Iterations and logarithms of formal automorphisms

Using the decomposition of an automorphism of the ring of formal power series in several variables, in a semisimple and a unipotent automorphism, I prove in this paper that an automorphism allows a continuous iteration if and only if it is the exponential of a derivation. This result implies a number of results recently obtained by Reich, Schwaiger, and Bucher.     

Degree functions and projectively full ideals in 2-dimensional rational singularities that can be desingularized by blowing up the unique maximal ideal

Let (R,m) be a 2-dimensional rational singularity with algebraically closed residue field and for which the associated graded ring is an integrally closed domain. According to Göhner, (R,m) satisfies condition (N): given a prime divisor v, there exists a unique complete m-primary ideal A_v in R with T(A_v)={v} and such that any complete m-primary ideal with unique Rees valuation v, is a power of A_v. We use the theory of degree functions developed by Rees and Sharp as well as some results about regular local rings, to investigate the degree coefficients d(A_v,v). As an immediate corollary, we find that for a simple complete m₁-primary ideal I₁ in an immediate quadratic transform (R₁,m₁) of (R,m); the inverse transform of I₁ in R is projectively full.     

Some stably tame polynomial automorphisms

We study the structure of length three polynomial automorphisms of R[X,Y] when R is a UFD. These results are used to prove that if SL_m(R[X₁,X₂,…,X_n])=E_m(R[X₁,X₂,…,X_n]) for all n≥0 and for all m≥3 then all length three polynomial automorphisms of R[X,Y] are stably tame.     

Cancellation of projective modules over polynomial extensions over a two-dimensional ring

Let R be a commutative Noetherian ring of dimension two with $1/2\in R$ and let $A=R[X_1,?,X_n]$. Let P be a projective A-module of rank 2. In this article, we prove that P is cancellative if ${\wedge }^2(P)\oplus A$ is cancellative.     

Projective generation of ideals in a certain ring

Let R be an affine domain of dimension $n\ge 3$ over a field of characteristic 0 and $D=R[X,Y]/(XY)$. Let $I?D$ be a local complete intersection ideal of height n such that $\mu (I/I^2)=n$. This paper examines under what condition I is surjective image of a projective D-module of rank n.     

A locally Nagata PID that is not Nagata

It has previously been shown by an example of dimension one that a locally excellent integral domain need not be excellent or even Nagata. A related example showed that a locally excellent normal domain of dimension at least two need not be excellent, though that example is Nagata. Here we present a locally excellent principal ideal domain which is neither excellent nor Nagata. A second example is a two dimensional locally excellent regular ring which is Japanese but not Nagata.     

On ring derivations and quadratic functionals

Summary In this note the general form of ring derivations on a function algebra is obtained. In particular, we give a very simple proof of the following result of Nandakumar: a ring derivation on a function algebra is trivial provided that the Choquet boundary of the algebra contains a dense sequentially non-isolated set. Using this theorem we prove a result concerning representability of quadratic functionals by bilinear forms.This work was supported by the Research Council of Slovenia.     

Strong Krull primes and flat modules

There are several theorems describing the intricate relationship between flatness and associated primes over commutative Noetherian rings. However, associated primes are known to act badly over non-Noetherian rings, so one needs a suitable replacement. In this paper, we show that the behavior of strong Krull primes most closely resembles that of associated primes over a Noetherian ring. We prove an analogue of a theorem of Epstein and Yao characterizing flat modules in terms of associated primes by replacing them with strong Krull primes. Also, we partly generalize a classical equational theorem regarding flat base change and associated primes in Noetherian rings. That is, when associated primes are replaced by strong Krull primes, we show containment in general and equality in many special cases. One application is of interest over any Noetherian ring of prime characteristic. We also give numerous examples to show that our results fail if other popular generalizations of associated primes are used in place of strong Krull primes.     

The symmetric group given by a Gröbner basis

We present a Gröbner basis associated with the symmetric group of degree n, which is determined by a strong generating set of the symmetric group and is defined by means of a term ordering with the elimination property.     

A class of retracts of polynomial algebras

We show that all retractions of polynomial algebras with sparse homogeneous parts are conjugate to the canonical ones, and as a consequence, the corresponding retracts are isomorphic to polynomial algebras.     

Quasi-socle ideals and Goto numbers of parameters

Goto numbers for certain parameter ideals Q in a Noetherian local ring (A,m) with Gorenstein associated graded ring are explored. As an application, the structure of quasi-socle ideals I=Q:m^q (q≥1) in a one-dimensional local complete intersection and the question of when the graded rings are Cohen-Macaulay are studied in the case where the ideals I are integral over Q.     

Duality and vanishing of generalized local cohomology

In this paper we study local duality and the vanishing and non-vanishing of generalized local cohomology. In particular for a local ring R and a finitely generated R-module N, the category $N^\bot$ of finitely generated R-modules M with $H^{i}_{\frak m}(M, N) = 0$ for all i depth N is studied.Received: 31 October 2002     

Quasi-socle ideals in a Gorenstein local ring

This paper explores the structure of quasi-socle ideals I=Q:m² in a Gorenstein local ring A, where Q is a parameter ideal and m is the maximal ideal in A. The purpose is to answer the problems as to when Q is a reduction of I and when the associated graded ring is Cohen-Macaulay. Wild examples are explored.     

Bounds on Castelnuovo-Mumford regularity for generalized Cohen-Macaulay graded rings

Supported by a grant from the Centre de Recerca Matematica, Institut d'Estudis Catalans (Spain)     

Euler class groups and a theorem of Roitman

We define the Euler class group of a polynomial algebra in lower codimension and prove an analogue of a result of Roitman (on projective modules) for the Euler class groups.     

Homology of perfect complexes

It is proved that the sum of the Loewy lengths of the homology modules of a finite free complex F over a local ring R is bounded below by a number depending only on R. This result uncovers, in the structure of modules of finite projective dimension, obstructions to realizing R as a closed fiber of some flat local homomorphism. Other applications include, as special cases, uniform proofs of known results on free actions of elementary abelian groups and of tori on finite CW complexes. The arguments use numerical invariants of objects in general triangulated categories, introduced here and called levels. They allow one to track, through changes of triangulated categories, homological invariants like projective dimension, as well as structural invariants like Loewy length. An intermediate result sharpens, with a new proof, the New Intersection Theorem for commutative algebras over fields. Under additional hypotheses on the ring R stronger estimates are proved for Loewy lengths of modules of finite projective dimension.     

Autour def(x)f(y) −f(xy)

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