Categoria: 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.

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.

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.

Ferran Cedó (Universitat Autònoma de Barcelona)

Indecomposable solutions of the Yang-Baxter equation of square-free cardinality

Abstract: Let $p_1,\dots,p_n$ be distinct prime numbers. Let $m_1,\dots,m_n$ be positive integers such that $m_1+\cdots+m_n>n$ . In previous joint work with J. Okni\'{n}ski, we proved that there exist simple involutive non-degenerate set-theoretic solutions $(X,r)$ of the Yang-Baxter equation with $|X|=p_1^{m_1}\cdots p_n^{m_n}$. A natural question is asked: If $n>1$, is there a simple involutive non-degenerate set-theoretic solution $(X,r)$ of the Yang-Baxter equation with $|X|=p_1\cdots p_n$?

In this talk, I will answer this question.

This is joint work with J. Okni\'{n}ski

Wolfgang Pitsch (Universitat Autònoma de Barcelona)

Witt group and Maslov index

Abstract: The main subjects of this talk will be $W(k)$, the Witt group over a field $k$, and the Maslov index of three Lagrangians in a symplectic space, which is an invariant, originally introduced in topology, taking values in $W(k)$. I will show how the machinery of Sturm sequences and Sylvester matrices developed by Barge-Lannes can be used to prove that the equivalence class of Maslov’s 2-cocycle, associated to the homonymous index, is trivial modulo $I^2$, with $I$ being the fundamental ideal of $W(k)$.

Román Álvarez (Universitat Autònoma de Barcelona)

Non-Finitely Generated Projective Modules over Integral Group Rings

Abstract: We introduce a relative version of the big projective modules introduced by Bass, which is an example of a non-finitely generated projective module. We develop the general theory of I-big projective modules introduced by Pavel Príhoda (2010). We inquire more deeply in a correspondence between countably generated projective modules over a ring R and finitely generated projective modules over a ring R modulo an ideal I and generalize it into an equivalence of categories as it is done by Herbera-Príhoda-Wiegand in a recent preprint (2020). Finally, we approach I-big projective modules over well-known rings in order to give an explicit example of the construction of non-finitely generated projective modules over the integral group ring ZA5, where A5 denotes the alternating group on 5 letters.