction

Let(R,ᒊ)be a one-dimensional local Cohen-Macaulay ring and letRbe the integral closure ofRin its field of quotients. If we assume thatRis analytically irreducible, i.e., thatRis a DVR (with a valuationv) and a finitely generated R-module, then the values of the elements ofRform a numerical semigroup v(R) = {v(a) | a ∈ R, a = 0} = {s0 = 0, s1, . . . , sr−1, sr,→}, where s0 < s1 < · · ·< sr and any integerx,x ≥ sr is inv(R)and the conductor C=(R:R)is not zero.

If we further assume thatR is residually rational, i.e., thatk, the residue field ofR, is isomorphic to the residue field ofR, thenr = R(R/C)and a sequence ofr natural numbers(t1, . . . , tr)is naturally associated toR,ti = R(ᑾ⁻_i¹/ᑾ_i⁻₋¹₁), whereᑾ_i = {x ∈ R | v(x) ≥ si}. This sequence of natural numbers associated to the ring was for the first time considered by Matsuoka in [8]. As in [1] we call the sequence(t¹, . . . , tr)the type sequence ofR, t.s.(R) for short. In particular the lengtht1 = R(ᒊ⁻¹/R) is the Cohen Macaulay type ofRand it turns out that_r

i=1ti = R(R/R). A typical example of an analytically irreducible and residually rational ring is the ring of an algebraic curve singlularity with one branch.

It is well known that, for each one-dimensional local Cohen Macaulay ring with finite integral closure, the lengthR(R/R)is bounded below and above in the following way:

R(R/C)+t−1≤R(R/R)≤R(R/C)t where againC =(R:R)andt is the CM type ofR.

Received 19 October 2010.

The first inequality depends on the existence of a canonical ideal (cf. [2, Lemma 19 (e)]) and the second is proved in [3, Theorem 1]. If the ringRis Gorenstein, i.e., of CM type 1, then the two inequalities become equalities andR(R/C) = R(R/R). The rings which realize the minimal length for R(R/R), i.e., such thatR(R/C)+t −1=R(R/R)have been introduced in [2] with the name of almost Gorenstein and recently revealed an interest in a geometric context (cf. [7]). On the other hand the rings which realize the maximal length forR(R/R)were characterized in [3] and also studied in [6].

In the analytically irreducible and residually rational case, there is a strict relation between the type sequence of R and the length R(R/R). It is not surprising that the almost Gorenstein rings are characterized by a type se- quence of the form(t,1,1, . . . ,1)and those which realize the maximal length are characterized by a type sequence of the form(t, t, . . . , t), cf. [1] and [5, Theorem 1.7].

This paper deals with the analytically irreducible non-residually rational case. We have still a numerical semigroupv(R)of values, butk, the residue field ofR, is not isomorphic toK, the residue field ofR. The almost Gorenstein rings are characterized by a type sequence of the form(t, n¹, . . . , nr+l)and the rings of maximal length by a type sequence of the form(t, tn¹, . . . , tnr+l), whereniare the dimensions of certaink-vector subspaces ofKdefined below.

As usual, if_ᑾand_ᑿare fractional ideals ofR, then_ᑾ: _ᑿ:= {x ∈Q(R)| xᑿ ⊆ ᑾ}, whereQ(R) is the field of quotients ofR,ᑾ⁻¹ = R : ᑾ andᑾis divisorial ifR:(R:ᑾ))=ᑾ.

2. The result

In all this paperRis a one-dimensional local analytically irreducible not ne- cessarily residually rational ring. So the integral closureRis a DVR andRhas an associated semigroup of values:

(1) v(R)= {s0=0, s1, . . . , sr−1, sr =c,→},

Denote byXthe generator of the maximal ideal ofRand define theconductor of the ring as the natural numberNsuch thatR:R=X^NR. Note thatN ≥c, thus we can setN = sr+l = sr +l =c+lfor somel ∈^N. In the residually rational case we haveN =c. Thus in order to extend the definition of the type sequence to the non-residually rational case, some care is needed. As we shall see, in the general case the “right” definition will consist of a sequence ofr+l numbers.

Let us see the details. Consider the ideals ofRdefined as (2) ᑾi = {x∈R|v(x)≥si}, i∈ {0, . . . , r +l}.

It is evident thatᑾ0 =R,ᑾ1= ᒊandᑾ_r+l = R : R. Moreover, we have the following chain of inclusions:

(3) ᑾ_r+l ⊂ · · · ⊂ᑾ0=ᑾ⁻₀¹=R⊆ᑾ⁻₁¹⊆ · · · ⊆ᑾ⁻_r+l¹,

Note that whereas on the left side we have strict inclusions, on the right side, a priori, some of the inclusions could be equalities.

