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We study the first Hilbert coefficient (after the multiplicity) $e_1$ of a local ring $(A,m). $ Under various circumstances, it is also called the {bf Chern number} of the local ring $A.$ Starting from the work of D.G. Northcott in the 60s, several results have been proved which give some relationships between the Hilbert coefficients, but always assuming the Cohen-Macaulayness of the basic ring. Recent papers of S. Goto, K. Nishida, A. Corso and W. Vasconcelos pushed the interest toward a more general setting. In this paper we extend an upper bound on $e_1$ proved by S. Huckaba and T. Marley. Thus we get the Cohen-Macaulayness of the ring $A$ as a consequence of the extremal behavior of the integer $e_1.$ The result can be considered a confirm of the general philosophy of the paper of W. Vasconcelos where the Chern number is conjectured to be a measure of the distance from the Cohen-Macaulyness of $A.$ This main result of the paper is a consequence of a nice and perhaps unexpected property of superficial elements. It is essentially a kind of Sally machine for local rings. In the last section we describe an application of these results, concerning an upper bound on the multiplicity of the Sally module of a good filtration of a module which is not necessarily Cohen-Macaulay. It is an extension to the non Cohen-Macaulay case of a result of Vaz Pinto.
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In this paper we introduce an effective method to construct rational deformations between couples of Borel-fixed ideals. These deformations are governed by flat families, so that they correspond to rational curves on the Hilbert scheme. Looking globally at all the deformations among Borel-fixed ideals defining points on the same Hilbert scheme, we are able to give a new proof of the connectedness of the Hilbert scheme and to introduce a new criterion to establish whenever a set of points defined by Borel ideals lies on a common component of the Hilbert scheme. The paper contains a detailed algorithmic description of the technique and all the algorithms are made available.
Let $(A,mathfrak{m})$ be an analytically unramified formally equidimensional Noetherian local ring with $ depth A geq 2$. Let $I$ be an $mathfrak{m}$-primary ideal and set $I^*$ to be the integral closure of $I$. Set $G^*(I) = bigoplus_{ngeq 0} (I^n)^*/(I^{n+1})^*$ be the associated graded ring of the integral closure filtration of $I$. We prove that $ depth G^*(I^n) geq 2$ for all $n gg 0$. As an application we prove that if $A$ is also an excellent normal domain containing an algebraically closed field isomorphic to $A/m$ then there exists $s_0$ such that for all $s geq s_0$ and $J$ is an integrally closed ideal emph{strictly} containing $(mathfrak{m}^s)^*$ then we have a strict inequality $mu(J) < mu((mathfrak{m}^s)^*)$ (here $mu(J)$ is the number of minimal generators of $J$).
Let R be a local domain, v a valuation of its quotient field centred in R at its maximal ideal. We investigate the relationship between R^h, the henselisation of R as local ring, and {~v}, the henselisation of the valuation v, by focussing on the recent result by de Felipe and Teissier referred to in the title. We give a new proof that simplifies the original one by using purely algebraic arguments. This proof is moreover constructive in the sense of Bishop and previous work of the authors, and allows us to obtain as a by-product a (slight) generalisation of the theorem by de Felipe and Teissier.
Hochsters theta invariant is defined for a pair of finitely generated modules on a hypersurface ring having only an isolated singularity. Up to a sign, it agrees with the Euler invariant of a pair of matrix factorizations. Working over the complex numbers, Buchweitz and van Straten have established an interesting connection between Hochsters theta invariant and the classical linking form on the link of the singularity. In particular, they establish the vanishing of the theta invariant if the hypersurface is even-dimensional by exploiting the fact that the (reduced) cohomology of the Milnor fiber is concentrated in odd degrees in this situation. In this paper, we give purely algebra
Let $A$ be a semigroup whose only invertible element is 0. For an $A$-homogeneous ideal we discuss the notions of simple $i$-syzygies and simple minimal free resolutions of $R/I$. When $I$ is a lattice ideal, the simple 0-syzygies of $R/I$ are the binomials in $I$. We show that for an appropriate choice of bases every $A$-homogeneous minimal free resolution of $R/I$ is simple. We introduce the gcd-complex $D_{gcd}(bf b)$ for a degree $mathbf{b}in A$. We show that the homology of $D_{gcd}(bf b)$ determines the $i$-Betti numbers of degree $bf b$. We discuss the notion of an indispensable complex of $R/I$. We show that the Koszul complex of a complete intersection lattice ideal $I$ is the indispensable resolution of $R/I$ when the $A$-degrees of the elements of the generating $R$-sequence are incomparable.
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