Let $(R,𝔪)$ be a commutative Noetherian regular local ring of dimension $d$ and $I$ be a proper ideal of $R$ such that ${\mathrm{mAss}}_R(R/I)={\mathrm{Assh}}_R\left(I\right)$. It is shown that the $R$-module $H_I^{\mathrm{ht}\left(I\right)}\left(R\right)$ is $I$-cofinite if and only if $\mathrm{cd}(I,R)=\mathrm{ht}\left(I\right)$. Also we present a sufficient condition under which this condition the $R$-module $H_I^i\left(R\right)$ is finitely generated if and only if it vanishes.

Let R be a Noetherian ring and I an ideal of R. Let M be an I-cofinite and N a finitely generated R-module. It is shown that the R-modules $To{r}_i^R(N,M)$ are I-cofinite for all i ≥ 0 whenever dim Supp(M) ≤ 1 or dim Supp(N) ≤ 2. This immediately implies that if I has dimension one (i.e., dim R/I = 1) then the R-modules $To{r}_i^R(N,H_I^j\left(M\right))$ are I-cofinite for all i,j ≥ 0. Also, we prove that if R is local, then the R-modules $To{r}_i^R(N,M)$ are I-weakly cofinite for all i ≥ 0 whenever dim Supp(M) ≤ 2 or dim Supp(N) ≤ 3. Finally, it is shown that...

Let denote an ideal in a Noetherian ring R, and M a finitely generated R-module. We introduce the concept of the cohomological dimension filtration $={M_i}_{i=0}^c$, where c = cd(,M) and $M_i$ denotes the largest submodule of M such that $cd(,M_i)\le i$. Some properties of this filtration are investigated. In particular, if (R,) is local and c = dim M, we are able to determine the annihilator of the top local cohomology module $H_{}^c\left(M\right)$, namely $An{n}_R\left(H_{}^c\left(M\right)\right)=An{n}_R(M/M_{c-1})$. As a consequence, there exists an ideal of R such that $An{n}_R\left(H_{}^c\left(M\right)\right)=An{n}_R(M/H{⁰}_{}\left(M\right))$. This generalizes the main results...

We use combinatorics to describe the topology of a singular irreducible plane curve germ f = 0 under small perturbation of parameters.

Discuterò una costruzione geometrica, fatta insieme a De Concini, di una modificazione di una configurazione di sottospazi che trasforma i sottospazi in un divisore a incroci normali. Inoltre nel caso di iperpiani questa costruzione è legata alla generalizzazione della equazione di Kniznik-Zamolodchikov ed alla teoria dei nodi, per i sistemi di radici produce dei modelli particolarmente interessati.

Alexander and Hirschowitz determined the Hilbert function of a generic union of fat points in a projective space when the number of fat points is much bigger than the greatest multiplicity of the fat points. Their method is based on a lemma which determines the limit of a linear system depending on fat points approaching a divisor.Other Hilbert functions were computed previously by Nagata. In connection with his counter-example to Hilbert’s fourteenth problem, Nagata determined the Hilbert function...

Nous déduisons de la formule du conducteur, conjecturée par S. Bloch, celle de P. Deligne exprimant, dans le cas d'une singularité isolée, la dimension totale des cycles évanescents en fonction du nombre de Milnor. En particulier, la formule de Deligne est établie en dimension relative un; en appendice, on généralise cet énoncé au cas d'un lieu singulier propre.