Homological Theory of Representations

Modern developments in representation theory rely heavily on homological methods. This book for advanced graduate students and researchers introduces these methods from their foundations up and discusses several landmark results that illustrate their power and beauty. Categorical foundations include abelian and derived categories, with an emphasis on localisation, spectra, and purity. The representation theoretic focus is on module categories of Artin algebras, with discussions of the representation theory of finite groups and finite quivers. Also covered are Gorenstein and quasi-hereditary algebras, including Schur algebras, which model polynomial representations of general linear groups, and the Morita theory of derived categories via tilting objects. The final part is devoted to a systematic introduction to the theory of purity for locally finitely presented categories, covering pure-injectives, definable subcategories, and Ziegler spectra. With its clear, detailed exposition of important topics in modern representation theory, many of which were unavailable in one volume until now, it deserves a place in every representation theorist's library.

Gorenstein n-X -Injective and n - X-Flat Modules with Respect to a Special Finitely Presented Module

Let [Formula: see text] be a ring, [Formula: see text] a class of [Formula: see text]-modules and [Formula: see text] an integer. We introduce the concepts of Gorenstein [Formula: see text]-[Formula: see text]-injective and [Formula: see text]-[Formula: see text]-flat modules via special finitely presented modules. Besides, we obtain some equivalent properties of these modules on [Formula: see text]-[Formula: see text]-coherent rings. Then we investigate the relations among Gorenstein [Formula: see text]-[Formula: see text]-injective, [Formula: see text]-[Formula: see text]-flat, injective and flat modules on [Formula: see text]-[Formula: see text]-rings (i.e., self [Formula: see text]-[Formula: see text]-injective and [Formula: see text]-[Formula: see text]-coherent rings). Several known results are generalized to this new context.

On generalized algebraic theories and categories with families

We give a syntax independent formulation of finitely presented generalized algebraic theories as initial objects in categories of categories with families (cwfs) with extra structure. To this end, we simultaneously define the notion of a presentation Σ of a generalized algebraic theory and the associated category CwFΣ of small cwfs with a Σ-structure and cwf-morphisms that preserve Σ-structure on the nose. Our definition refers to the purely semantic notion of uniform family of contexts, types, and terms in CwFΣ. Furthermore, we show how to syntactically construct an initial cwf with a Σ-structure. This result can be viewed as a generalization of Birkhoff’s completeness theorem for equational logic. It is obtained by extending Castellan, Clairambault, and Dybjer’s construction of an initial cwf. We provide examples of generalized algebraic theories for monoids, categories, categories with families, and categories with families with extra structure for some type formers of Martin-Löf type theory. The models of these are internal monoids, internal categories, and internal categories with families (with extra structure) in a small category with families. Finally, we show how to extend our definition to some generalized algebraic theories that are not finitely presented, such as the theory of contextual cwfs.

Normal subgroups of SimpHAtic groups

A group is SimpHAtic if it acts geometrically on a simply connected simplicially hereditarily aspherical (SimpHAtic) complex. We show that finitely presented normal subgroups of the SimpHAtic groups are either: finite, or of finite index, or virtually free. This result applies, in particular, to normal subgroups of systolic groups. We prove similar strong restrictions on group extensions for other classes of asymptotically aspherical groups. The proof relies on studying homotopy types at infinity of groups in question. We also show that non-uniform lattices in SimpHAtic complexes (and in more general complexes) are not finitely presentable and that finitely presented groups acting properly on such complexes act geometrically on SimpHAtic complexes. In Appendix we present the topological two-dimensional quasi-Helly property of systolic complexes.

Non-Archimedean volumes of metrized nef line bundles

Let $L$ be a line bundle on a proper, geometrically reduced scheme $X$ over a non-trivially valued non-Archimedean field $K$. Roughly speaking, the non-Archimedean volume of a continuous metric on the Berkovich analytification of $L$ measures the asymptotic growth of the space of small sections of tensor powers of $L$. For a continuous semipositive metric on $L$ in the sense of Zhang, we show first that the non-Archimedean volume agrees with the energy. The existence of such a semipositive metric yields that $L$ is nef. A second result is that the non-Archimedean volume is differentiable at any semipositive continuous metric. These results are known when $L$ is ample, and the purpose of this paper is to generalize them to the nef case. The method is based on a detailed study of the content and the volume of a finitely presented torsion module over the (possibly non-noetherian) valuation ring of $K$.

Relations between Control Mechanisms for Sequential Grammars1

We extend and refine previous results within the general framework for regulated rewriting based on the applicability of rules in sequential grammars [3]. Besides the well-known control mechanisms as control graphs, matrices, permitting and forbidden rules, partial order on rules, and priority relations on rules we also consider the new variant of activation and blocking of rules as investigated in [1, 2, 4]. Moreover, we exhibit special results for strings and multisets as well as for arrays in the general variant defined on Cayley grids of finitely presented groups. Especially we prove that array grammars defined on Cayley grids of finitely presented groups using #-context-free array productions together with control mechanisms as control graphs, matrices, permitting and forbidden rules, partial order on rules, priority relations on rules, or activation and blocking of rules have the same computational power as such array grammars using arbitrary array productions.

Tensor products of finitely presented functors

We provide explicit constructions for various ingredients of right exact monoidal structures on the category of finitely presented functors. As our main tool, we prove a multilinear version of the universal property of so-called Freyd categories, which in turn is used in the proof of correctness of our constructions. Furthermore, we compare our construction with the Day convolution of arbitrary additive functors. Day convolution always yields a closed monoidal structure on the category of all additive functors. In contrast, right exact monoidal structures for finitely presented functor categories are not necessarily closed. We provide a necessary criterion for being closed that relies on the underlying category having weak kernels and a so-called finitely presented prointernal hom structure. Our results are stated in a constructive way and thus serve as a unified approach for the implementation of tensor products in various contexts.