Ideals In Regular Local Rings Research Articles

Vanishing of Avramov obstructions for products of sequentially transverse ideals

Two ideals I and J are called transverse if I ∩ J = I J . We show that the obstructions defined by Avramov for products of (sequentially) transverse ideals in regular local rings are always 0. In particular, we compute an explicit free resolution and Koszul homology for all such ideals. Moreover, we construct an explicit trivial Massey operation on the associated Koszul complex and hence (by Golod's construction) a minimal free resolution of the residue field over the quotient defined by the product of transverse ideals. We conclude with questions about the existence of associative multiplicative structures on the minimal free resolution of the quotient defined by products of transverse ideals.

Extended plus closure in complete local rings

The (full) extended plus closure was developed as a replacement for tight closure in mixed characteristic rings. Here it is shown by adapting André's perfectoid algebra techniques that, for complete local rings that have F-finite residue fields, this closure has the colon-capturing property. In fact, more generally, if R is a (possibly ramified) complete regular local ring of mixed characteristic that has an F-finite residue field, I and J are ideals of R, and the local domain S is a finite R-module, then (IS:J)⊆(I:J)Sepf. A consequence is that all ideals in regular local rings are closed, a fact which implies the validity of the direct summand conjecture and the Briançon–Skoda theorem in mixed characteristic.

Stability properties of powers of ideals in regular local rings of small dimension

Let $(R,\mathfrak{m})$ be a regular local ring or a polynomial ring over a field, and let $I$ be an ideal of $R$ which we assume to be graded if $R$ is a polynomial ring. Let astab$(I)$ resp. $\overline{\rm astab}(I)$ be the smallest integer $n$ for which Ass$(I^n)$ resp. Ass$(\overline{I^n})$ stabilize, and dstab$(I)$ be the smallest integer $n$ for which depth$(I^n)$ stabilizes. Here $\overline{I^n}$ denotes the integral closure of $I^n$. We show that astab$(I)=\overline{\rm astab}(I)={\rm dstab}(I)$ if dim$\,R\leq 2$, while already in dimension $3$, astab$(I)$ and $\overline{\rm astab}(I)$ may differ by any amount. Moreover, we show that if dim$\,R=4$, then there exist ideals $I$ and $J$ such that for any positive integer $c$ one has ${\rm astab}(I)-{\rm dstab}(I)\geq c$ and ${\rm dstab}(J)-{\rm astab}(J)\geq c$.

On the Cohen–Macaulayness of the conormal module of an ideal

In the present paper we investigate a question stemming from a long-standing conjecture of Vasconcelos: given a generically complete intersection perfect ideal I in a regular local ring R, when is it true that the Cohen–Macaulayness of I/I2 (or R/I2) implies that R/I is Gorenstein? This property is known to hold for licci ideals and, essentially, squarefree monomial ideals. We show that a positive answer actually holds for every monomial ideal. We then give a positive answer for several special classes of ideals and provide application to algebroid curves with low multiplicity. We also exhibit prime ideals in regular local rings and homogeneous level ideals in polynomial rings for which the answer is negative and use them to show the sharpness of our main result, as they lie in the first class of ideals not covered by it. The homogeneous examples have been found thanks to the help of J.C. Migliore. As a by-product, we exhibit several classes of Cohen–Macaulay ideals whose square is not Cohen–Macaulay. Our methods work both in the homogeneous and in the local settings.

Integrally closed ideals in regular local rings of dimension two

Let I be an integrally closed m -primary ideal of a two-dimensional regular local ring ( R , m , k ) . By Zariski’s unique factorization theorem, the ideal I factors as I 1 s 1 ⋯ I h s h , where the s i are positive integers and the I i are distinct simple integrally closed ideals. For each integer i with 1 ≤ i ≤ h , let V i denote the unique Rees valuation ring associated with I i , and let v i denote the Rees valuation associated with V i . Assume that the field k is relatively algebraically closed in the residue field k ( v i ) of V i . Let J = ( a , b ) R be a reduction of I and let ( a b ) v i ¯ denote the image of a b in the residue field k ( v i ) of V i . We prove that [ k ( v i ) : k ( ( a b ) v i ¯ ) ] = s i . Assume that ( e , f ) R is a reduction of I i . We prove that there exists a minimal generating set { g e s i , g e s i − 1 f , … , g f s i , ξ s i + 1 , … , ξ r } for I , where g ∈ ∏ j ≠ i I j s j with g V i = ( ∏ j ≠ i I j s j ) V i and v i ( ξ p ) > v i ( I ) for p with s i + 1 ≤ p ≤ r . There exists Q i ∈ Min ( m R [ I t ] ) such that V i = R [ I t ] Q i ∩ Q ( R ) . We prove that the quotient ring R [ I t ] / Q i is normal Cohen–Macaulay with minimal multiplicity at its maximal homogeneous ideal with this multiplicity being s i . In particular, R [ I t ] / Q i is regular if and only if s i = 1 .

