Román Álvarez (Universitat Autònoma de Barcelona)
Package Deal Theorems for Localizations over h-local Domains
Abstract: Let $R$ be a commutative ring with total ring of fractions $Q$, let $\Lambda$ be a (not necessarily commutative) $R$-algebra, and let $M$ be a finitely generated right $\Lambda$-module. For each maximal ideal $m$ of $R$, consider a (not necessarily finitely generated) $\Lambda_m$-submodule $X(m)$ of $M_m$. For which such families is there a $\Lambda$-submodule $N$ of $M$ such that $N_m=X(m)$? This question was answered by Levy-Odenthal for $R$ a commutative Noetherian ring of Krull dimension 1 under two consistency hypotheses:
1. $X(m)=M_m$ for almost all maximal ideals $m$ of $R$;
2. $X(m)\otimes Q=M\otimes Q$ for all maximal ideals $m$ of $R$.
They called these kinds of results Package Deal Theorems.
In this talk, I will give a version of this result for a larger class of domains, namely $h$-local domains, which were introduced by Matlis in the 1960s. $h$-local domains are commutative domains with the property that each non-zero element is contained in only finitely many maximal ideals of the ring and each non-zero prime ideal is contained in a unique maximal ideal of the ring. From results of previous work of Herbera-Příhoda and the original techniques of Levy-Odenthal, I will conclude with a Package Deal Theorem for traces of projective modules over $h$-local domains.