The following facts about the idealsᑾ_i are well known, but we recall them for the convenience of the reader:

Proposition2.1.For every i ∈ {0, . . . , r +l}, the ideals defined above have the following properties.

1. ᑾ⁻_r+l¹ =R; 2. ᑾ_i is divisorial.

3. Ifi >0then_ᑾ⁻_i ¹=ᑾ⁻_i−¹1and henceR(ᑾ_i⁻¹/ᑾ⁻_i−¹1)≥1.

Proof. 1. Asᑾ_r+l =X^NR, we have

ᑾ⁻_r+l¹ =R :X^NR=X^−N(R :R)

=X^−NX^NR=R.

(4)

2. AsR=R:ᑾ⁻_r+l¹, we have thatRis divisorial as a fractional ideal ofR. It follows thatX^hRis divisorial for everyh∈^N. This shows thatᑾ_iis divisorial, since

(5) ᑾ_i =R∩X^siR.

3. Ifi >0, then bothᑾi andᑾi−1are divisorial. Thus, ifR:ᑾi−1=R:ᑾi, thenᑾ_i−1=R:(R :ᑾ_i−1)=R:(R :ᑾ_i)=ᑾ_i, which is contradiciton.

Now we are ready to give our definition of type sequence of the ringR. For everyi∈ {1,2, . . . , r +l}let

(6) ti(R):=R(ᑾ⁻_i ¹/ᑾ⁻_i−¹₁).

We call the sequence of numbers(t1(R), t2(R), . . . , tr+l(R))thetype sequence ofR, and we denote it by t.s.(R).

As in the residually rational case we have that (7) t¹(R)=R(ᒊ⁻¹/R)=:t(R) which is the Cohen-Macaulay type ofR.

Note that for every 1 ≤ i ≤ r +l,ᒊᑾ_i−1 ⊆ ᑾ_i and soᑾ_i−1/ᑾ_i, is a k- vector space; let us denote it byVR(si−1)). Since the inclusion_ᑾi ⊂ ᑾi−1is

strict,VR(si−1)=0 and hence the numberni−1:=dim_kVR(si−1)is a positive integer. These vector spaces were considered also in [4] and can be defined not only for the ringRbut also for any fractional ideal ofR. LetF be such an ideal andi∈^N. Then

(8) F (i):= {x ∈F |v(x)≥i}

is a fractional ideal ofR and we haveF (i) ⊆ F (j) for every i ≥ j. The R-modulesF (i)/F (i+1)are also vector spaces overk, and we denote them byVF(i).

As we have outlined in the introduction, these vector spaces are very im- portant for studying lengths for the analytically ireducible rings which are not residually rational. IfE⊆F are fractional ideals ofR, then, in the residually rational case,R(F/E)=#{v(F )\v(E)}, cf. [8, Proposition 1]. In the non- residually rational case we use the dimensions of the previous defined vector spaces as it was proved in [9].

Proposition2.2 ([9, Proposition 11]).LetEandFbe two fractional ideals ofRsuch thatE⊆F ⊆R. Then there exists ans ∈^Nsuch that

(9) R(F/E)=

s−1

r=0

[dim_k(VF(r))−dim_k(VE(r))].

We recall also another result which in this form appears in [9] and in fact it is an adapted version of [4, Proposition 3.5]. Observe that if V and W are two k-vector subspaces of K, wherek ⊆ K is a field extension, then (V :W):= {x ∈K|xW ⊆V}is also ak-vector subspace ofK.

Lemma 2.3 ([9, Lemma 3]). Let k ⊆ K be an extension of fields with n=dim_kK <∞and letV ⊂Kbe ann−1-dimensionalk-vector subspace ofK. Then for everyk-vector subspaceW ⊆Kwe have

(10) dim_k(V :W)+dim_k(W)=n.

In order to prove our main theorem, we need the next result on the dimen- sions of previous defined vector spaces related to the fractional ideals ᑾ⁻_i ¹, i∈ {1, . . . , r+l}.

Lemma2.4. Letn=dim_kK, wherekis the residue field ofRandKis the residue field ofR. Then:

1. dim_k(V_ᑾi⁻¹(N−1−si−1))=n; 2. dim_k(V_ᑾi−1⁻¹(N−1−si−1))≤n−ni−1;

Proof. Let us prove the first assertion. Fix ani ∈ {1, . . . , r+l}. Then for anyγ ∈Kwe have that

(11) γ X^{N−1−si−1}ᑾ_i ⊆X^{N−1+(si−si−1)}R⊆X^NR⊆R.

