Tilting Theory and Functor Categories II. Generalized Tilting

In this paper we continue the project of generalizing tilting theory to the category of contravariant functors $\mathrm{Mod}(\mathcal{C})$ , from a skeletally small preadditive category $\mathcal{C}$ to the category of abelian groups, initiated in [15]. We introduce the notion of a generalized tilting category $\mathcal{T}$ , and we concentrate here on extending Happel’s theorem to $\mathrm{Mod}(\mathcal{C})$ ; more specifically, we prove that there is an equivalence of triangulated categories $\mathcal{D}^{b}( \mathrm{Mod}(\mathcal{C}))\cong \mathcal{D}^{b}(\mathrm{Mod}(\mathcal{T}))$ . We then add some restrictions on our category $\mathcal{C}$ , in order to obtain a version of Happel’s theorem for the categories of finitely presented functors. We end the paper proving that some of the theorems for artin algebras, relating tilting with contravariantly finite categories proved in Auslander and Reiten (Adv Math 12(3):306–366, 1974; Adv Math 86(1):111–151, 1991), can be extended to the category of finitely presented functors $\mathrm{mod}(\mathcal{C})$ , with $\mathcal{C}$ a dualizing variety.     

Constructive finite free resolutions

Northcott’s book Finite Free Resolutions (1976), as well as the paper (J. Reine Angew. Math. 262/263:205–219, 1973), present some key results of Buchsbaum and Eisenbud (J. Algebra 25:259–268, 1973; Adv. Math. 12: 84–139, 1974) both in a simplified way and without Noetherian hypotheses, using the notion of latent nonzero divisor introduced by Hochster. The goal of this paper is to simplify further the proofs of these results, which become now elementary in a logical sense (no use of prime ideals, or minimal prime ideals) and, we hope, more perspicuous. Some formulations are new and more general than in the references (J. Algebra 25:259–268, 1973; Adv. Math. 12: 84–139, 1974; Finite Free Resolutions 1976) (Theorem 7.2, Lemma 8.2 and Corollary 8.5).     

Combinatorial Hopf algebra of superclass functions of type D

We provide a Hopf algebra structure on the space of superclass functions on the unipotent upper triangular group of type D over a finite field based on a supercharacter theory constructed by André and Neto in J. Algebra 305, 394–429 (2006) and 322 (2009). Also, we make further comments with respect to types B and C. Type A was explored by M. Aguiar et al. in Adv. Math. 229 (2012); thus this paper is a contribution to understand combinatorially the supercharacter theory of the other classical Lie types.     

Bounding the number of points on a curve using a generalization of Weierstrass semigroups

In this article we use techniques from coding theory to derive upper bounds for the number of rational places of the function field of an algebraic curve defined over a finite field. The used techniques yield upper bounds if the (generalized) Weierstrass semigroup (J Pure Appl Algebra 207(2), 243–260, 2006) for an n-tuple of places is known, even if the exact defining equation of the curve is not known. As shown in examples, this sometimes enables one to get an upper bound for the number of rational places for families of function fields. Our results extend results in (J Pure Appl Algebra 213(6), 1152–1156, 2009).     

Further variations on the theme of FC-groups

Groups that are FC, or more generally satisfy any of the weakenings of the FC-condition considered in de Giovanni (Serdica Math. J. 28:241?C254, 2002) and Robinson et?al. (J. Algebra 326:218?C226, 2011), have local systems consisting of normal finite-by-nilpotent subgroups. Apart from generalizing results from de Giovanni (Serdica Math. J. 28:241?C254, 2002) and Robinson et al. (J. Algebra 326:218?C226, 2011) to the more general context of locally (normal and finite-by-nilpotent) groups, we partially settle an open problem raised in Robinson et?al. (J. Algebra 326:218?C226, 2011) concerning the isomorphism of maximal p-subgroups, but in this more general setting of locally (normal and finite-by-nilpotent) groups.     

The numerical range of positive operators on Hilbert lattices

We show symmetry properties of the numerical range of positive operators on Hilbert lattices. These results generalise the respective properties for positive matrices shown in Li et al. (Linear Algebra Appl 350:1–23, 2002) and Maroulas et al. (Linear Algebra Appl 348:49–62, 2002). Similar assertions are also valid for the block numerical range of positive operators.     

Additive Results of the Drazin Inverse for Bounded Linear Operators

Given two bounded linear operators $P$ and $Q$ on a Banach space the formula for the Drazin inverse of $P+Q$ is given, under the assumptions $P^2 Q+PQ^2=0$ and $P^3 Q=PQ^3=0$ . In particular, some recent results arising in Drazin (Am Math Mon 65:506–514, 1958), Hartwig et al. (Linear Algebra Appl 322:207–217, 2001) and Castro-González et al. (J Math Anal Appl 350:207–215, 2009) are extended.     

Hall Algebras of Odd Periodic Triangulated Categories

We define the Hall algebra associated to any triangulated category under some finiteness conditions with odd periodic translation functor T. This generalizes the results in Toën (Duke Math J 135(3):587–615, 2006) and Xiao and Xu (Duke Math J 143(2):357–373, 2008).     

Koplienko Trace Formula

Koplienko (Sib Math J 25(5): 735–743, 1984) gave a trace formula for perturbations of self-adjoint operators by operators of Hilbert–Schmidt class ${\mathcal{B}_2(\mathcal{H})}$ . Recently Gesztesy et?al. (Basics Z Mat Fiz Anal Geom 4(1):63–107, 2008) gave an alternative proof of the trace formula when the operators involved are bounded. In this article, we give a still another proof and extend the formula for unbounded case by reducing the problem to a finite dimensional one as in the proof of Krein trace formula by Voiculescu (On a Trace Formula of M. G. Krein. Operator Theory: Advances and Applications, vol. 24, pp. 329–332. Birkhauser, Basel, 1987), Sinha and Mohapatra (Proc Indian Acad Sci (Math Sci) 104(4):819–853, 1994).     

