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Fiber Invariants of Projective Morphisms and Regularity of Powers of Ideals
Sankhaneel Bisui
,
Huy Tài Hà
,
Abu Chackalamannil Thomas
Abstract
We introduce an invariant, associated to a coherent sheaf of graded modules over a projective morphism of schemes, which controls when sheaf cohomology can be passed through the given morphism. We then use this invariant to estimate the stability indexes of the regularity and $a^*$-invariant of powers of homogeneous ideals. Specifically, for an equigenerated homogeneous ideal $I$ in a standard graded algebra over a Noetherian ring, we give bounds for the smallest values of power $q$ starting from which $a^*(I^q)$ and $\mathrm{reg}(I^q)$ become linear functions.