arXiv subjects

Anne-Marie Simon

Publications and source records attributed to Anne-Marie Simon.

3 recordsLinked to original sources

arXiv2020

Torsion of injective modules and weakly pro-regular sequences

Let $R$ a commutative ring, $\mathfrak{a} \subset R$ an ideal, $I$ an injective $R$-module and $S \subset R$ a multiplicatively closed set. When $R$ is Noetherian it is well-known that the $\mathfrak{a}$-torsion sub-module $Γ_{\mathfrak{a}}(I)$, the factor module $I/Γ_{\mathfrak{a}}(I)$ and the localization $I_S$ are again injective $R$-modules. We investigate these properties in the case of a commutative ring $R$ by means of a notion of relatively-$\mathfrak{a}$-injective $R$-modules. In particular we get another characterization of weakly pro-regular sequences in terms of relatively injective modules. Also we present examples of non-Noetherian commutative rings $R$ and injective $R$-modules for which the previous properties do not hold. Moreover, under some weak pro-regularity conditions we obtain results of Mayer-Vietoris type.

math.AC↗

arXiv2007

Complete intersections of dimension zero: variations on a theme of Wiebe

Wiebe's criterion, which recognizes complete intersections of dimension zero among the class of noetherian local rings, is revisited and exploited in order to provide information on what we call C.I.0-ideals (those such that the corresponding quotient is a complete intersection of dimension zero) and also on chains of C.I.0-ideals. A correspondence is established between C.I.0-ideals and a certain kind of matrices which we call $x$-nice, and a chain of C.I.0-ideals corresponds to a factorization of some $x$-nice matrix. When the local ring $A$ itself is a complete intersection of dimension zero, a C.I.0-ideal is necessarily of the form $(0:bA)$ for some $b\in A$. Some criteria are provided to recognize whether an ideal $(0:bA)$ is C.I.0 or not. When $y$ is a minimal generator of the maximal ideal of $A$, it is also proved that the ideals $yA$ and $(0:yA)$ are C.I.0 simultaneously and that this is the case exactly when the ideal $(0:yA)$ is principal. These C.I.0-ideals of the form $(0:yA)$, $y$ being a minimal generator of the maximal ideal, are investigated. They are of interest because the smallest nonnull C.I.0-ideal in a strict chain of C.I.0-ideals of the maximal length is necessarily of that form, and their existence has some implications for a realization of the ring, i.e. for the way the ring can can be viewed as a quotient of a regular local ring.

math.AC↗

arXiv2003

Stiffness of finite free resolutions and the Canonical Element Conjecture

Over a noetherian local ring certain minimal finite free resolutions possess a property which we call stiffness. This calls to mind the Buchsbaum-Eisenbud criterion for exactness. Yet we only prove stiffness over equicharacteristic rings. However, Hochster's Canonical Element Conjecture is shown to be true for every ring with a fixed prime residual characteristic, precisely when every resolution over each Gorenstein ring of this type is stiff.

math.AC↗