Tilting Theoretical Approach to Moduli Spaces Over Preprojective Algebras

We apply tilting theory over preprojective algebras Λ to the study of moduli spaces of Λ-modules. We define the categories of semistable modules and give equivalences, so-called reflection functors, between them by using tilting modules over Λ. Moreover we prove that the equivalence induces an isomorphism of K-schemes between moduli spaces. In particular, we study the case when the moduli spaces are related to Kleinian singularities, and generalize some results of Crawley-Boevey (Am J Math 122:1027–1037, 2000).     

Auslander–Buchweitz Approximation Theory for Triangulated Categories

We introduce and develop an analogous of the Auslander–Buchweitz approximation theory (see Auslander and Buchweitz, Societe Mathematique de France 38:5–37, 1989) in the context of triangulated categories, by using a version of relative homology in this setting. We also prove several results concerning relative homological algebra in a triangulated category $\mathcal{T},$ which are based on the behavior of certain subcategories under finiteness of resolutions and vanishing of Hom-spaces. For example: we establish the existence of preenvelopes (and precovers) in certain triangulated subcategories of $\mathcal{T}.$ The results resemble various constructions and results of Auslander and Buchweitz, and are concentrated in exploring the structure of a triangulated category $\mathcal{T}$ equipped with a pair $(\mathcal{X},\omega),$ where $\mathcal{X}$ is closed under extensions and ω is a weak-cogenerator in $\mathcal{X},$ usually under additional conditions. This reduces, among other things, to the existence of distinguished triangles enjoying special properties, and the behavior of (suitably defined) (co)resolutions, projective or injective dimension of objects of $\mathcal{T}$ and the formation of orthogonal subcategories. Finally, some relationships with the Rouquier’s dimension in triangulated categories is discussed.     

Layer Lengths, Torsion Theories and the Finitistic Dimension

Let $\mathcal{C}$ be a length-category. Generalizing the Loewy length, we propose the layer length associated with a torsion theory, which is a new measure for objects of $\mathcal{C}$ . As an application, we use the layer lengths and the Igusa–Todorov function to get a theorem (see Theorem 6.4) having as corollaries the main results of Huard et al. (Bull Lond Math Soc 41:367–376, 2009) and Wang (Commun Algebra 22(7):419–449, 1994).     

Multivariate Generalized Marshall–Olkin Distributions and Copulas

The multivariate generalized Marshall–Olkin distributions, which include the multivariate Marshall–Olkin exponential distribution due to Marshall and Olkin (J Am Stat Assoc 62:30–41, 1967) and multivariate Marshall–Olkin type distribution due to Muliere and Scarsini (Ann Inst Stat Math 39:429–441, 1987) as special cases, are studied in this paper. We derive the survival copula and the upper/lower orthant dependence coefficient, build the order of these survival copulas, and investigate the evolution of dependence of the residual life with respect to age. The main conclusions developed here are both nice extensions of the main results in Li (Commun Stat Theory Methods 37:1721–1733, 2008a, Methodol Comput Appl Probab 10:39–54, 2008b) and high dimensional generalizations of some results on the bivariate generalized Marshall–Olkin distributions in Li and Pellerey (J Multivar Anal 102:1399–1409, 2011).     

A Characterization of the Locally Finite Networks Admitting Non-Constant Harmonic Functions of Finite Energy

We characterize the locally finite networks admitting non-constant harmonic functions of finite energy. Our characterization unifies the necessary existence criteria of Thomassen (J Comb Theory, Ser B 49:87?C102, 1990) and of Lyons and Peres (2011) with the sufficient criterion of Soardi (1991). We also extend a necessary existence criterion for non-elusive non-constant harmonic functions of finite energy due to Georgakopoulos (J Lond Math Soc, 2010).     

Tilting and cluster tilting for quotient singularities

We shall show that the stable categories of graded Cohen–Macaulay modules over quotient singularities have tilting objects. In particular, these categories are triangle equivalent to derived categories of finite dimensional algebras. Our method is based on higher dimensional Auslander–Reiten theory, which gives cluster tilting objects in the stable categories of (ungraded) Cohen–Macaulay modules.     

Growth,entropy and commutativity of algebras satisfying prescribed relations

In 1964, Golod and Shafarevich found that, provided that the number of relations of each degree satisfies some bounds, there exist infinitely dimensional algebras satisfying the relations. These algebras are called Golod–Shafarevich algebras. This paper provides bounds for the growth function on images of Golod–Shafarevich algebras based upon the number of defining relations. This extends results from Smoktunowicz and Bartholdi (Q J Math. doi:10.1093/qmath/hat005 2013) and Smoktunowicz (J Algebra 381:116–130, 2013). Lower bounds of growth for constructed algebras are also obtained, permitting the construction of algebras with various growth functions of various entropies. In particular, the paper answers a question by Drensky (A private communication, 2013) by constructing algebras with subexponential growth satisfying given relations, under mild assumption on the number of generating relations of each degree. Examples of nil algebras with neither polynomial nor exponential growth over uncountable fields are also constructed, answering a question by Zelmanov (2013). Recently, several open questions concerning the commutativity of algebras satisfying a prescribed number of defining relations have arisen from the study of noncommutative singularities. Additionally, this paper solves one such question, posed by Donovan and Wemyss (Noncommutative deformations and flops, ArXiv:1309.0698v2 [math.AG]).