Therefore we have

(12) γ X^{N−1−si−1} ∈R:ᑾ_i =ᑾ⁻_i ¹,

for everyγ ∈ K. ThusKX^{N−1−si−1} ⊆ R : _ᑾi. It follows thatV_ᑾi⁻¹(N −1− si−1)Kand this is of dimensionnoverk.

Now we can prove the second assertion. It is easy to see that (13) γ X^{N−1−si−1}ᑾ_i−1⊆R ⇐⇒γ VR(si−1)⊆VR(N−1) which is of course further equivalent toγ ∈VR(N−1):VR(si−1).

Thus we have that:

(14) γ ∈V_ᑾ⁻¹_i−1(N−1−si−1)⇐⇒γ ∈(VR(N−1):VR(si−1)) Then we can conclude that:

(15) dim_k(V_ᑾ⁻¹_i−1(N−1−si−1))=dim_k(VR(N−1):VR(si−1)).

AsVR(N−1)is a proper subspace ofK, we can find ak-vector subspace U ⊂Kof codimension 1 such thatVR(N−1)⊆U and so

dim_k(VR(N−1):VR(si−1))≤dim_k(U :VR(si−1))

=n−dim_k(VR(si−1))

=n−ni−1, (16)

where for the first equality we have used Lemma 2.3.

We shall see now certain upper and lower bounds forti(R), which generalize [8, Proposition 3].

Proposition2.5. For everyi ∈ {1, . . . , r+l}: (17) ni−1≤ti(R)≤t(R) ni−1.

Proof. To show the upper bound we shall use [3, Lemma 1]. This affirms that, if we have two ideals of the ringR,I1andI2such thatI1⊆I2andI2/I1

is a simpleR-module, then

(18) R(R:I1/R:I2)≤t(R).

We are using this result for our ideals ᑾ_i ⊆ ᑾ_i−1. The R-module ᑾ_i−1/ᑾ_i is not simple, but it is in fact ak-vector space of finite dimension equal toni−1. Then we can apply [3, Lemma 1]ni−1times and we conclude the proof for the upper bound.

Now we want to show the lower bound. Using Proposition 2.2 we have that (19) R(ᑾ_i⁻¹/ᑾ⁻_i−¹1)≥dim_k(V_ᑾ⁻¹_i (N−1−si−1))−dim_k(V_ᑾ⁻¹_i−1(N−1−si−1)).

By Lemma 2.4, we get:

(20) ti(R):=R(ᑾ⁻_i ¹/ᑾ⁻_i−¹1)≥n−(n−ni−1)=ni−1.

Similarly to the residually rational case, we can characterize rings of min- imal and maximal length by their type sequences.

Theorem2.6.Letni =dim_k(ᑾ_i/ᑾ_i−1). Then 1. Ris almost Gorenstein if and only if

(21) t.s.(R)=(t(R), n¹, n², . . . , nr+l−1).

2. Ris of maximal length if and only if

(22) t.s.(R)=(t(R), t(R)n1, t(R)n2, . . . , t(R)nr+l−1).

Proof. Recall that the almost Gorenstein property means (cf. [2, Defini- tion-Proposition 20]) that

(23) R(R/R)=R(R/ᑾ_r+l)+t(R)−1. Equation (23) is equivalent to:

(24) ^r+l

i=1

ti(R)=^r+l−

1

i=0

ni+t(R)−1

Asn0 = dim_k(VR(s0)) = dim_k(R/ᒊ) = 1 andt1(R) = t(R), the previous equation is equivalent to:

(25) ^r+l

i=2

ti(R)=^r+l−

1

i=1

ni.

Thus, the conclusion follows using Proposition 2.5 which claims thatni−1≤ti, so the previous equality holds if and only ifti =ni−1for everyi, 2≤i ≤r+l.

The ringRis of maximal length if and only if (26) R(R/R)=t(R)R(R/ᑾ_r+l).

And this is further equivalent to (27)

r+l

i=1

ti(R)=t(R)

r+l−1 i=0

ni =

r+l−1 i=0

t(R)ni.

By Proposition 2.5,ti ≤t(R)ni−1for everyi∈ {1, . . . , r+l−1}, and we know also thatn0 = 1, so the previous equality holds if and only ifti = t(R)ni−1

for everyi ∈ {1, . . . , r+l}.

As a consequence of Theorem 2.6, 1, and of the fact that a Gorenstein (Kunz) ring is an almost Gorenstein ring of type 1 (2, respectively), we can draw the following conclusion.

Corollary2.7.

