Type sequences of one-dimensional local analytically irreducible rings
aa r X i v : . [ m a t h . A C ] O c t Type sequences of one-dimensional localanalytically irreducible rings
Valentina BarucciDip. di Matematica, Universit`a di Roma “La Sapienza”,P.le Aldo Moro 2, Roma-00185, Italye-mail: [email protected]
Ioana Cristina S¸erbane-mail: [email protected]
Abstract
We extend the notion of type sequence to rings that are not necessarilyresidually rational. Using this invariant we characterize different types ofrings as almost Gorenstein rings and rings of maximal length.
Let ( R, m ) be a one-dimensional local Cohen-Macaulay ring and let R be theintegral closure of R in its field of quotients. If we assume that R is analyticallyirreducible, i.e. that R is a DVR (with a valuation v ) and a finitely generated R -module, then the values of the elements of R form a numerical semigroup v ( R ) = { v ( a ) | a ∈ R, a = 0 } = { s = 0 , s , . . . , s r − , s r , →} , where s < s < · · · < s r and any integer x , x ≥ s r is in v ( R ) and the conductor C = ( R : R ) isnot zero.If we further assume that R is residually rational, i.e. that k , the residue fieldof R , is isomorphic to the residue field of R , then r = ℓ R ( R/C ) and a sequenceof r natural numbers ( t , . . . , t r ) is naturally associated to R , t i = ℓ R ( a − i / a − i − ),where a i = { x ∈ R | v ( x ) ≥ s i } . This sequence of natural numbers associated tothe ring was for the first time considered by Matsuoka in [8]. As in [1] we callthe sequence ( t , . . . , t r ) the type sequence of R , t.s.( R ) for short. In particularthe length t = ℓ R ( m − /R ) is the Cohen Macaulay type of R and it turns outthat P ri =1 t i = ℓ R ( R/R ). A typical example of an analytically irreducible andresidually rational ring is the ring of an algebraic curve singlularity with onebranch.It is well known that, for each one-dimensional local Cohen Macaulay ringwith finite integral closure, the length ℓ R ( R/R ) is bounded under and above in1he following way: ℓ R ( R/C ) + t − ≤ ℓ R ( R/R ) ≤ ℓ R ( R/C ) t where again C = ( R : R ) and t is the CM type of R .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 ring R isGorenstein, i.e. of CM type 1, then the two inequalities become equalities and ℓ R ( R/C ) = ℓ R ( R/R ). The rings which realize the minimal length for ℓ R ( R/R ),i.e. such that ℓ R ( R/C ) + t − ℓ R ( R/R ) have been introduced in [2] withthe name of almost Gorenstein and recently revealed an interest in a geometriccontext (cf.[7]). On the other hand the rings which realize the maximal lengthfor ℓ R ( R/R ) were characterized in [3] and also studied in [6].In the analytically irreducible and residually rational case, there is a strictrelation between the type sequence of R and the length ℓ R ( R/R ). It is not sur-prising that the almost Gorenstein rings are characterized by a type sequenceof the form ( t, , , . . . ,
1) and those which realize the maximal length are char-acterized by a type sequence of the form ( t, t, . . . , t ), cf. [1] and [5, Theorem1.7].This paper deals with the analytically irreducible non-residually rationalcase. We have still a numerical semigroup v ( R ) of values, but k , the residuefield of R , is not isomorphic to K , the residue field of R . The almost Gorensteinrings are characterized by a type sequence of the form ( t, n , . . . , n r + l ) and therings of maximal length by a type sequence of the form ( t, tn , . . . , tn r + l ), where n i are the dimensions of opportune k -vector subspaces of K .As usual, if a and b are fractional ideals of R , then a : b := { x ∈ Q ( R ) | x b ⊆ a } , where Q ( R ) is the field of quotients of R , a − = R : a and a is divisorial if R : ( R : a )) = a . In all this paper R is a one-dimensional local analytically irreducible not nec-essarily residually rational ring. So the integral closure R is a DVR and R hasan associated semigroup of values: v ( R ) = { s = 0 , s , . . . , s r − , s r = c, →} , (1)Denote by X the generator of the maximal ideal of R and define the conductor of the ring as the natural number N such that R : R = X N R . Note that N ≥ c ,thus we can set N = s r + l = s r + l = c + l for some l ∈ N . In the residuallyrational case we have N = c . Thus in order to extend the definition of the typesequence to the non-residually rational case, some care is needed. As we shall2ee, in the general case the “right” definition will consist of a sequence of r + l numbers.Let us see the details. Consider the ideals of R defined as a i = { x ∈ R | v ( x ) ≥ s i } , i ∈ { , . . . , r + l } . (2)It is evident that a = R , a = m and a r + l = R : R . Moreover, we have thefollowing chain of inclusions: a r + l ⊂ · · · ⊂ a = a − = R ⊆ a − · · · ⊆ a − r + l , (3)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 ideals a i are well known, but we recall themfor the convenience of the reader: Proposition 1.
