# Generalized Multiplicities, Reductions of Ideals, and Depth of Blowup Algebras

#### Abstract

Let R be a Cohen-Macaulay local ring with infinite residue field. We define the notion of Goto-minimal j-multiplicity for ideals of maximal analytic spread and we study the interplay among this new notion, the notion of minimal j-multiplicity of Polini-Xie, and the Cohen-Macaulayness of the blowup algebras of ideals satisfying certain residual assumptions. We also provide a bound on the reduction number of ideals of Goto-minimal j-multiplicity having either Cohen-Macaulay associated graded algebra, or linear decay in the depth of their powers. Over a polynomial ring k[x1,..., xd], we prove a characterization of the j- and epsilon-multiplicities of a monomial ideal as normalized volumes of certain regions. Our results are extensions of Teissier's theorem on Hilbert-Samuel multiplicities of m-primary monomial ideals. We show that every lex-segment ideal L generated in one degree t is a normal ideal. If k in an infinite field, we also prove that L(x1,..., xd) d(t-2)+g-t+1 is contained in the core of L, i.e., the intersection of all its minimal reductions, where g is the height of L.

#### Degree

Ph.D.

#### Advisors

Ulrich, Purdue University.

#### Subject Area

Mathematics

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