Categoria: Seminars

Publicat a

Seminar (Operator Algebras)

Laurent Cantier (Universitat Autònoma de Barcelona)

The Cu$_1$-semigroup as an invariant for K$_1$-obstruction cases

Abstract: The aim of this talk is to explicitly shows that the unitary Cuntz semigroup, defined using the Cuntz semigroup and the K$_1$ group, strictly contains more information than the latter invariants alone. To that end, we construct two C*-algebras, distinguished by their unitary Cuntz semigroup, whose K-Theory and Cu-semigroup are isomorphic. Both A and B, constructed as inductive limits of NCCW 1-algebras, are non-simple unital separable C∗-algebras of stable rank one with K$_1$-obstructions. This shows that a likewise invariant is necessary in order to extend classification results of C*-algebras by means of Cuntz semigroup to the non trivial K$_1$ group case.

Publicat a

Seminar (Operator Algebras)

Ado Dalla Costa (Universidade Federal de Santa Catalina)

Free actions of groups on separated graphs and their associated C*-algebras

Abstract: I will report on joint work with Alcides Buss and Pere Ara on the study of free actions of groups on separated graphs and explain how this structure reflects on the level of their associated C*-algebras. We prove a version of the Gross-Tucker theorem in this context and show how this can be used to describe the various C*-algebras attached to separated graphs carrying a free action. All this leads to certain Landstad-type duality theorems involving these algebras.

Publicat a

Seminar (Operator Algebras)

Pere Ara (Universitat Autònoma de Barcelona)

The inverse semigroup of a separated graph

Abstract: For a directed graph $E$, the graph semigroup $S(E)$ was defined by Ash and Hall in 1975. The graph semigroup $S(E)$ is an inverse semigroup, and has been studied by many authors in connection with the theories of graph C*-algebras, Leavitt path algebras, and topological groupoids. For a separated graph $(E,C)$, the direct analogue of $S(E)$ is not an inverse semigroup in general. However, we will introduce an inverse semigroup $IS(E,C)$ for each separated graph, which produces the same graph semigroup $S(E)$ as above in the non-separated case. We will develop a normal form of the elements of $IS(E,C)$ in close analogy to the Scheiblich normal form for elements of the free inverse semigroup.

This is joint work in progress with Alcides Buss and Ado Dalla Costa, both from Universidade Federal de Santa Catarina (Brazil).

Publicat a

Seminar (Ring Theory)

Dolors Herbera (Universitat Autònoma de Barcelona)

Torsion free modules over commutative domains of Krull dimension 1

Abstract: Let $R$ be a commutative domain. Let $\mathcal F$ be the class of $R$-modules that are infinite direct sums of finitely generated torsion-free modules. In the talk we will discuss the question of whether $\mathcal F$ is closed under direct summands.

If $R$ is local of Krull dimension 1, $\mathcal F$ being closed under direct summands is equivalent to saying that any indecomposable, finitely generated torsion-free module has local endomorphism ring.

For the global case, we show also in the case of Krull dimension 1 that the property on $\mathcal F$ is inherited by the localization at a maximal ideal. Moreover, there is an interesting relation between ranks of indecomposable modules over such localizations.

The machinery we use to prove these results was explained in Roman Álvarez’s talk, in the previous session of the seminar.

Time permitting, we will also discuss the property `being locally a direct summand’ versus `being a direct summand’ in the setting of our problem. The results we obtain allows us to give a complete answer to the initial problem in some particular cases.

The talk is based on a joint work with Roman Álvarez and Pavel Příhoda.

Publicat a

Seminar (Ring Theory)

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.

Publicat a

Seminar (Ring Theory)

Carles Casacuberta (Universitat de Barcelona)

Homotopy reflectivity is equivalent to the weak Vopenka principle

Abstract: It is well known that the existence of homotopical localization with respect to every (possibly proper) class of maps between spaces or spectra is implied by suitable large-cardinal axioms. However, no concluding evidence had been given that the existence of such localizations could not be proved in ZFC. Using a recent result of Trevor Wilson, we prove that the existence of localizations with respect to classes of maps of spaces or spectra is equivalent to the weak Vopenka principle, stating that there is no full embedding of the opposite category of ordinals into any locally presentable category. In fact we prove that the weak Vopenka principle is equivalent to the claim that every colocalizing subcategory of the homotopy category of any stable locally presentable model category is reflective. This is joint work with Javier Gutiérrez.