# Some Trends in Algebra 2011

A conference on module theory and its relations to algebraic geometry, category theory,
commutative algebra, homotopy theory, logic, and representation theory.

Prague, September 6 - 10, 2011

Conference e-mail : sta@karlin.mff.cuni.cz

The conference was supported by the Grant Agency of the Czech Republic, project no. 201/09/0816, and by the DAAD-AVCR bilateral project MEB 101005.

## Program

9:30-10:25 Coffee break 11:00-11:55 12:00-12:55 Lunch 15:00-15:55 Coffee break 16:30-17:25
Tuesday Lutz Struengmann
(University of Duisburg-Essen)
Pathological examples of dual groups
Daniel Herden
(University of Duisburg-Essen)
Constructing absolutely rigid coloured trees
Katrin Leistner
(University of Duisburg-Essen)
Classes of partially isomorphic modules
Claudia Metelli
(Universita' di Napoli)
Finite rank Butler groups
Pavel Ruzicka
(Charles University in Prague)
Abelian groups with a minimal generating set
Wednesday Pedro Guil Asensio
Model category structures arising from Drinfeld vector bundles
Jan Trlifaj
(Charles University in Prague)
Restricted Drinfeld vector bundles are local
Generalized Flat Cotorsion modules. Applications
Gabriela Olteanu
(Babes-Bolyai University Cluj-Napoca)
Idempotents In Rational Group Algebras
Pavel Prihoda
(Charles University in Prague)
Packages and generalized lattices
Thursday Jan Stovicek
(Charles University in Prague)
Flat Mittag-Leffler modules over countable rings
Manuel Cortes Izurdiaga
Strict Mittag-Leffler modules
David Pospisil
(Charles University in Prague)
(Co)tilting classes over commutative noetherian rings
Boat trip (16:00-17:00) Conference dinner (19:30-...)
Friday Simion Breaz
(Babes-Bolyai University, Cluj-Napoca)
Commuting properties of Hom and Ext functors
Septimiu Crivei
(Babes-Bolyai University, Cluj-Napoca)
One-sided exact categories
Flaviu Pop
(Babes-Bolyai University, Cluj-Napoca)
Natural dualities between abelian categories
Problem session

## Abstracts and Presentations

SpeakerTitleAbstractPresentation
Simion Breaz Commuting properties of Hom and Ext functors I'll present results about modules (or abelian groups) $M$ such that the covariant or contravariant Hom and Ext functors induced by $M$ preserve or invert some direct products or sums. In particular, I'll prove that $\Hom_R(-,M)$ preserves direct products if only if $M$ is the trivial module and I'll describe the structure of modules $M$ over hereditary rings such that the functors $\Ext(M,-)$ preserve direct sums. Download PDF
Manuel Cortes Izurdiaga Strict Mittag-Leffler modules A module $M$ is strict Mittag Leffler if for each finite subset $X$ of $M$ there exist morphisms $f:M \rightarrow G$ and $g:G \rightarrow M$ such that $G$ is finitely presented and $fg$ is the identity on $X$. We shall give new properties of strict Mittag-Leffler and we shall characterize them in terms of free realizations of positive primitive formulas. Download PDF
Septimiu Crivei One-sided exact categories One-sided exact categories appear naturally as instances of Grothendieck pretopologies. In an additive setting they are given by considering the one-sided part of Keller's axioms defining Quillen's exact categories. We study one-sided exact additive categories and a stronger version defined by adding the one-sided part of Quillen's obscure axiom''. We show that some homological results, such as the Short Five Lemma and the $3\times 3$ Lemma, can be proved in our context. This is a joint work with Silvana Bazzoni. Download PDF
Sergio Estrada Generalized Flat Cotorsion modules. Applications Flat cotorsion modules play an important role in homological algebra and also in the theory of purity of finitely accessible additive categories. They are defined as the kernel of the Flat cotorsion pair. In the talk we will consider the kernel of an arbitrary cotorsion pair and study some properties of it. After that we will generalize this situation by considering the full subcategory of both cofibrant and fibrant objects in an abelian model category structure. We will see that this category admits an easy model structure, and that its associated homotopy category (which is its stable category) is equivalent to the homotopy category of the original abelian model category. We will illustrate the construction with examples in concrete categories. Download PDF
Pedro Guil Asensio Model category structures arising from Drinfeld vector bundles We present a general construction of model category structures on the category $Ch(Qco(X))$ of unbounded chain complexes of quasi-coherent sheaves on a semi-separated scheme $X$. The construction is based on making compatible the filtrations of individual modules of sections at open affine subsets of $X$. It does not require closure under direct limits as previous methods. We apply it to describe the derived category $\mathbb D (Qco(X))$ via various model structures on $Ch(Qco(X))$. As particular instances, we recover recent results on the flat model structure for quasi-coherent sheaves. Our approach also includes the case of (infinite-dimensional) vector bundles, and of restricted flat Mittag-Leffler quasi-coherent sheaves, as introduced by Drinfeld. Finally, we prove that the unrestricted case does not induce a model category structure as above in general. (Joint work with Sergio Estrada, Mike Prest, and Jan Trlifaj).
Daniel Herden Constructing absolutely rigid coloured trees Absolute constructions have been the topic of a number of recent papers, where a mathematical object, notion or property is called absolute if it is preserved under extensions of the underlying universe of set theory. In this talk we want to revisit the set theoretic foundations for absolute constructions in general. Based on classical work by Silver, Nash-Williams and Shelah we give a new short combinatorial proof showing the first Erdos cardinal $\kappa(\omega)$ to be the sharp upper bound for the cardinality of an absolutely rigid family of coloured trees, where we will focus on the construction of such a family for cardinals $<\kappa(\omega)$. Download PDF
Katrin Leistner Classes of partially isomorphic modules A classical aim in abelian group theory is the search for classification theorems. Often these make use of numerical invariants. Beginning with Barwise and Eklof (1970), some classification theorems were generalized with the help of model-theoretic techniques. The obtained isomorphy, however, is partial. For countable groups, Barwise showed that this is equivalent to general isomorphy. But how strict a classification is partial isomorphy if we consider the uncountable case? To answer this question, we provide under V=L a class of partially isomorphic modules, classified by their Ulm-invariants, and investigate the structure of its members. Download PDF
Claudia Metelli Finite rank Butler groups A B(n)-group W is a torsionfree Abelian group that is the sum of a finite number of pure rank one subgroups tied by n independent linear relations. It is thus determined by a linear choice - the relations - and by an order-theoretical choice - the isomorphism types of the rank one groups. I will show how the linear structure mainly determines the order-theoretical (poset) structure; and how this holds also for the pure inclusion of W into a completely decomposable group (Butler's Theorem). This is joint work with Clorinda De Vivo. Download PDF
Lutz Struengmann Pathological examples of dual groups In this talk we discuss three results concerning dual groups of subgroups of the Baer-Specker group. Firstly, we construct one of size $2^{\aleph_0}$, which is not a dual group, and hence strongly non-reflexive. In contrast, due to Gobel and Pokutta, under Martin's axiom every subgroup strictly smaller than the continuum is actually a dual group. Secondly, we show that the latter is not a theorem of ZFC, as adding $\aleph_1$ many Cohen reals to the ground model, we obtain a model of ZFC in which there is a non-dual subgroup of size $\aleph_1$. However, the continuum may be large. Thirdly, we construct using Martin's axiom a subgroup whose $$n$$th dual is not an $$(n+1)$$st dual for any $$n$$. Together our results solve two questions from the book by Eklof and Mekler. Download PDF