On finitely presented functors, Auslander algebras and almost split sequences

This thesis consists of two parts: In Part A we study the category of finitely presented functors and use it to determine the representation type of the Auslander Algebra of Aq = K-algebra , denoted Rq (K is a field). This is possible because the category of finitely generated modules over Rq, mod Rq , is equivalent to the category of finitely presented functors from (mod Aq)ᵒᵖ to Mod k. Part A finishes with the construction of the Auslander-Reiten quiver of Rq in case q = 3. Part B deals with the construction of almost split sequences in the category mod° Δ of lattices over an R-order Δ , where R is a complete discrete rank 1 valuation ring. In the first chapter of part B we give a description of some unpublished work by J. A. Green who permitted me to include it in this thesis. This work contains a method to construct a short exact sequence 0-> N-> E-> S-> 0 in a way which gives an explicit expression for the subfunctor Im( ,g) of ( ,S) , and shows that the construction of almost split sequences can be viewed as a particular case of this problem. In the second chapter of part B we continue this work by deducing a "trace formula" which provides a practical way of dealing with a certain step of the construction of almost split sequences in mod° Δ. Then we consider the particular case where Δ is the group ring

Liftings of formal groups and the Artinian completion of BP

Let BP denote the localization at υn, of the Brown-Peterson spectrum (associated to the prime p). There is a related ring spectrum E(n) with homotopy ring (as a quotient ring of in fact the cohomology theory E(n)*( ) is determined via a Conner-Floyd type isomorphism from on finite complexes, and moreover E(n) and BP are in the same Bousfield class (see [2, 14]). Although it is known (essentially from [17]) that BP cannot be a product of suspensions of E(n) in a multiplicative sense, D. Ravenel conjectured that such a splitting might occur after suitable completion of these spectra (see the introduction to [14]). This question was the original motivation of the present paper; however in proving Ravenel's conjecture we were naturally led to the consideration of some fundamental results in the theory of liftings of formal group laws and ‘change of ring' results for Ext groups occurring in connection with the work of [10, 11, 12

Computations in the derived module category

This thesis is centred around computations in the derived module category of finitely generated lattices over the integral group ring of a finite group G. Building upon the representability of the cohomology functor in the derived module category in dimensions greater than 0, we give a new characterisation of the cohomology of lattices in terms of their G-invariants, only having the syzygies of the trivial lattice to keep track of dimension. With the example of the dihedral group of order 6 we show that this characterisation significantly simplifies computations in cohomology. In particular, we determine the Bieberbach groups, that is, the fundamental groups of compact at Riemannian manifolds, with dihedral holonomy group of order 6. Furthermore, we give an interpretation of the cup product in the derived module category and show that it arises naturally as the composition of morphisms. Inspired by the graded-commutativity of the cup product in singular cohomology we give a sufficient condition for the cohomology ring of a lattice to be graded-commutative in dimensions greater than 0

Higher Monodromy

For a given category C and a topological space X, the constant stack on X with stalk C is the stack of locally constant sheaves with values in C. Its global objects are classified by their monodromy, a functor from the Poincare groupoid of X to C. In this paper we recall these notions from the point of view of higher category theory and then define the 2-monodromy of a locally constant stack with values in a 2-category as a 2-functor from the homotopy 2-groupoid into the 2-category. We show that 2-monodromy classifies locally constant stacks on a reasonably well-behaved space X. As an application, we show how to recover from this classification the cohomological version of a classical theorem of Hopf, and we extend it to the non abelian case.Comment: 43 pages. This is a revised version of our preprint RIMS 1432 (11-2003

An Introduction To Homological Algebra

In this master's thesis we develop homological algebra using category theory. We develop basic properties of abelian categories, triangulated categories, derived categories, derived functors, and t-structures. At the end of most oft the chapters there is a short section for notes which guide the reader to further results in the literature. Chapter 1 consists of a brief introduction to category theory. We define categories, functors, natural transformations, limits, colimits, pullbacks, pushouts, products, coproducts, equalizers, coequalizers, and adjoints, and prove a few basic results about categories like Yoneda's lemma, criterion for a functor to be an equivalence, and criterion for adjunction. In chapter 2 we develop basics about additive and abelian categories. Examples of abelian categories are the category of abelian groups and the category of R-modules over any commutative ring R. Every abelian category is additive, but an additive category does not need to be abelian. In this chapter we also introduce complexes over an additive category, some basic diagram chasing results, and the homotopy category. Some well known results that are proven in this chapter are the five lemma, the snake lemma and functoriality of the long exact sequence associated to a short exact sequence of complexes over an abelian category. In chapter 3 we introduce a method, called localization of categories, to invert a class of morphisms in a category. We give a universal property which characterizes the localization up to unique isomorphism. If the class of morphisms one wants to localize is a localizing class, then we can use the formalism of roofs and coroofs to represent the morphisms in the localization. Using this formalism we prove that the localization of an additive category with respect to a localizing class is an additive category. In chapter 4 we develop basic properties of triangulated categories, which are also additive categories. We prove basic properties of triangulated categories in this chapter and show that the homotopy category of an abelian category is a triangulated category. Chapter 5 consists of an introduction to derived categories. Derived categories are special kind of triangulated categories which can be constructed from abelian categories. If A is an abelian category and C(A) is the category of complexes over A, then the derived category of A is the category C(A)[S^{-1}], where S is the class consisting of quasi-isomorphisms in C(A). In this chapter we prove that this category is a triangulated category. In chapter 6 we introduce right and left derived functors, which are functors between derived categories obtained from functors between abelian categories. We show existence of right derived functors and state the results needed to show existence of left derived functors. At the end of the chapter we give examples of right and left derived functors. In chapter 7 we introduce t-structures. T-structures allow one to do cohomology on triangulated categories with values in the core of a t-structure. At the end of the chapter we give an example of a t-structure on the bounded derived category of an abelian category