Primary decomposition of commutative monoid congruences is insensitive to certain features of primary decomposition in commutative rings. These features are captured by the more refined theory of mesoprimary decomposition of congruences, introduced here complete with witnesses and associated prime objects. The combinatorial theory of mesoprimary decomposition lifts to arbitrary binomial ideals in monoid algebras. The resulting binomial mesoprimary decomposition is a new type of intersection decomposition for binomial ideals that enjoys computational efficiency and independence from ground field hypotheses. Binomial primary decompositions are easily recovered from mesoprimary decomposition.

Let $L$ be a finite field extension of ${\mathbb{Q}}_p$ and let $G$ be the group of $L$-rational points of a split connected reductive group over $L$. We view $G$ as a locally $L$-analytic group with Lie algebra $\mathfrak{g}$. The purpose of this work is to propose a construction which extends the localization of smooth $G$-representations of P. Schneider and U. Stuhler to the case of locally analytic $G$-representations. We define a functor from admissible locally analytic $G$-representations with prescribed infinitesimal character to a category of equivariant sheaves on the Bruhat–Tits building of $G$. For smooth representations, the corresponding sheaves are closely related to the sheaves of Schneider and Stuhler. The functor is also compatible, in a certain sense, with the localization of $\mathfrak{g}$-modules on the flag variety by A. Beilinson and J. Bernstein.

Soit $\Pi$ une représentation unitaire de ${GL}_2\left({\mathbb{Q}}_p\right)$, topologiquement de longueur finie. Nous décrivons la sous-représentation ${\Pi }^{an}$ de ses vecteurs localement analytiques, et sa filtration par rayon d’analyticité, en termes du $\left(\phi ,\Gamma \right)$-module qui lui est associé via la correspondance de Langlands locale $p$-adique, et nous en déduisons que le complété universel de ${\Pi }^{an}$ n’est autre que $\Pi$.

Let $\Pi$ be a unitary representation of ${GL}_2\left({\mathbb{Q}}_p\right)$, topologically of finite length. We describe the subrepresentation ${\Pi }^{an}$ made of its locally analytic vectors, and its filtration by radius of analyticity, in terms of the $\left(\phi ,\Gamma \right)$-module attached to $\Pi$ via the $p$-adic local Langlands correspondence, and we deduce that the universal completion of ${\Pi }^{an}$ is $\Pi$ itself.

We prove that if a quasitilted algebra is tame, then the associated moduli spaces are products of projective spaces. Together with an earlier result of Chindris this gives a geometric characterization of the tame quasitilted algebras. In our proof we use knowledge of the representation theory of the tame quasitilted algebras and a construction of semi-invariants as determinants.