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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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Ultrafilters and their monads
Hladil, Josef ; Slávik, Alexander (advisor) ; Růžička, Pavel (referee)
Generalising the notion of an ultrafilter to structured sets, we construct the ultrafilter monad in the categories of partially ordered sets and finitely colourable graphs. This is done similarly to codensity monads, knowing that the codensity monad of the inclusion of finite sets into sets is the ultrafilter monad. We derive an equivalent definition of an ultrafilter on an object applicable for general graphs, also giving rise to a monad. We show that ultrafilters on a poset can be completely characterised in terms of suprema or infima of directed subsets when the poset has only finite antichains. We attempt to classify algebras over the poset ultrafilter monad; our results completely classify the algebras with all antichains finite as posets with a particular compact Hausdorff topology. 1
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Classes of modules arising in algebraic geometry
Slávik, Alexander ; Trlifaj, Jan (advisor) ; Iacob, Alina (referee) ; Shaul, Liran (referee)
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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Classes of modules arising in contemporary algebraic geometry
Slávik, Alexander ; Trlifaj, Jan (advisor) ; Šťovíček, Jan (referee)
In the setting of Noetherian or Dedekind domains, we investigate the properties of very flat and contraadjusted modules. These are the modules from the respective classes in the cotorsion pair (VF, CA) generated by the set of all modules of the form R[s−1 ]. Furthermore, we introduce the concept of locally very flat modules and pursue the analogy of their relation to very flat modules and the relation between projective and flat Mittag-Leffler modules. It is shown that for Noetherian domains, the class of all very flat modules is covering, if and only if the class of all locally very flat modules is precovering, if and only if the spectrum of the ring is finite; for domains of cardinality less than 2ω , this is further equivalent to the class of all contraadjusted modules being enveloping.
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Set-theoretic methods in module theory
Slávik, Alexander ; Trlifaj, Jan (advisor) ; Žemlička, Jan (referee)
A class of modules is called deconstructible if it coincides with the class of all S-filtered modules for some set of modules S. Such classes provide a convenient setting for construction of approximations. We prove that for any deconstructible class C the class of all modules possessing a C-resolution is deconstructible and the same holds for the classes of mod ules with bounded C-resolution dimension. Furthermore, we study the lo cally F-free modules; a sufficient condition on the class F is given for the class of all locally F-free modules to be closed under transfinite exten sions. This enables us to show that there are many non-trivial examples of non-deconstructible classes, generalizing the recent result of D. Herbera and J. Trlifaj concerning the non-deconstructibility of the class of all flat Mittag-Leffler modules over a non-right perfect ring.
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