Markus Brodmann

The asymptotic nature of the analytic spread

In (3), corollary, p. 373) Burch gives the following inequality for the analytic spread l ( I ) of an ideal I of a noetherian local ring ( R , m ): In this paper we shall improve this by showing that the number min depth ( R / I n ) may be replaced by the asymptotic value of depth ( R / I n ) for large n (which exists) (see Section (2)). By its definition (see (6), def. 3)) the analytic spread is of asymptotic nature, i.e. depends on the modules I n / mI n = U n only for large n . We shall prove a stronger result, Section (4), which also shows the asymptotic nature of l ( I ). This result might be interesting for itself, particularly as it is not of local nature. Once Section (4) is proved and once we know that depth ( R / I n ) is asymptotically constant (which turns out to be an easy consequence of ( 1 ), (1)), our improved inequality is easily established: Indeed, replacing R by R / xR where x is regular with respect to almost all modules ( R / I n ), we perform a change which affects only finitely many of the modules U n (see Section (8)).

A finiteness result for associated primes of local cohomology modules

We show that the first non-finitely generated local cohomology module Hi a (M ) of a finitely generated module M over a noetherian ring R with respect to an ideal a ⊆ R has only finitely many associated primes.

Cohomological patterns of coherent sheaves over projective schemes

Abstract We study the sets P(X, F )={(i,n)∈ N 0 × Z | H i (X, F (n))≠0} , where X is a projective scheme over a noetherian ring R 0 and where F is a coherent sheaf of O X -modules. In particular we show that P(X, F ) is a so called tame combinatorial pattern if the base ring R 0 is semilocal and of dimension ⩽1. If X= P d R 0 is a projective space over such a base ring R 0 , the possible sets P(X, F ) are shown to be precisely all tame combinatorial patterns of width ⩽ d . We also discuss the “tameness problem” for arbitrary noetherian base rings R 0 and prove some stability results for the R 0 -associated primes of the R 0 -modules H i (X, F (n)) .

On annihilators and associated primes of local cohomology modules

We establish the Local-global Principle for the annihilation of local cohomology modules over an arbitrary commutative Noetherian ring R at level 2. We also establish the same principle at all levels over an arbitrary commutative Noetherian ring of dimension not exceeding 4. We explore interrelations between the principle and the Annihilator Theorem for local cohomology, and show that, if R is universally catenary and all formal fibres of all localizations of R satisfy Serres condition (Sr), then the Annihilator Theorem for local cohomology holds at level r over R if and only if the Local-global Principle for the annihilation of local cohomology modules holds at level r over R. Moreover, we show that certain local cohomology modules have only finitely many associated primes. This provides motivation for the study of conditions under which the set ⋃m,n∈NAss(M/(xm,yn)M) (where M is a finitely generated R-module and x,y∈R) is finite: an example due to M. Katzman shows that this set is not always finite; we provide some sufficient conditions for its finiteness.

Arithmetic properties of projective varieties of almost minimal degree

We study the arithmetic properties of projective varieties of almost minimal degree, that is of non-degenerate irreducible projective varieties whose degree exceeds the codimension by precisely

Local cohomology over homogeneous rings with one-dimensional local base ring

2

Local cohomology of certain Rees- and form-rings, I

. We notably show, that such a variety

Finiteness of ideal transforms

X \subset {\mathbb{P}}^r

On the dimension and multiplicity of local cohomology modules

is either arithmetically normal (and arithmetically Gorenstein) or a projection of a variety of minimal degree

Associated primes of graded components of local cohomology modules

\tilde {X} \subset {\mathbb{P}}^{r + 1}