Tilting equivalences: From hereditary algebras to symmetric groups

We study Koszul duality for finite dimensional hereditary algebras, and various generalisations to trivial extension algebras, to Schur algebras, to doubles of Schur bialgebras, and to deformations of doubles of Schur bialgebras. We describe applications to the modular representation theory of symmetric groups.     

Nilpotent subgroups of self equivalences of torsion spaces

LetX be a finitep-torsion based connected nilpotent CW-complex. We give a criterion of a subgroup of ε(X), the group of self equivalences ofX, to be a nilpotent group, in terms of its action onE _*(X), whereE is a CW-spectrum, satisfying some technical conditions.     

Tilting theory and cluster combinatorics

We introduce a new category C, which we call the cluster category, obtained as a quotient of the bounded derived category D of the module category of a finite-dimensional hereditary algebra H over a field. We show that, in the simply laced Dynkin case, C can be regarded as a natural model for the combinatorics of the corresponding Fomin-Zelevinsky cluster algebra. In this model, the tilting objects correspond to the clusters of Fomin-Zelevinsky. Using approximation theory, we investigate the tilting theory of C, showing that it is more regular than that of the module category itself, and demonstrating an interesting link with the classification of self-injective algebras of finite representation type. This investigation also enables us to conjecture a generalisation of APR-tilting.     

Tilting via torsion pairs and almost hereditary noetherian rings

We generalize the tilting process by Happel, Reiten and Smalø to the setting of finitely presented modules over right coherent rings. Moreover, we extend the characterization of quasi-tilted artin algebras as the almost hereditary ones to all right noetherian rings.     

Differential torsion theory

Differential torsion theories are introduced and it is shown that for a hereditary torsion theory τ every derivation on an R-module M has a unique extension to its module of quotients if and only if τ is a differential torsion theory. Dually, we show that when τ is cohereditary, every derivation on M can be lifed uniquely to its module of coquotients.     

Tilting theory and the finitistic dimension conjectures

Let be a right noetherian ring and let be the class of all finitely presented modules of finite projective dimension. We prove that findim iff there is an (infinitely generated) tilting module such that pd and . If is an artin algebra, then can be taken to be finitely generated iff is contravariantly finite. We also obtain a sufficient condition for validity of the First Finitistic Dimension Conjecture that extends the well-known result of Huisgen-Zimmermann and Smalø.

     

Rook-by-rook rook theory: Bijective proofs of rook and hit equivalences

Suppose μ and ν are integer partitions of n, and N>n. It is well known that the Ferrers boards associated to μ and ν are rook-equivalent iff the multisets [μ_i+i:1iN] and [ν_i+i:1iN] are equal. We use the Garsia–Milne involution principle to produce a bijective proof of this theorem in which non-attacking rook placements for μ are explicitly matched with corresponding placements for ν. One byproduct is a direct combinatorial proof that the matrix of Stirling numbers of the first kind is the inverse of the matrix of Stirling numbers of the second kind. We also prove q-analogues and p,q-analogues of these results. We also use the Garsia–Milne involution principle to show that for any two rook boards B and B^′, if B and B^′ are bijectively rook-equivalent, then B and B^′ are bijectively hit-equivalent.     

Support variety theory is invariant under singular equivalences of Morita type

Using a new equivalent definition of support varieties in the sense of Snashall and Solberg [23], we show that both the (Fg) condition and support varieties are preserved under singular equivalences of Morita type. In particular, support variety theory is invariant under stable equivalences of Morita type.     

Hereditary torsion theory of pseudo-regular S-systems

The concept of pseudo-regular S-system is introduced. A torsion theory $\tau _u = \left( {U_S ,\bar U_S } \right)$ with its torsion class U _S consisting of pseudo-regular S-systems, is constructed. Its corresponding quasi-filter is completely described. A number of results on the relations between τ _u and two special torsion theories, the stable and Lambek torsion theories, are obtained.     

Relativistic theory of gravity and a torsion field

The fundamentals of gravity theory are stated in a Minkowski space with an effective nonzero-torsion Riemann-Cartan space-time, which is more general than the Riemannian space. The theory presented thus includes a torsion field of the Einstein-Cartan type in the general concept of the relativistic theory of gravity. Expressions for the metric and canonical energy-momentum tensors of the gravitational field and nongravitational matter in the Minkowski space are found. Noncoordinate gauge transformations are introduced under which the variation of the density of the gravitational Lagrangian is a divergence expression. Translated from Teoreticheskaya i Matematischeskaya Fizika, Vol. 118, No. 1, pp. 126–132, January, 1999.     

Deformations and derived equivalences

Suppose and are block algebras of finite groups over a complete local commutative Noetherian ring whose residue field is a field of positive characteristic. We prove that a split-endomorphism two-sided tilting complex (as introduced by Rickard) for the derived categories of bounded complexes of finitely generated modules over , resp. , preserves the versal deformation rings of bounded complexes of finitely generated modules over , resp. .

     

Extending modules relative to a torsion theory

An R-module M is said to be an extending module if every closed submodule of M is a direct summand. In this paper we introduce and investigate the concept of a type 2 τ-extending module, where τ is a hereditary torsion theory on Mod-R. An R-module M is called type 2 τ-extending if every type 2 τ-closed submodule of M is a direct summand of M. If τ _I is the torsion theory on Mod-R corresponding to an idempotent ideal I of R and M is a type 2 τ _I -extending R-module, then the question of whether or not M/MI is an extending R/I-module is investigated. In particular, for the Goldie torsion theory τ _G we give an example of a module that is type 2 τ _G -extending but not extending.     

Semiartinian modules relative to a torsion theory

In the first section we generalize the concept of the socle of a module by replacing simples with τ-simple modules for a hereditary torsion theory τ. The second section is concerned with the τ-Loewy series, and finally these general results are applied in the section 3 to the notions of τ-semiartinian rings and modules.     

Derived equivalences and Gorenstein algebras

We introduce a notion of Gorenstein R-algebras over a commutative Gorenstein ring R. Then we provide a necessary and sufficient condition for a tilting complex over a Gorenstein R-algebra A to have a Gorenstein R-algebra B as the endomorphism algebra and a construction of such a tilting complex. Furthermore, we provide an example of a tilting complex over a Gorenstein R-algebra A whose endomorphism algebra is not a Gorenstein R-algebra.