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To do homological algebra with unbounded chain complexes one needs to first find a way of constructing resolutions. Spal-tenstein solved this problem for chain complexes of R-modules by truncating further and further to the left, resolving the pieces, and gluing back the partial resolutions. Our aim is to give a homotopy theoretical interpretation of this procedure, which may be extended to a relative setting. We work in an arbitrary abelian category A and fix a class of "injective objects" I. We show that Spaltenstein's construction can be captured by a pair of adjoint functors between unbounded chain complexes and towers of non-positively graded ones. This pair of adjoint functors forms what we call a Quillen pair and the above process of truncations, partial resolutions, and gluing, gives a meaningful way to resolve complexes in a relative setting up to a split error term. In order to do homotopy theory, and in particular to construct a well behaved relative derived category D(A;I), we need more: the split error term must vanish. This is the case when I is the class of all injective R-modules but not in general, not even for certain classes of injectives modules over a Noetherian ring. The key property is a relative analogue of Roos's AB4*-n axiom for abelian categories. Various concrete examples such as Gorenstein homological algebra and purity are also discussed., QC 20190611

L'objectiu principal d'aquesta tesis és l'estudi de la conjectura de Perron. Aquesta conjectura afirma que certa funcio sobre el grup de Torelli mod p amb valors en Z/p és un invariant d'esferes d'homologia modul p. Per tal d'abordar l'estudi d'aquesta conjectura, en aquesta tesis primer estudiem les esferes d'homologia modul p, les esferes d'homologia racional, i donem un criteri per determinar quan una esfera d'homologia racional té un split de Heegaard amb aplicació d'enganxament un element del grup de Torelli mod p, el cual ve donat pel nucli de la representació Symplectica modul p del mapping class group. A continuació estenem els resultats de l'article ''Trivial cocycles and invariants of homology 3-spheres'' obtenint una construcció d'invariants amb valors a un grup abelià sense restriccions, a partir d'una família adecuada de 2-cocycles sobre el grup de Torelli. En particular, expliquem la influencia de l'invariant de Rohlin en la perdua de la unicitat en tal construcció. Posteriorment, utilitzant les mateixes eines, obtenim una construcció d'invariants esferes d'homologia racional que tenen un split de Heegaard amb aplicació d'enganxament un element del grup de Torelli mod p, a partir d'una família adecuada de 2-cocycles sobre el grup de Torelli modul p. A més, al llarg d'aquesta construcció obtenim un invariant d'esfers d'homologia modul p el qual no apareix en la literatura. Finalment, demostrem que la conjectura de Perrron és falsa donant una obstrucció que be donada pel fet que la primera classe caracteristica dels fibrats de superfícies reduida modul p no és nula., The main objective of this thesis is the study Perron's conjecture. This conjecture affirms that some function on the group of Torelli mod p, with values in Z/p, is an invariant of mod p homology spheres. In order to study this conjecture, in this thesis we first study the mod p homology spheres, the rational homology spheres and we give a criterion to determine whenever a rational homology sphere has a Heegaard splitting with gluing map an element of the Torelli group mod p, which is the group given by the kernel of the Symplectic representation modulo p of the mapping class group. Next, we extend the results of the article ''Trivial cocycles and invariants of homology 3-spheres'' obtaining a construction of invariants with values to an abelian group without restrictions, from a suitable family of 2-cocycles on the Torelli group. In particular, we explain the influence of the invariant of Rohlin in the lost of uniqueness in such construction. Later, using the same tools, we obtain a construction of invariants of rational homology spheres that have a Heegaard splitting with gluing map an element of the mod p Torelli group, from a suitable family of 2-coccycles on the mod p Torelli group. In addition, throughout this construction we obtain an invariant of mod p homology spheres which does not appear in the literature. Finally, we prove that Perrron's conjecture is false providing an obstruction that is given by the fact that the first characteristic class of surface bundles reduced modulo p does not vanish.

For a commutative ring R, we exploit localization techniques and point-free topology to give an explicit realization of both the Zariski frame of R (the frame of radical ideals in R) and its Hochster dual frame as lattices in the poset of localizing subcategories of the unbounded derived category D(R). This yields new conceptual proofs of the classical theorems of Hopkins-Neeman and Thomason. Next we revisit and simplify Balmer’s theory of spectra and supports for tensor triangulated categories from the viewpoint of frames and Hochster duality. Finally we exploit our results to show how a coherent scheme (X, OX) can be reconstructed from the tensor triangulated structure of its derived category of perfect complexes.

We study classes of modules over a commutative ring which allow to do homological algebra relative to such a class. We classify those classes consisting of injective modules by certain subsets of ideals. When the ring is Noetherian the subsets are precisely the generization closed subsets of the spectrum of the ring., QC 20130506

We show that H-spaces with finitely generated cohomology, as an algebra or as an algebra over the Steenrod algebra, have homotopy exponents at all primes. This provides a positive answer to a question of Stanley., QC 20100525

We characterize the class of homotopy pull-back squares by means of elementary closure properties. The so called Puppe theorem which identifies the homotopy fiber of certain maps constructed as homotopy colimits is a straightforward consequence. Likewise we characterize the class of squares which are homotopy pull-backs "up to Bousfield localization". This yields a generalization of Puppe's theorem which allows us to identify the homotopy type of the localized homotopy fiber. When the localization functor is homological localization this is one of the key ingredients in the group completion theorem., QC 20111006