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.

The powers of unmixed bipartite edge ideals

In this paper, we study the depth and the Castelnuovo–Mumford regularity of the powers of edge ideals which are unmixed and whose underlying graphs are bipartite. In particular, we prove that the depth of the powers of the edge ideal stabilizes when the exponent is the same as half the number of vertices in the underlying connected bipartite graph. We also define the idea of “drop” in the sequence of depth of powers of ideals. Further, we show that the sequence of depth of the powers of such edge ideals may have any number of “drops”. In the process of proving these results we put forward some interesting examples and some questions for future research. As for regularity, we establish a formula for the regularity of the powers of such edge ideals in terms of the regularity of the edge ideal itself.

Tight closure of powers of ideals and tight hilbert polynomials

Let (R, ) be an analytically unramified local ring of positive prime characteristic p. For an ideal I, let I* denote its tight closure. We introduce the tight Hilbert function $$H_I^*\left( n \right) = \Im \left( {R/\left( {{I^n}} \right)*} \right)$$ and the corresponding tight Hilbert polynomial $$P_I^*\left( n \right)$$, where I is an m-primary ideal. It is proved that F-rationality can be detected by the vanishing of the first coefficient of $$P_I^*\left( n \right)$$. We find the tight Hilbert polynomial of certain parameter ideals in hypersurface rings and Stanley-Reisner rings of simplicial complexes.