For every i ∈ { , . . . , r + l } , the ideals defined above have thefollowing properties.1. a − r + l = R ;2. a i is divisorial.3. If i > then a − i = a − i − and hence ℓ R ( a − i / a − i − ) ≥ .Proof. . As a r + l = X N R , we have a − r + l = R : X N R = X − N ( R : R )= X − N X N R = R. (4)2 . As R = R : a − r + l , we have that R is divisorial as a fractional ideal of R . Itfollows that X h R is divisorial for every h ∈ N . This shows that a i is divisorial,as a i = R ∩ X s i R. (5)3 . If i >
0, then both a i and a i − are divisorial. Thus, if R : a i − = R : a i ,then a i − = R : ( R : a i − ) = R : ( R : a i ) = a i , which is contradiciton.Now we are ready to give our definition of type sequence of the ring R . Forevery i ∈ { , , . . . , r + l } let t i ( R ) := ℓ R ( a − i / a − i − ) . (6)We call the sequence of numbers ( t ( R ) , t ( R ) , . . . , t r + l ( R )) the type sequence of R , and we denote it by t . s . ( R ). 3s in the residually rational case we have that t ( R ) = ℓ R ( m − /R ) =: t ( R ) (7)which is the Cohen-Macaulay type of R .Note that for every 1 ≤ i ≤ r + l , ma i − ⊆ a i and so a i − / a i , is a k -vectorspace; let us denote it by V R ( s i − )). Since the inclusion a i ⊂ a i − is strict, V R ( s i − ) = 0 and hence the number n i − := dim k V R ( s i − ) is a positive integer.These vector spaces were considered also in [4] and can be defined not only forthe ring R but also for any fractional ideal of R . Let F be such an ideal and i ∈ N . Then F ( i ) := { x ∈ F | v ( x ) ≥ i } (8)is a fractional ideal of R and we have F ( i ) ⊆ F ( j ) for every i ≥ j . The R − modules F ( i ) /F ( i + 1) are also vector spaces over k , and we denote themby V F ( 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 notresidually rational. If E ⊆ F are fractional ideals of R , then, in the residuallyrational case, ℓ 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 vectorspaces as it was proved in [9]. Proposition 2. [9, Proposition 11]
Let E and F be two fractional ideals of R such that E ⊆ F ⊆ R . Then there exists an s ∈ N such that ℓ R ( F/E ) = s − X r =0 [dim k ( V F ( r )) − dim k ( V E ( r ))] . (9)We recall also another result which in this form appears in [9] and in fact itis an adapted version of [4, Proposition 3.5]. Observe that if V and W are two k -vector subspaces of K , where k ⊆ K is a field extension, then ( V : W ) := { x ∈ K | xW ⊆ V } is also a k -vector subspace of K . Lemma 3. [9, Lemma 3]
Let k ⊆ K be an extension of fields with n = dim k K < ∞ and let V ⊂ K be an n − -dimensional k -vector subspace of K . Then forevery k -vector subspace W ⊆ K we have dim k ( V : W ) + dim k ( W ) = n. (10)In order to prove our main theorem, we need the next result on the di-mensions of previous defined vector spaces related to the fractional ideals a − i , i ∈ { , . . . r + l } . 4 emma 4. Let n = dim k K , where k is the residue field of R and K is theresidue field of R . Then:1. dim k ( V a − i ( N − − s i − )) = n ;2. dim k ( V a − i − ( N − − s i − )) ≤ n − n i − ;Proof. Let us prove the first assertion. Fix an i ∈ { , . . . , r + l } . Then for any γ ∈ K we have that γX N − − s i − a i ⊆ X N − s i − s i − ) R ⊆ X N R ⊆ R. (11)Therefore we have γX N − − s i − ∈ R : a i = a i − , (12)for every γ ∈ K . Thus KX N − − s i − ⊆ R : a i . It follows that V a i − ( N − − s i − ) ≃ K and this is of dimension n over k .Now we can prove the second assertion. It is easy to see that γX N − − s i − a i − ⊆ R ⇐⇒ γV R ( s i − ) ⊆ V R ( N −
1) (13)which is of course further equivalent to γ ∈ V R ( N −
1) : V R ( s i − ).Thus we have that: γ ∈ V a − i − ( N − − s i − ) ⇐⇒ γ ∈ ( V R ( N −
1) : V R ( s i − )) (14)Then we can conclude that:dim k ( V a − − ( N − − s i − )) = dim k ( V R ( N −
1) : V R ( s i − )) . (15)As V R ( N −
1) is a proper subspace of K , we can find a k -vector subspace U ⊂ K of codimension 1 such that V R ( N − ⊆ U and sodim k ( V R ( N −
1) : V R ( s i − )) ≤ dim k ( U : V R ( s i − ))= n − dim k ( V R ( s i − )) == n − n i − , (16)where for the first equality we have used Lemma 3.We shall see now certain upper and lower bounds on t i ( R ), which generalize[8, Proposition 3]. Proposition 5.