A vanishing theorem for finitely supported ideals in regular local rings

A cohomological vanishing property is proved for finitely supported ideals in an arbitrary d-dimensional regular local ring. (Such vanishing implies some refined Briancon-Skoda-type results, not otherwise known in mixed char- acteristic.) It follows that the adjoint e I of a finitely supported ideal I has order ord e I = sup(ordI +1 −d, 0), and that taking adjoints of finitely supported ideals commutes with taking strict transforms at infinitely near points. In particular, e I is also finitely supported.

Maximally differential ideals in regular local rings

It is shown that ifA is a regular local ring andI is a maximally differential ideal inA, thenI is generated by anA-sequence.

Integrally Closed Modules and Their Divisors

ABSTRACT There is a beautiful theory of integral closure of ideals in regular local rings of dimension two, due to Zariski, several aspects of which were later extended to modules. Our goal is to study integral closures of modules over normal domains by attaching divisors/determinantal ideals to them. They will be of two kinds: the ordinary Fitting ideal and its divisor, and another ‘determinantal’ ideal obtained through Noether normalization. They are useful to describe the integral closure of some class of modules and to study the completeness of the modules of Kähler differentials.

Full ideals in regular local rings

Paper withdrawn by author after original electronic posting date of May 22, 2003 and prior to preparation of the printed issue.

The $a$-invariant and Gorensteinness of graded rings associated to filtrations of ideals in regular local rings

Let A be a regular local ring and let F = {F n } n ∈ z be a filtration of ideals in A such that R(F) = ○+ n>0 F n is a Noetherian ring with dimR(F) = dimA + 1. Let G(F) = ○+ n>0 F n /F n+1 and let a(G(F)) be the a-invariant of G(F). Then the theorem says that F 1 is a principal ideal and F n = F n 1 for all n ∈ Z if and only if G(F) is a Gorenstein ring and a(G(F)) = -1. Hence a(G(F)) ≤ -2, if G(F) is a Gorenstein ring, but the ideal F 1 is not principal.

Quadratic transform of complete ideals in regular local rings

Quadratic transform of complete ideals in regular local rings

The Positivity of Intersection Multiplicities and Symbolic Powers of Prime Ideals

Serre's nonnegativity conjecture for intersection multiplicities has recently been proven by O. Gabber. In this paper we investigate Serre's positivity conjecture using the methods which he developed. We show in particular that the positivity conjecture has implications for properties of symbolic powers of prime ideals in regular local rings.

Unique factorization of *-products of one-fibered monomial ideals

Let R k[x 1,x 2,x 3] be a 3-dimensional polynomial ring over a field k.We show that no non-trivial relations exist among *-products of complete one-fibered (x 1,x 2,x 3)-primary monomial ideals. This is connected with questions about how Zariski's work on the unique factorization of complete ideals in regular local rings of dimension two might be generalized to higher dimensions.

Symbolic squares in regular local rings

In this paper we prove for several classes of ideals in regular local rings of equicharacteristic 0 that the symbolic square of the ideal contains no minimal generator of the ideal. Our techniques come from residual intersections, integral closures of ideals, and differentials.

Adjoints of ideals in regular local rings

The adjoint of an ideal I in a regular local ring R is the R-ideal adj(I):=H^0(Y, I\omega_Y), where f:Y -> Spec(R) is a proper birational map with Y nonsingular and IO_Y invertible, and \omega_f is a canonical relative dualizing sheaf. (Such an f is supposed to exist.) The basic conjecture is that I.adj(I^n)=adj(I^{n+1}) whenever n is >= the analytic spread of I. This is a strong version of the Briancon-Skoda theorem, implying all other known versions. It follows from the (conjectural) vanishing of H^i(Y,\omega_Y) for all i>0. When R is essentially of finite type over a char. 0 field, that vanishing has been deduced by Cutkosky from Kodaira vanishing. It also holds whenever dim.R = 2; and in this case we can say considerably more about adjoints, tying in e.g., with classical material on adjoint curves and conductors.

Almost complete intersections and factorial rings

It is our purpose in this paper to investigate the relationship between two classes of ideals in regular local rings R: prime ideals P such that R/P is factorial, and prime ideals Q which are minimally generated by htQ + 1 elements. These latter ideals are called almost complete intersections. Peskine and Szpiro [ 121 say two unmixed ideals Z and J in a regular local ring R are (geometrically) linked if Z and .Z have no common minimal prime ideals and if there is an R-sequence xi ,..., x, such that (xi ,..., x,) = Zn.Z. We will delete the word “geometricaly” throughout this paper; linkage will always be used in the sense above. Murthy [ 1 l] essentially showed that if R is a regular local ring and P is a prime ideal such that R/P is factorial, then P is linked to an almost complete intersection. In Section 1 we provide a new and simple proof of this result using a generalization of a theorem of Hartshorne [2] concerning the connectedness of Spec(R). Recall that a ring R is said to satisfy Serre’s condition S, if the following condition holds:

On Unmixed Ideals in Regular Local Rings

On Unmixed Ideals in Regular Local Rings