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Filtrations of modules
Dubov, Mykyta ; Trlifaj, Jan (advisor) ; Šaroch, Jan (referee)
The aim of this thesis is to familiarise the reader with the concept of a C-filtration, to show properties of special instances of C-filtrations and to construct structures corresponding to general C-filtrations. The first chapter deals with the definition of a C-filtration, a transfinite composition series and its length, the semiartinian module, the socle-sequence and its length, a proof of the Jordan-Hölder Theorem for transfinite composition series and a proof of the properties relevant to general semiartinian modules. In the second chapter, the concept of a closed subset of an ordinal and its properties are defined and proved, this concept is key to the proof of Hill's Lemma, which for a general C-filtration provides a complete, distributive, dense sublattice containing the given C-filtration. The last third chapter deals with the dual concept to the concept of the socle-sequence, the decreasing Loewy series. We prove the equality of their lengths for modules of finite length and analyze the relations of their lengths for the case of general modules.
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Flat Relative Mittag-Leffler Modules and Approximations
Ben Yassine, Asmae ; Trlifaj, Jan (advisor) ; Cortés Izurdiaga, Manuel (referee) ; Herbera Espinal, Dolors (referee)
Práce představuje hlavní výsledky naší společné práce s vedoucím mé práce na aproximacích modulů, s primárním důrazem na třídu plochých relativně Mittag-Lefflerových modulů, Zariského lokalitu kvazikoherentních svazků spo- jených s touto třídou, a dualizaci aproximací. Nejprve charakterizujeme třídy DQ skládající se ze všech plochých relativně Mittag-Lefflerových modulů z hlediska jejich lokální struktury. Dále ukážeme, že Enochsova domněnka platí pro všechny třídy DQ. Tyto výsledky jsou aplikovány na speciální případ f-projektivních modulů. Naše studium se pak rozšiřuje na ascent a descent pro relativní verze Mittag-Lefflerovy vlastnosti vzhledem k plochým a věrně plochým homomorfismům komutativních okruhů. Toto zk- oumání vede k výsledkům, jako je Zariského lokalita lokálně f-projektivních kvazi-koherentních svazků pro všechna schémata, a pro každé n ≥ 1, Zariského lokalita n-Drinfeldových vektorových bandlů pro všechna lokálně noetherovská schémata. Nakonec se zaměříme na obecné aproximační třídy modulů a zk- oumáme možnosti dualizace v závislosti na uzávěrových vlastnostech těchto tříd. Zatímco některé důkazy se snadno dualizují, jiné vyžadují existenci velkých kardinálů:...
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Classes of modules arising in algebraic geometry
Slávik, Alexander ; Trlifaj, Jan (advisor)
This thesis summarises the author's results in representation theory of rings and schemes, obtained with several collaborators. First, we show that for a quasicompact semiseparated scheme X, the derived category of very flat quasicoherent sheaves is equivalent to the derived category of flat quasicoherent sheaves, and if X is affine, this is further equivalent to the homotopy category of projectives. Next, we prove that if R is a commutative Noetherian ring, then every countably generated flat module is quite flat, i.e., a direct summand of a transfinite extension of localizations of R in countable multiplicative subsets. Further, we investigate the relations between the geometric and categorical purity in categories of sheaves; we give a characterization of indecomposable geometric pure-injectives in both the quasicoherent and non-quasicoherent case. In partic- ular, we describe the Ziegler spectrum and its geometric part for the category of quasicoherent sheaves on the projective line over a field. The final result is the equivalence of the following statements for a quasicompact quasiseparated scheme X: (1) the category QCoh(X) of all quasicoherent sheaves on X has a flat generator; (2) for every injective object E of QCoh(X), the internal Hom functor into E is exact; (3) for some injective...