For every i ∈ { , . . . , r + l } : n i − ≤ t i ( R ) ≤ t ( R ) n i − . (17)5 roof. For showing the upper bound we shall use [3, Lemma 1]. This affirmsthat, if we have two ideals of the ring R , I and I such that I ⊆ I and I /I is a simple R − module, then ℓ R ( R : I /R : I ) ≤ t ( R ) . (18)We are using this result for our ideals a i ⊆ a i − . The R − module a i − / a i isnot simple, but it is in fact a k − vector space of finite dimension equal to n i − .Then we can apply [3, Lemma 1] n i − times and we conclude the proof for theupper bound.Now we want to show the lower bound. Using Proposition 2 we have that ℓ R ( a − / a − − ) ≥ dim k ( V a − ( N − − s i − )) − dim k ( V a − − ( N − − s i − )) . (19)By Lemma 4, we get: t i ( R ) := ℓ R ( a − / a − − ) ≥ n − ( n − n i − ) = n i − . (20)Similarly to the residually rational case, we can characterize rings of minimaland maximal length by their type sequences. Theorem 6.
Let n i = dim k ( a i / a i − ) . Then1. R is almost Gorenstein if and only if t . s . ( R ) = ( t ( R ) , n , n , . . . , n r + l − ) . (21) R is of maximal length if and only if t . s . ( R ) = ( t ( R ) , t ( R ) n , t ( R ) n , . . . , t ( R ) n r + l − ) . (22) Proof.
Recall that the almost Gorenstein property means (cf. [2, Definition-Proposition 20]) that ℓ R ( R/R ) = ℓ R ( R/ a r + l ) + t ( R ) − . (23)Equation (23) is equivalent to: r + l X i =1 t i ( R ) = r + l − X i =0 n i + t ( R ) − n = dim k ( V R ( s )) = dim k ( R/ m ) = 1 and t ( R ) = t ( R ), the previousequation is equivalent to: r + l X i =2 t i ( R ) = r + l − X i =1 n i . (25)Thus, the conclusion follows using Proposition 5 which claims that n i − ≤ t i ,so the previous equality holds if and only if t i = n i − for every i , 2 ≤ i ≤ r + l .The ring R is of maximal length if and only if ℓ R ( R/R ) = t ( R ) ℓ R ( R/ a r + l ) . (26)And this is further equivalent to r + l X i =1 t i ( R ) = t ( R ) r + l − X i =0 n i = r + l − X i =0 t ( R ) n i . (27)By Proposition 5, t i ≤ t ( R ) n i − for every i ∈ { , . . . , r + l − } , and we knowalso that n = 1, so the previous equality holds if and only if t i = t ( R ) n i − forevery i ∈ { , . . . , r + l } .As a consequence of Theorem 6,1, and of the fact that a Gorenstein (Kunz)ring is an almost Gorenstein ring of type 1 (2, respectively), we can draw thefollowing conclusion. Corollary 7. R is Gorenstein if and only if t . s . ( R ) = (1 , n , . . . , n r + l − ) ,2. R is Kunz if and only if t . s . ( R ) = (2 , n , . . . , n r + l − ) . Note that the proof for Gorensteiness could have been also obtained in adirect manner. Indeed, if R is Gorenstein, then the ring itself is a canonicalideal, and so for every i ∈ { , . . . , r + l } , t i ( R ) = ℓ R (( R : a i ) /R : a i − )= ℓ R (( R : ( R : a i − )) / ( R : ( R : a i ))) = ℓ R ( a i − / a i )= dim k V R ( s i − ) = n i − . (28) Example 1 . Consider the following subring of the ring of power series Q ( √ , √ X ]]. R = Q + X Q ( √ , √
3) + X Q ( √ , √
3) + X Q ( √
2) + X Q ( √ , √ X ]]7n this example, k = Q and K = Q ( √ , √ n = 1 , n = 4 , n = 4 , n = 2. Moreover, since a − = Q ( √
2) + X Q ( √ , √ X ]] a − = Q ( √
2) + X Q ( √ , √ X ]], a − = Q ( √
2) + X Q ( √ , √ X ]], a − = Q ( √ , √ X ]],we get t = t = 3 , t = 4 , t = 4 , t = 2, so that the type sequence is( t, n , n , n ) and the ring is almost Gorenstein. Example 2
A ring of maximal length. R = R + X i R + X R + X C [[ X ]]Here k = R , K = C , n = n = n = 1 and since a − = R + X i R + X C [[ X ]] a − = R + X C [[ X ]] a − = C [[ X ]]we get t = t = 5 , t = 5 , t = 5, so that the type sequence is ( t, tn , tn )and the ring is of maximal length.The examples above are generalized semigroup rings, GSR for short, i.e.rings of the form k + XV + · · · + X N − V N − + X N K [[ X ]]where V i are k -vector subspaces of K . To every one-dimensional analyticallyirreducible ring R can be associated a GSR e R , as in [9]. More precisely e R isthe subring of K [[ X ]] defined, with the notation of the previous section, as e R := X i ≥ V R ( i ) X i That is in fact the generalization of the way of associating to R , in the residuallyrational case, the semigroup ring k [[ S ]], where S = v ( R ). Observe howeverthat the type sequence of R and its associated GSR is not always the same.For example, if R = k [[ X , X + X , X ]], with characteristic of k unequal to2, then the associated GSR is k [[ X , X , X , X ]], which has type sequence(3 , , , R is (2 , , , a − contains no element with value 2, but a − contains X − X . However we canprove that: Proposition 8.
