×

Geometry of chain complexes and outer automorphisms under derived equivalence. (English) Zbl 1036.16003

One of the main results of this paper is that, if \(A\) is a finite dimensional algebra over an algebraically closed field, the identity component \(\text{Out}(A)^0\) of the algebraic group of outer automorphisms of \(A\) is invariant under derived equivalence. (The authors note that this invariance theorem was also proved independently and with different methods by R. Rouquier.) This generalizes a result of Brauer on Morita invariance of \(\text{Out}(A)^0\), and also a result on the tilting-cotilting of \(\text{Out}(A)^0\) due to F. Guil-Asensio and M. Saorín [Arch. Math. 76, No. 1, 12-19 (2001; Zbl 1036.16017)]. The derived equivalence was introduced by J. Rickard [J. Lond. Math. Soc., II. Ser. 39, No. 3, 436-456 (1989; Zbl 0642.16034)], and the paper under review is a contribution to the extensive literature on exhibiting invariants under derived equivalence. Note that derived invariance fails for the group \(\operatorname{Aut}(A)\) of all algebra automorphisms of \(A\), as well as for the full Picard group. The authors also obtain in the paper that the derived Picard group \(\text{DPic}(A)\) contains only finitely many two-sided tilting complexes of fixed total dimension which are pairwise non-isomorphic when viewed as one-sided complexes over \(A\).

MSC:

16E05 Syzygies, resolutions, complexes in associative algebras
18E30 Derived categories, triangulated categories (MSC2010)
18G35 Chain complexes (category-theoretic aspects), dg categories
16P10 Finite rings and finite-dimensional associative algebras
16G10 Representations of associative Artinian rings
16W20 Automorphisms and endomorphisms
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Frank W. Anderson and Kent R. Fuller, Rings and categories of modules, 2nd ed., Graduate Texts in Mathematics, vol. 13, Springer-Verlag, New York, 1992. · Zbl 0765.16001
[2] M. Auslander and Sverre O. Smalø, Preprojective modules over Artin algebras, J. Algebra 66 (1980), no. 1, 61 – 122. · Zbl 0477.16013 · doi:10.1016/0021-8693(80)90113-1
[3] Armand Borel, Linear algebraic groups, 2nd ed., Graduate Texts in Mathematics, vol. 126, Springer-Verlag, New York, 1991. · Zbl 0726.20030
[4] E. Cline, B. Parshall, and L. Scott, Derived categories and Morita theory, J. Algebra 104 (1986), no. 2, 397 – 409. · Zbl 0604.16025 · doi:10.1016/0021-8693(86)90224-3
[5] J.A. de la Peña, Tame algebras. Some fundamental notions, Universität Bielefeld. Ergänzungsreihe 95-010, 1995.
[6] A. Fröhlich, The Picard group of noncommutative rings, in particular of orders, Trans. Amer. Math. Soc. 180 (1973), 1 – 45. · Zbl 0278.16016
[7] F. Guil Asensio and M. Saorín, On automorphism groups induced by bimodules, Arch. Math. (Basel) 76 (2001), 12-19. CMP 2001:07 · Zbl 1036.16017
[8] Dieter Happel, Triangulated categories in the representation theory of finite-dimensional algebras, London Mathematical Society Lecture Note Series, vol. 119, Cambridge University Press, Cambridge, 1988. · Zbl 0635.16017
[9] B. Keller, Invariance and localization for cyclic homology of DG algebras, J. Pure Appl. Algebra 123 (1998), 223-273. · Zbl 0890.18007
[10] Hanspeter Kraft, Geometrische Methoden in der Invariantentheorie, Aspects of Mathematics, D1, Friedr. Vieweg & Sohn, Braunschweig, 1984 (German). · Zbl 0569.14003
[11] Helmut Lenzing and Hagen Meltzer, The automorphism group of the derived category for a weighted projective line, Comm. Algebra 28 (2000), no. 4, 1685 – 1700. · Zbl 0965.16008 · doi:10.1080/00927870008826922
[12] Markus Linckelmann, Stable equivalences of Morita type for self-injective algebras and \?-groups, Math. Z. 223 (1996), no. 1, 87 – 100. · Zbl 0866.16004 · doi:10.1007/PL00004556
[13] R. David Pollack, Algebras and their automorphism groups, Comm. Algebra 17 (1989), no. 8, 1843 – 1866. · Zbl 0682.16022 · doi:10.1080/00927878908823824
[14] Jeremy Rickard, Morita theory for derived categories, J. London Math. Soc. (2) 39 (1989), no. 3, 436 – 456. · Zbl 0642.16034 · doi:10.1112/jlms/s2-39.3.436
[15] Jeremy Rickard, Derived equivalences as derived functors, J. London Math. Soc. (2) 43 (1991), no. 1, 37 – 48. · Zbl 0683.16030 · doi:10.1112/jlms/s2-43.1.37
[16] R. Rouquier, Groupes d’automorphismes et équivalences stables ou dérivées, Preprint.
[17] R. Rouquier and A. Zimmermann, Picard groups for derived module categories, Preprint. · Zbl 1058.18007
[18] P. Deligne, Cohomologie étale, Lecture Notes in Mathematics, Vol. 569, Springer-Verlag, Berlin-New York, 1977. Séminaire de Géométrie Algébrique du Bois-Marie SGA 41\over2; Avec la collaboration de J. F. Boutot, A. Grothendieck, L. Illusie et J. L. Verdier.
[19] Detlef Voigt, Induzierte Darstellungen in der Theorie der endlichen, algebraischen Gruppen, Lecture Notes in Mathematics, Vol. 592, Springer-Verlag, Berlin-New York, 1977 (German). Mit einer englischen Einführung. · Zbl 0374.14010
[20] A. Yekutieli, Dualizing complexes, Morita equivalence and the derived Picard group of a ring, J. London Math. Soc. 60 (1999), 723-746. CMP 2000:11 · Zbl 0954.16006
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.