1. Ris Gorenstein if and only if t.s.(R)=(1, n1, . . . , nr+l−1), 2. Ris Kunz if and only if t.s.(R)=(2, n1, . . . , nr+l−1).

Note that the proof for Gorensteiness could also have been obtained in a direct manner. Indeed, ifR is Gorenstein, then the ring itself is a canonical ideal, and so for everyi∈ {1, . . . , r +l},

ti(R)=R((R:ᑾ_i)/R:ᑾ_i−1)

=R((R:(R:ᑾ_i−1))/(R:(R :ᑾ_i)))=R(ᑾ_i−1/ᑾ_ᒆ)

=dim_kVR(si−1)=ni−1. (28)

3. Examples

Example 1. Consider the following subring of the ring of power series Q√

2,√ 3

[[X]].

R=^Q+X^3Q√ 2,√

3

+X^4Q√ 2,√

3

+X^5Q√ 2

+X^6Q√ 2,√

3 [[X]]

In this example,k = ^QandK = ^Q√ 2,√

3

. According to the notation of previous section, we haven0=1,n1=4,n2=4,n3=2. Moreover, since

ᑾ⁻₁¹=^Q√ 2

+X^3Q√ 2,√

3 [[X]], ᑾ⁻₂¹=^Q√

2

+X^2Q√ 2,√

3 [[X]], ᑾ⁻3¹=^Q√

2

+X^Q√ 2,√

3 [[X]], ᑾ⁻₄¹=^Q√

2,√ 3

[[X]],

we get t = t1 = 3, t2 = 4, t3 = 4, t4 = 2, so that the type sequence is (t, n1, n2, n3)and the ring is almost Gorenstein.

Example2. A ring of maximal length.

R =^R+X³i^R+X^6R+X^9C[[X]]

Herek=^R,K=^C,n⁰=n¹=n²=1 and since ᑾ⁻₁¹=^R+X³i^R+X^6C[[X]], ᑾ⁻2¹=^R+X^3C[[X]],

ᑾ⁻3¹=^C[[X]],

we gett =t1=5,t2=5,t3=5, so that the type sequence is(t, tn1, tn2)and the ring is of maximal length.

The examples above are generalized semigroup rings, GSR for short, i.e., rings of the form

k+XV¹+ · · · +X^N−1VN−1+X^NK[[X]]

whereViarek-vector subspaces ofK. To every one-dimensional analytically irreducible ringRcan be associated a GSRR, as in [9]. More preciselyRis the subring ofK[[X]] defined, with the notation of the previous section, as

R:=

i≥0

VR(i)Xⁱ

That is in fact the generalization of the way of associating toR, in the residually rational case, the semigroup ringk[[S]], where S = v(R). Observe however that the type sequence ofRand its associated GSR is not always the same. For example, ifR = k[[X⁴, X⁶+X⁷, X¹⁰]], with characteristic ofk unequal to 2, then the associated GSR isk[[X⁴, X⁶, X¹¹, X¹³]], which has type sequence (3,1,1,1). On the other hand the type sequence ofRis(2,2,1,1), in factᑾ⁻₁¹ contains no element with value 2, butᑾ⁻₂¹containsX²−X³. However we can prove that:

Proposition3.1.The ringRis almost Gorenstein if and only if the associ- ated GSRRis almost Gorenstein andtype(R)=type(R).

Proof. Let ω be a canonical ideal of R, R ⊆ ω ⊆ R (cf. [2] for the definition and the existence). Then, by [9, Theorem 17],ω =

i≥0Vω(i)Xⁱ is a canonical ideal ofR. Moreover, by [9, equation (24) and Lemma 16], we have:

(29) R(ω/R)=R(ω/R) ≥type(R) −1≥type(R)−1

By [2, Definition-Proposition 20] we get thatR is almost Gorenstein if and only ifR(ω/R) ≥ type(R)−1. Thus R is almost Gorenstein if and only if both inequalities of (29) are equalities, that is if and only if Ris almost Gorenstein and type(R)=type(R).

We shall give now an example of computing the type sequence of a ring which is not a GSR and the lengthR(R/R)is not minimal neither maximal.

Example3. LetR = ^R[[iX³+X⁴, X⁵, iX¹⁰+X¹¹, X¹⁶]]. For this ring k = ^Rand K = ^C. Observe that R is not a GSR, but we can compute its associated GSR. First let us try to compute the type sequence ofR. After some computations we can writeRas:

R =^R+(iX³+X⁴)^R+X^5R+(−X⁶+2iX⁷+X⁸)^R +(iX⁸+X⁹)^R+(−iX⁹+3iX¹¹+X¹²)^R+X^10R +(iX¹⁰+X¹¹)^R+(−X¹¹+2iX¹²)^R+(X¹²−2X¹⁴)^R +X^13R+(iX¹³+X¹⁴)^R+iX^14R+X^15C[[X]].

