J. K. Verma

Extended Rees algebras and mixed multiplicities

The aim of this paper is to study mixed multiplicities of ideals and use them to calculate multiplicities of certain homogeneous ideals of extended Rees algebras and to give a complete characterization of those parameter ideals whose extended Rees algebras are Cohen-Macaulay with minimal multiplicity at their maximal homogeneous ideals. Let (R, m) be a local ring of dimension d. Let I be an m-primary ideal. It is well known that the length of the artinian ring R/F, I(R/F), is a polynomial of degree d in r for all large r. The coefficient of nd/d ! in this polynomial, called the multiplicity of I, denoted by e(I), is a well-understood and useful invariant of/. Suppose J is another m-primary ideal. Then it is natural to consider l (R/FJ s) for positive integers r and s. It is proved in [B] that for large values of r and s, I(R/FJ ~) is given by a polynomial P(r, s) of total degree d in r and s. Moreover, the terms of total degree d in P(r, s) have the form

Rees algebras and mixed multiplicities

Let (R, m) be a local ring of positive dimension d and I and J two m-primary ideals of R. Let T denote the Rees algebra R[Jt] localized at the maximal homogeneous ideal (m, Jt). It is proved that e((I, Jt)T) = eo(IIJ) + el(I|J) + + ed_l(IIJ) where ei(IIJ), i = 0,1,...,d 1 are the first d mixed multiplicities of I and J. A formula due to Huneke and Sally concerning the multiplicity of the Rees algebra (of a complete zero-dimensional ideal of a two dimensional regular local ring) at its maximal homogeneous ideal is recovered.

Multigraded Rees algebras and mixed multiplicities

Let I1, I2,…, Ig be ideals of positive height in a local ring (R, m). Let I0 be m-primary. Set S = R[I1t1, I2t2,…, Igtg], where t1,…, tg are indeterminates and M = (m, I1t1, I2t2,…, Igtg). A formula is developed for the multiplicity of the ideal (I0, I1t1, I2t2,…, Igtg)SM in terms of mixed multiplicities of I 0, I1,…, Ig. This formula is used to prove that if R is Cohen-Macaulay of dimension 2 and I1 = ma1,…, Ig = mag for positive integers a1, a2,…, ag, then SM is Cohen-Macaulay with minimal multiplicity if and only if R has minimal multiplicity.

Hilbert polynomials and powers of ideals

The growth of Hilbert coefficients for powers of ideals are studied. For a graded ideal I in the polynomial ring S = K[x1, . . ., xn] and a finitely generated graded S-module M, the Hilbert coefficients ei(M/IkM) are polynomial functions. Given two families of graded ideals (Ik)k=0 and (Jk)k=0 with Jk Ik for all k with the property that JkJl Jk+l and IkIl Ik+l for all k and l, and such that the algebras and are finitely generated, we show the function k e0(Ik/Jk) is of quasi-polynomial type, say given by the polynomials P0,. . ., Pg-1. If Jk = Jk for all k, for a graded ideal J, then we show that all the Pi have the same degree and the same leading coefficient. As one of the applications it is shown that , if I is a monomial ideal. We also study analogous statements in the local case.

On fiber cones of m-primary ideals

Two formulas for the multiplicity of the fiber cone F(I) = L 1=0 I n/mIn of an m-primary ideal of a d-dimensional Cohen-Macaulay local ring (R, m) are derived in terms of the mixed mul- tiplicity ed 1(m|I), the multiplicity e(I), and superficial elements. As a consequence, the Cohen- Macaulay property of F(I) when I has minimal mixed multiplicity or almost minimal mixed multi- plicity is characterized in terms of the reduction number of I and lengths of certain ideals. We also characterize the Cohen-Macaulay and Gorenstein properties of fiber cones of m-primary ideals with a d-generated minimal reduction J satisfying l(I 2 / JI) = 1 or l(Im/ Jm) = 1.

JOINT REDUCTIONS OF COMPLETE IDEALS

The aim of this paper is to extend and unify several results concerning complete ideals in 2-dimensional regular local rings by using the theory of joint reductions and mixed multiplicities. The theory of complete ideals in a 2-dimensional regular local ring was developed by Zariski in his 1938 paper [Z]. This theory is presented in a simpler and general form in [ZS, Appendix 5] and [H2]. The Zariskis product theorem asserts that the product of complete ideals in a 2-dimensional regular local ring is again complete. Counterexamples to such a statement in 3-dimensional regular local rings have been given by Huneke in [HI, § 3]. A surprising generalization of Zariskis product theorem was obtained by Huneke and Sally in [H-S, Theorem 4.1]; Let (R, m) be a d-dimensional Cohen-Macaulay local ring satisfying (ί?2) Let ί be a height two complete ideals of analytic spread two. Then P is complete for all positive integers k. Inspired by this theorem, Huneke asked the following

Fiber cones of ideals with almost minimal multiplicity

Fiber cones of 0-dimensional ideals with almost minimal multiplicity in CohenMacaulay local rings are studied. Ratliff-Rush closure of filtration of ideals with respect to another ideal is introduced. This is used to find a bound on the reduction number with respect to an ideal. Rossi’s bound on reduction number in terms of Hilbert coefficients is obtained as a consequence. Sufficient conditions are provided for the fiber cone of 0-dimensional ideals to have almost maximal depth. Hilbert series of such fiber cones are also computed.

Mixed multiplicities of ideals versus mixed volumes of polytopes

The main results of this paper interpret mixed volumes of lattice polytopes as mixed multiplicities of ideals and mixed multiplicities of ideals as Samuels multiplicities. In particular, we can give a purely algebraic proof of Bernsteins theorem which asserts that the number of common zeros of a system of Laurent polynomial equations in the torus is bounded above by the mixed volume of their Newton polytopes.

Local cohomology of Rees algebras and Hilbert functions

Let I be an ideal primary to the maximal ideal in a local ring. We utilize two well-known theorems due to J.-P. Serre to prove that the difference between the Hilbert function and the Hilbert polynomial of I is the alternating sum of the graded pieces of the graded local cohomology (with respect to its positively-graded ideal) of the Rees ring of I. This gives new insight into the higher Hilbert coefficients of I. The result is inspired by one due to J. D. Sally in dimension two and is implicit in a paper by D. Kirby and H. A. Mehran, where very different methods are used.

Grothendieck–Serre formula and bigraded Cohen–Macaulay Rees algebras

Abstract The Grothendieck–Serre formula for the difference between the Hilbert function and Hilbert polynomial of a graded algebra is generalized for bigraded standard algebras. This is used to get a similar formula for the difference between the Bhattacharya function and Bhattacharya polynomial of two m -primary ideals I and J in a local ring (A, m ) in terms of local cohomology modules of Rees algebras of I and J. The cohomology of a variation of the Kirby–Mehran complex for bigraded Rees algebras is studied which is used to characterize the Cohen–Macaulay property of bigraded Rees algebra of I and J for two dimensional Cohen–Macaulay local rings.