Geometry of chain complexes and outer automorphisms under derived equivalence

Abstract

Abstract. The two main theorems proved here are as follows: If A is a finite dimensional algebra over an algebraically closed field, the identity component of the algebraic group of outer automorphisms of A is invariant under derived equivalence. This invariance is obtained as a consequence of the following generalization of a result of Voigt. Namely, given an appropriate geometrization Comp A d of the family of finite A-module complexes with fixed sequence d of dimensions and an "almost projective" complex X ∈ Comp A d , there exists a canonical vector space embedding, where G is the pertinent product of general linear groups acting on Comp A d , tangent spaces at X are denoted by T X (−), and X is identified with its image in the derived category D b (A-Mod).

Brauer graph algebras are closed under derived equivalence

“…We are going to show that derived equivalent symmetric stably biserial algebras have the same number of deformed loops using the identity component Out 0 (A) of the group of outer automorphisms. By [23,33], the group Out 0 (A) is invariant under derived equivalence as an algebraic group, for a finite dimensional algebra A over an 3) π l (β), π l (β ) are parallel for all l. In this case A = kQ/I is a caterpillar. For an arbitrary p i ∈ P, choose a presentation…”

“…The main technique, used in this paper, is the computation of the rank of the maximal torus T (A) of the identity component Out 0 (A) of the group of outer automorphisms for a symmetric stably biserial algebra A. The group Out 0 (A) is a derived invariant [23,33] used quite seldom. The only previous systematic application we know of is the proof of the fact that the number of arrows in the quiver of a gentle algebra is a derived invariant [1].…”

Derived equivalences of algebras

“… (1)The Hochschild ( co‐ ) homology and cyclic homology. In particular, the centers of rings ( see ) . (2)The number of non‐isomorphic simple modules if we are restricted to Artin algebras. (3)Finiteness of global ( or finitistic ) dimensions ( see ) . (4)The Cartan determinants, and the characteristic polynomials of Coxeter matrices if the Cartan matrices of Artin algebras are invertible ( see [, Lemma 4.1]; for a detailed proof, see [, Proposition 6.8.9] ) . (5)Algebraic ‐groups and ‐theory ( see ) . (6)Symmetry of algebras over an arbitrary field ( respectively, self‐injectivity of algebras over an algebraically closed field ) ( see ) . (7)Finite‐dimensional gentle algebras over a field ( see ) . (8)The identity component of the algebraic group of outer automorphisms of finite‐dimensional algebras ( see ) . …”

Automorphismes, graduations et catégories triangulées

“…Le résultat suivant aété obtenu indépendemment, et par des méthodes différentes, par Huisgen-Zimmermann et Saorìn [11]. Ce résultat avaitétéétabli pour deséquivalences dérivées particulières auparavant [9].…”

“…Nous déduisons de cette description de Out que la composante connexe Out 0 de l'identité est invariante paréquivalence de Morita (théorème 4.2), résultat dûà Brauer [16]. De manière similaire, on déduit l'invariance de Out 0 paréquivalence dérivée (théorème 4.6), résultat obtenu indépendamment, et par des méthodes différentes, par Huisgen-Zimmermann et Saorín [11]. Nous donnons aussi une version de ce résultat pour les géomètres : pour une variété projective lisse, le produit Pic 0 × Aut 0 est invariant paŕ equivalence de catégories dérivées (théorème 4.18).…”