The regular algebra of a poset

Ara, Pere

doi:10.1090/S0002-9947-09-04884-3

The regular algebra of a poset
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Trans. Amer. Math. Soc. 362 (2010), 1505-1546 Request permission

Abstract:

Let $K$ be a fixed field. We attach to each finite poset $\mathbb P$ a von Neumann regular $K$-algebra $Q_K(\mathbb P)$ in a functorial way. We show that the monoid of isomorphism classes of finitely generated projective $Q_K(\mathbb P)$-modules is the abelian monoid generated by $\mathbb P$ with the only relations given by $p=p+q$ whenever $q<p$ in $\mathbb P$. This extends the class of monoids for which there is a positive solution to the realization problem for von Neumann regular rings.

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