Thus for the ringRwe have: the conductor of the ringN = 15 andn0 = 1, n1 = 1, n2 = 1, n3 = 1, n4 = 1, n5 = 1, n6 = 2, n7 = 1, n8 = 1, n9=2,n10 =1. We compute now the inverses of the ideals which appear in the definition of the type sequence.

ᒊ⁻¹=^R+(iX³+X⁴)^R+X^5R+(−X⁶+2iX⁷+X⁸)^R

+(X⁷−2X⁹)^R+(iX⁸+X⁹)^R+(−iX⁹+3iX¹¹+X¹²)^R +X^10R+(iX¹⁰+X¹¹)^R+X^11R+X^12C[[X]].

ᑾ⁻₂¹=^R+(iX³+X⁴)^R+X^5R+(−X⁶+2iX⁷+X⁸)^R +(X⁷−2X⁹)^R+(iX⁸+X⁹)^R+X⁹i^R+X^10C[[X]]. ᑾ⁻₃¹=^R+(iX³+X⁴)^R+X^5R+(−X⁶+2iX⁷+X⁸)^R+X^7R

+(iX⁷+X⁸)^R+X⁸i^R+X^9C[[X]]. ᑾ⁻₄¹=^R+(iX³+X⁴)^R+X^5R+X^6R+X^7C[[X]]. ᑾ⁻₅¹=^R+(iX³+X⁴)^R+X^5R+X^6C[[X]]. ᑾ⁻₆¹=^R+(iX³+X⁴)^R+X^5C[[X]]. ᑾ⁻₇¹=^R+(iX²−X³)^R+X³i^R+X^4C[[X]]

ᑾ⁻₈¹=^R+X²i^R+X^3C[[X]].

ᑾ⁻₉¹=^R+X^2C[[X]]. ᑾ⁻₁₀¹=^R+X^C[[X]]. ᑾ⁻11¹=^C[[X]]=R.

Then:t1 = R(ᒊ⁻¹/R) = 3,t2 = 1,t3 = 2,t4 = 1,t5 = 1,t6 = 1,t7 = 3, t8=1,t9=1,t10=2,t11=1.

The associated GSR ofR isR= ^R[[iX³, X⁵, iX¹⁰, iX¹⁷]]. After compu- tations we have that the type sequence ofRis(t¹ = 3, t² = 1, t³ = 2, t⁴ = 1, t5= 1, t6=1, t7=3, t8= 1, t9=1, t10 =2, t11 =1). We observe that in this case the type sequence ofRis equal to the type sequence of its associated GSR.

REFERENCES

1. Barucci, V., Dobbs, D. E., and Fontana M.,Maximality properties in numerical semigroups and applications to one-dimensional analytically irreducible local domains, Mem. Amer.

Math. Soc. 125 (1997).

2. Barucci, V., and Fröberg, R.,One-dimensional almost Gorenstein rings, J. Algebra 188 (1997), 418–442.

3. Brown, W. C., and Herzog, J.,One dimensional local rings of maximal and almost maximal length, J. Algebra 151 (1992), 332–347.

4. Campillo, A., Delgado, F., and Kiyek, K., Gorenstein property and symmetry for one- dimensional local Cohen-Macaulay rings, Manuscripta Math. 83 (1994), 405–423.

5. D’Anna, M.,Canonical module and one-dimensional analytically irrreducible Arf domains, pp. 215–225 in: Commutative Ring Theory (Proc. Fés 1995), Lecture Notes in Pure and Appl. Math. 185, Dekker, New York 1997.

6. Delfino, D.,On the inequalityλ(R/R) ≤ t(R)λ(R/ᑝ)for one-dimensional local rings, J. Algebra 169 (1997), 332–342.

7. Kleiman, S., Martins, R. V.,The canonical model of a singular curve, Geom. Dedicata 139 (2009), 139–166.

8. Matsuoka, T.,On the degree of singularity of one-dimensional analytically irreducible No- etherian local rings, J. Math. Kyoto Univ. 11 (1971), 485–494.

9. ¸Serban, I. C.,On canonical ideals of one-dimensional local analytically irreducible rings, Comm. Algebra 36 (2008), 4263–4275.

DIPARTIMENTO DI MATEMATICA UNIVERSITÀ DI ROMA “LA SAPIENZA”

P.LE ALDO MORO 2 00185 ROMA ITALY

E-mail:barucci@mat.uniroma1.it ioanase@yahoo.com