Kazuhiko Kurano

Numerical equivalence defined on Chow groups of Noetherian local rings

In the present paper, we define a notion of numerical equivalence on Chow groups or Grothendieck groups of Noetherian local rings, which is an analogue of that on smooth projective varieties. Under a mild condition, it is proved that the Chow group modulo numerical equivalence is a finite dimensional ℚ-vector space, as in the case of smooth projective varieties. Numerical equivalence on local rings is deeply related to that on smooth projective varieties. For example, if Grothendieck’s standard conjectures are true, then a vanishing of Chow group (of local rings) modulo numerical equivalence can be proven. Using the theory of numerical equivalence, the notion of numerically Roberts rings is defined. It is proved that a Cohen–Macaulay local ring of positive characteristic is a numerically Roberts ring if and only if the Hilbert–Kunz multiplicity of a maximal primary ideal of finite projective dimension is always equal to its colength. Numerically Roberts rings satisfy the vanishing property of intersection multiplicities. We shall prove another special case of the vanishing of intersection multiplicities using a vanishing of localized Chern characters.

Adams operations, localized Chern characters, and the positivity of Dutta multiplicity in characteristic 0

The positivity of the Dutta multiplicity of a perfect complex of A-modules of length equal to the dimension of A and with homology of finite length is proven for homomorphic images of regular local rings containing a field of characteristic zero. The proof uses relations between localized Chern characters and Adams operations.

The Positivity of Intersection Multiplicities and Symbolic Powers of Prime Ideals

Serres nonnegativity conjecture for intersection multiplicities has recently been proven by O. Gabber. In this paper we investigate Serres 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.

Gorenstein Isolated Quotient Singularities of Odd Prime Dimension are Cyclic

In this article, we shall prove that Gorenstein isolated quotient singularities of odd prime dimension are cyclic. In the case where the dimension is bigger than 1 and is not an odd prime number, then there exist Gorenstein isolated noncyclic quotient singularities.

On relations on minors of generic symmetric matrices

Let R be a commutative ring with 1, and fix an integer n B 1. Suppose X, are variables with 1 d i<j< n. We denote by S = R[X] the polynomial ring over R with n(n + 1)/2 variables X,. Assume X, = X,, when 1 d i <j< n. Then (X,) is the generic (n x n)-symmetric matrix with entries in S. For a positive integer p such that 1 dp<n, let J, be the ideal of S defined by all p-minors of (X,). The rings S/J, have been the classical objects of intensive study. For instance, by the first and the second fundamental theorems [2, 111, it is well known that when an orthogonal group acts on a certain polynomial ring in a certain way, the ring of invariants is described in the form of S/J,. Furthermore it has been proved that when R is Cohen-Macaulay, S/J, is also Cohen-Macaulay and the equality depth(J,) = proj dim,(S/J,) = (n -p + 1 )(n -p + 2)/2 holds [7]. In [S], Jbzetiak, Pragacz, and Weyman constructed the minimal free resolution of S/J, for any n and p, when R contains the rational number field (the minimal free resolution means the resolution such that its entries of boundary maps do not have constant terms). But over an arbitrary commutative ring R minimal free resolutions have not been obtained in general. In the cases where p = 1 and p = n, we can construct the minimal free resolutions of S/J,, from the Koszul complex. Further, when p = n 1, minimal free resolutions of S/J, were explicitly constructed in [3, 41. In this article, we prove that the relation module of minors of (X,) is generated by degree 0 and 1 relations on minors, and explicitly describe those relations. Moreover, by using this result, we prove the existence of minimal free resolutions of S/J, when p = n 2. The basic idea is the same as that of [6], but we use plethysm formulas in place of Cauchy’s formula used in [6]. 388 0021-8693/89

On the vanishing and the positivity of intersection multiplicities over local rings with small non-complete intersection loci

3.00

The canonical module of a Cox ring

Throughout this paper A is a commutative Noetherian ring of dimension d with the maximal ideal m and we assume that there exists a regular local ring S such that A is a homomorphic image of S , i.e., A = S/I for some ideal I of S . Furthermore we assume that A is equi-dimensional, i.e., dim A = dim A / for any minimal prime ideal of A . We put .

Hilbert–Kunz Functions over Rings Regular in Codimension One

In this paper, we shall describe the graded canonical module of a Noetherian multi-section ring of a normal projective variety. In particular, in the case of the Cox ring, we prove that the graded canonical module is a graded free module of rank one with the shift of degree

On Chow Groups of G-Graded Rings

K_X

Todd Classes of Affine Cones of Grassmannians

. We shall give two kinds of proofs. The first one utilizes the equivariant twisted inverse functor developed by the first author. The second proof is down-to-earth, that avoids the twisted inverse functor.