The ring R is almost Gorenstein if and only if the associatedGSR e R is almost Gorenstein and type( R ) = type( e R ) . roof. Let ω be a canonical ideal of R , R ⊆ ω ⊆ R (cf. [2] for the definitionand the existence). Then, by [9, Theorem 17], e ω = P i ≥ V ω ( i ) X i is a canonicalideal of e R . Moreover, by [9, equation (24) and Lemma 16], we have: ℓ R ( ω/R ) = ℓ e R ( e ω/ e R ) ≥ type( e R ) − ≥ type( R ) − R is almost Gorenstein if and onlyif ℓ R ( ω/R ) ≥ type( R ) −
1. Thus R is almost Gorenstein if and only if bothinequalities of (29) are equalities, that is if and only if e R is almost Gorensteinand type( R ) = type( e R ).We shall give now an example of computing the type sequence of a ringwhich is not a GSR and the length ℓ R ( R/R ) is not minimal nor maximal.
Example 3
Let R = R [[ iX + X , X , iX + X , X ]]. For this ring k = R and K = C . Observe that R is not a GSR, but we can compute itsassociated GSR. First let us try to compute the type sequence of R . Aftersome computations we can write R as: R = R + ( iX + X ) R + X R + ( − X + 2 iX + X ) R + ( iX + X ) R + ( − iX +3 iX + X ) R + X R + ( iX + X ) R + ( − X + 2 iX ) R + ( X − X ) R + X R + ( iX + X ) R + iX R + X C [[ X ]].Thus for the ring R we have: the conductor of the ring N = 15 and n = 1, n = 1, n = 1, n = 1, n = 1, n = 1, n = 2, n = 1, n = 1, n = 2, n = 1.We compute now the inverses of the ideals which appear in the definition of thetype sequence. m − = R + ( iX + X ) R + X R + ( − X + 2 iX + X ) R + ( X − X ) R + ( iX + X ) R + ( − iX + 3 iX + X ) R + X R + ( iX + X R + X R + X C [[ X ]]. a − = R + ( iX + X ) R + X R + ( − X + 2 iX + X ) R + ( X − X ) R + ( iX + X ) R + X i R + X C [[ X ]]. a − = R + ( iX + X ) R + X R + ( − X + 2 iX + X ) R + X R + ( iX + X ) R + X i R + X C [[ X ]]. a − = R + ( iX + X ) R + X R + X R + X C [[ X ]]. a − = R + ( iX + X ) R + X R + X C [[ X ]]. a − = R + ( iX + X ) R + X C [[ X ]]. a − = R + ( iX − X ) R + X i R + X C [[ X ]] a − = R + X i R + X C [[ X ]]. a − = R + X C [[ X ]]. a − = R + X C [[ X ]]. a − = C [[ X ]] = R .Then: t = ℓ R ( m − /R ) = 3, t = 1, t = 2, t = 1, t = 1, t = 1, t = 3, t = 1, t = 1, t = 2, t = 1.The associated GSR of R is e R = R [[ iX , X , iX , iX ]]. After computations9e have that the type sequence of e R is ( t = 3 , t = 1 , t = 2 , t = 1 , t = 1 , t =1 , t = 3 , t = 1 , t = 1 , t = 2 , t = 1). We observe that in this case the typesequence of R is equal to the type sequence of its associated GSR. References [1] V. Barucci, D.E. Dobbs, M. Fontana:
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