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The Gabriel-Roiter measure in representation theory
Krasula, Dominik ; Šťovíček, Jan (advisor) ; Trlifaj, Jan (referee)
The Gabriel-Roiter measure is a module-theoretic invariant, defined in 1972 by P. Gabriel. It is an order-preserving function that refines a composition length of a module by also taking lengths of indecomposable submodules into account. We calculate all Gabriel-Roiter measures for finite-length representa- tions of an orientation of a Dynkin graph D4 and an orientation of a Euclidean graph ˜A3. In 2007, H. Krause proposed a combinatorial definition of the Gabriel-Roiter measure based on other length functions instead of composition length. We study these alternative Gabriel-Roiter measures on thin representations of quiv- ers whose underlying graph is a tree. 1
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Inverse limits in module categories
Menčík, Matouš ; Trlifaj, Jan (advisor) ; Šaroch, Jan (referee)
For a class of modules C, we study the class lim ←− C of modules that can be obtained as inverse limits of modules from C. In particular, we investigate how additional properties of the class C are reflected by properties of the class lim ←− C. We also address the question of whether for a given module M, every inverse limit of products of M is an inverse limit of finite products of M. We provide examples of modules for which the answer is positive, negative, and for which there is a reason to believe that it depends on additional set-theoretic assumptions. 1
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Groebner bases
Mašková, Kristýna ; Trlifaj, Jan (advisor) ; Příhoda, Pavel (referee)
Gröbner basis is a particular kind of a generating set of an ideal in the polynomial ring S = K[x1, . . . , xn]. This notion is based upon the concept of a monomial order. We define these concepts and present Buchberger's criterion, that enables us to effectively verify whether a generating set is a Gröbner basis. We introduce Buchberger's algorithm, that produces a Gröbner basis from a finite set of generators. We consider a special case of linear homogeneous ideals, where Gröbner basis can be computed simply by the Gaussian elimination. Finally, we extend this theory to submodules of free modules and briefly indicate how to use Gröbner bases to prove Hilbert's syzygy theorem. 1
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R-projectivity
Fuková, Kateřina ; Trlifaj, Jan (advisor) ; Žemlička, Jan (referee)
A module over a ring R is R-projective if it is projective relative to R. This module- theoretic notion is dual to the notion of an R-injective module that plays a key role in the classic Baer's Criterion for Injectivity. This Thesis is concerned with the validity of dual version of Baer's Criterion. It also introduces a concept of projectivity in a general category-theoretic setting. DBC is known to hold for all perfect rings. However, DBC either fails or it is undecid- able in ZFC for non-perfect rings. In this Thesis we deal with the subclass of non-perfect rings, which are small, regular, semiartinian and have primitive factors artinian. Trlifaj showed that there is an extension of ZFC in which DBC holds for such rings. Especially, it is enough to consider extension of ZFC in which the weak version of Jensen's Diamond Principle holds. This combinatorial principle is known as the Weak Diamond Principle. Apart from an overview of the properties of rings mentioned above and introduction of the necessary set-theoretic notions, the Thesis also contains a proof of this new result by Trlifaj published in the paper "Weak diamond, weak projectivity, and transfinite extensions of simple artinian rings" in the J. Algebra in 2022. 1
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Vychylující teorie pro kvazikoherentní svazky
Čoupek, Pavel ; Šťovíček, Jan (advisor) ; Trlifaj, Jan (referee)
We introduce the definition of 1-cotilting object in a Grothendieck category and investigate its relation to the analogue of the standard definition of 1-cotilting module. The 1-cotilting quasi-coherent sheaves on a Noetherian scheme are stud- ied in particular: using the classification of hereditary torsion pairs in the category of quasi-coherent sheaves on a Noetherian scheme X, to each hereditary torsion- free class F that is generating we assign a 1-cotilting quasi-coherent sheaf whose 1-cotilting class is F. This provides a family of pairwise non-equivalent 1-cotilting quasi-coherent sheaves which are parametrized by specialization closed subsets of X avoiding the set of associated points of a chosen generator of the category of quasi-coherent sheaves. In many cases (e.g. for separated schemes), this set of avoided points can be chosen as the set of associated points of the scheme. 1
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Modules over Gorenstein rings
Pospíšil, David ; Trlifaj, Jan (advisor) ; Příhoda, Pavel (referee) ; Herbera Espinal, Dolors (referee)
Title: Modules over Gorenstein rings Author: David Pospíšil Department: Department of Algebra Supervisor: Prof. RNDr. Jan Trlifaj, DSc. Supervisor's e-mail address: trlifaj@karlin.mff.cuni.cz Abstract: The dissertation collects my actual contributions to the clas- sification of (co)tilting modules and classes over Gorenstein rings. Com- pared with the original intent we get a more general result in classification of (co)tilting classes namely for general commutative noetherian rings (see the third paper in this dissertation). The dissertation consists of an introduction and three papers with coauthors. The first paper (published in Contemp. Math.) contains a classification of all (co)tilting modules and classes over 1- Gorenstein commutative rings. The second paper (published in J. Algebra) contains a classification of all tilting classes over regular rings of Krull dimen- sion 2 and also a classification of all tilting modules in the local case. Finally the third paper (preprint) contains a classification of all (co)tilting classes and also torsion pairs over general commutative noetherian rings. All these classi- fications are in terms of subsets of the spectrum of the ring and by associated prime ideals of modules. Keywords: (co)tilting module, (co)tilting class, torsion pair, Gorenstein ring, regular ring,...
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