Hans C. I will indicate why it is necessary to develop a cohomology theory for Lie-Rinehart pairs A,L that encompasses also the case when the the A-module underlying the Lie algebra L is non-projective. Simion Breaz Babes-Bolyai Univ. Lidie Angeleri Univ. They are all infinite dimensional, and it turns out that they are determined by a tilting module, or dually, by a cotilting module. This also leads to classification results for indecomposable pure-injective modules.

John Baldwin Univ. We focus on the use of theories and syntactical properties of theories in roles 2 and 3. Formal methods enter both into the classification of theories and the study of definable set of a particular model. We regard a property of a theory in first or second order logic as virtuous if the property has mathematical consequences for the theory or for models of the theory.

For first order logic, categoricity is trivial. One can lay out a schema with a few parameters depending on the theory which describes the structure of any model of any theory categorical in uncountable power. Similar schema for the decomposition of models apply to other theories according to properties defining the stability hierarchy. We consider discussions on method by Kashdan, and Bourbaki as well as such logicians as Hrushovski and Shelah.

Moritz Groth Radboud Univ. It adresses the problem that the rather crude passage from model categories to homotopy categories results in a serious loss of information. In the stable context, the typice defects of triangulated categories non-functoriality of cone constru-ction, lack of homotopy colimits can be seen as a reminiscent of this fact.

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- Modules over Non-Noetherian Domains!
- Shout, Sister, Shout!: The Untold Story of Rock-and-Roll Trailblazer Sister Rosetta Tharpe?
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The simple but surprisingly powerful idea behind a derivator is that instead one should form homotopy catego-ries of various diagram categories and also keep track of the calculus of homotopy Kan extensions. In this talk we cover some basics of derivators culminating in a sketch proof that stable derivators provide an enhancement of triangulated categories.

Possibly more important than this result itself are the techniques developed along the way as they lay the foundations for further research directions. The aim of this talk is to hopefully advertise derivators as a convenient, 'weakly terminal' approach to axiomatic homotopy theory. I will outline how this can be used to study the structure of compactly generated torsion pairs in triangulated categories and to classify compactly generated co-t-structures in triangulated categories.

Its goal is to relate representations of absolute Galois groups of number fields to certain analytic objects, automorphic representations. For example, a special case of this correspondence was proved by Wiles and Taylor as the main step of the proof of Fermat's Last Theorem.

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In the talk, we will first give an overview and a motivation for the Langlands program, keeping the prerequisited minimal. Then we will discuss some recent asymptotic results on the number of various types of Galois and automorphic representations. We consider the set Spec R of prime ideals of R as a partially ordered set, ordered by inclusion.

In this talk, we describe prime spectra for some two-dimensional rings of polynomials and power series. Shelah's proof can be adapted to various other settings, including abelian groups, R-modules, filtered R-modules, colorings of infinite graphs, and transversals of systems of sets. It hasn't been clear whether there is a single statement unifying these applications and their various conclusions. This talk is a preliminary report on joint work with J.

Rosicky on what seems to be the "most comprehensive" form of singular compactness. The proof is an adaptation of one of Shelah's arguments to the setting of accessible categories and functors. We consider the ways in which they represent a shift in the direction of category theory, leading to the peculiar no-mans-land between abstract model theory and accessible categories currently being explored by T.

Beke, J. Rosicky, and myself. Finally, we consider a few surprising results concerning AECs that arise as simple appli-cations of ideas from the realm of accessible categories.

## Modules Over Non Noetherian Domains Mathematical Surveys And Monographs

In this talk, I will explain both constructions, describe their functorial properties, and explain their links. We also show that the collection of co-t-structures admitted by the category of perfect complexes is very restricted joint work with J. Recently, flat Mittag-Leffler modules over countable non-perfect rings were shown not to form a precovering class in ZFC.

We find a different proof of the latter fact which, combined with infinite dimensional tilting theory, makes it possible to trace the phenomenon to all countable hereditary artin algebras R of infinite representation type: We prove that the class of all locally Baer R-modules is not precovering joint work with A.

