A generalization of Hartshorne's connectedness theorem

In this paper, we use local cohomology theory to present some results about connectedness property of prime spectrum of modules. In particular, we generalize the Hartshorne's connectedness theorem.

Limit behavior of the rational powers of monomial ideals

We investigate the rational powers of ideals. We find that in the case of monomial ideals, the canonical indexing leads to a characterization of the rational powers yielding that symbolic powers of squarefree monomial ideals are indeed rational powers themselves. Using the connection with symbolic powers techniques, we use splittings to show the convergence of depths and normalized Castelnuovo–Mumford regularities. We show the convergence of Stanley depths for rational powers, and as a consequence of this, we show the before-now unknown convergence of Stanley depths of integral closure powers. Additionally, we show the finiteness of asymptotic associated primes, and we find that the normalized lengths of local cohomology modules converge for rational powers, and hence for symbolic powers of squarefree monomial ideals.

-modules on rigid analytic spaces III: weak holonomicity and operations

We develop a dimension theory for coadmissible $\widehat {\mathcal {D}}$ -modules on rigid analytic spaces and study those which are of minimal dimension, in analogy to the theory of holonomic $\mathcal {D}$ -modules in the algebraic setting. We discuss a number of pathologies contained in this subcategory (modules of infinite length, infinite-dimensional fibres). We prove stability results for closed immersions and the duality functor, and show that all higher direct images of integrable connections restricted to a Zariski open subspace are coadmissible of minimal dimension. It follows that the local cohomology sheaves $\underline {H}^{i}_Z(\mathcal {M})$ with support in a closed analytic subset $Z$ of $X$ are also coadmissible of minimal dimension for any integrable connection $\mathcal {M}$ on $X$ .