
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 pureinjectives in both the quasicoherent and nonquasicoherent 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...


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 pureinjectives in both the quasicoherent and nonquasicoherent 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...


Settheoretic 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 Sfiltered 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 Cresolution is deconstructible and the same holds for the classes of mod ules with bounded Cresolution dimension. Furthermore, we study the lo cally Ffree modules; a sufficient condition on the class F is given for the class of all locally Ffree modules to be closed under transfinite exten sions. This enables us to show that there are many nontrivial examples of nondeconstructible classes, generalizing the recent result of D. Herbera and J. Trlifaj concerning the nondeconstructibility of the class of all flat MittagLeﬄer modules over a nonright perfect ring.