Ivo Dell'Ambrogio Univ. Even better, we can use this to provide conceptual links on Morita equivalence, Picard groups, the Grothendieck group and working relative to a base commutative ring the notion of Azumaya algebra. Greg Stevenson Univ. Among these, there is a class of categories of infinite Dynkin type A, which admit a rather simple combinatorial description. Adam-Christiaan van Roosmalen Univ. Luigi Salce Univ.

We will explain how to construct a cohomology theory governing deformations of algebras of specific types associative, commutative, Lie, Poisson , and of diagrams of these algebras. We will give many examples of deformation cohomology for algebras, morphisms of algebras and, more generally, diagrams of morphisms of algebras. We will mention several applications of absorption and then discuss the following open problem: Given a finite relational structure A and a subset B of A, is it decidable if B is an absorbing subuniverse? We provide an affirmative answer in the case when A has bounded width i.

Our proof mimics the proof of Zadori's conjecture by Barto: the idea is to encode the problem as an instance of CSP A. Christopher C.

In the conjecture was proved for rings of Krull dimension at most 2, but it is still open in general. Recently tilting classes over commutative noetherian rings have been classified in terms of specialization closed subsets of the Zariski spectrum and the Auslander-Bridger transpose. In this talk we will show that existence of maximal Cohen-Macaulay modules yields an alternative description of the tilting classes, and derive some consequences for the structure of these modules using approximation theory.

Martin Zeman Univ. In a joint result with Caicedo we show that if M is any proper class innner model that interprets the cardinal successor function correctly on a proper class of regular cardinals, then M is able to figure out the core model, modulo some random information.

We also give local version of the result and an application in descriptive set theory. Frantisek Marko Pen.

### Table of Contents

Alexander Zubkov Omsk State Ped. Alex Martsinkovsky North Eastern Univ. Phill Schultz Univ. G has finite rank, i. G is self-small, i. Gabor Braun Univ. These provide many pathological examples found by several authors.

**optanpasecfits.tk**

## Modules over non-Noetherian domains | Open Library

All these are variations of the same construction, namely, a standard method to represent endomorphism rings adapted to dual groups. I will explain this technique, so the audience will be able to come up with its own pathological dual groups. Only basic knowledge of abelian groups is required to understand the lecture. The classification is in terms of certain subsets of the Zariski spectrum of the corresponding ring. To give a context, the tilting and cotilting modules in the talk are generalized infinitely generated versions of classical finite dimensional tilting and cotilting modules from representation theory.

The main application of classical co tilting modules is that they give a precise description of homological similarities between different algebras in terms of so-called derived equivalences. Although one can to some extent generalize these results even to our setting, the corresponding theory is in its infancy and will be mentioned only marginally. I will rather focus on the classification and some examples.

In general, it is not possible - for example any reduced monoid with an order unit can express a direct sum decompositions of finitely generated projective modules over a suitable ring. Even for some very classical examples it is not easy to give a classification of projective modules - for integers in number fields it means exactly to determine its ideal class groups.

For semilocal rings, the situation is rather different. Facchini and D. Herbera and I obtained a similar result for countably generated projective modules over noetherian semilocal rings. But the non-noetherian case is still not understood well. In this talk I will discuss examples mostly due to D.

## Neat submodules over integral domains

Herbera that can be obtained using pullbacks and a remarkable example of Gerasimov and Sakhaev - a semilocal ring having a non-finitely generated projective module with finitely generated factor modulo its Jacobson radical. The category of operads is non-abelian category carrying a model structure. Thus resolutions cofibrant replacements of operads are of interest.

We review several results on construction and significance of these resolutions. In particular, we explain how resolution in the abelian category of operadic mod ules can be used to find a first order approximation for H. We show that the left cotorsion envelope of R is finitely generated if and only if R is a semiperfect cotorsion ring. Our proof is based on set-theoretical counting arguments. We also discuss some possible extensions of this result as well as its connection with other open questions.

Joint work with Dilek Pusat.