Dg algebras with enough idempotents, their dg modules and their derived categories

Manuel Saorín

Abstract


We develop the theory dg algebras with enough idempotents and their dg modules and show their equivalence with that of small dg categories and their dg modules. We introduce the concept of dg adjunction and show that the classical covariant tensor-Hom and contravariant Hom-Hom adjunctions of modules over associative unital algebras are extended as dg adjunctions between categories of dg bimodules. The corresponding adjunctions of the associated triangulated functors are studied, and we investigate when they are one-sided parts of bifunctors which are triangulated on both variables. We finally show that, for a dg algebra with enough idempotents, the perfect left and right derived categories are dual to each other.

Keywords


Dg algebra, dg module, dg category, dg functor, dg adjunction, homotopy category, derived category, derived functor

Full Text:

PDF

References


begin{thebibliography}{88}

bibitem{AF}

Frank W. Anderson and Kent R. Fuller, {sc Rings and categories of modules}, 2nd edition, Grad. Texts Maths {bf 13}, Springer-Verlag (1992).

bibitem{B}

Thomas B"uhler, {em Exact categories}, Expo. Math. {bf 28}(1) (2010), 1-69

bibitem{D1}

Yuriy A. Drozd, {em Tame an wild matrix problems}, in Representations and quadratic forms, Kiev 1979; english translation: Amer. Math. Soc. Transl. {bf 128}(2) (1986), 31-55.

bibitem{D2}

Yuriy A. Drozd, {em Tame and wild matrix problems}, in Representation Theory II, Proc. Confer. Ottawa 1979; V. Dlab and P. Gabriel (edts.). Springer Lect. Notes Math. {bf 832} (1980), 242-258.

bibitem{G}

Pierre Gabriel, {em Des cat'egories ab'eliennes}, Bull. Soc. Math. France {bf 90} (1962), 323-448.

bibitem{GOR}

Natalia S. Golovaschuk, Serge Ovsienko and Andrei V. Rojter, {emph On the schurian DGC}, Matrix problems, IM AN USSR, Kiev (1977), 162-165.

bibitem{H}

Dieter Happel, {sc Triangulated Categories in the Representation Theory

of Finite Dimensional Algebras}, London Math. Soc. Lect. Note Ser. {bf 119}.

Cambridge University Press 1988.

bibitem{HS}

Peter Hilton, Urs Stammbach, {sc A Course in Homological Algebra}, 2nd edition.

Grad. Texts Math. {bf 4}, Springer-Verlag (1971).

bibitem{KS}

Masaki Kashiwara and Pierre Shapira, {sc Categories and sheaves}, Grundl. Math. Wiss {bf 332}, Springer-Verlag (2006)

bibitem{K1} Bernhard Keller, {em Deriving DG categories}, Ann.

Sci. 'Ecole Norm. Sup {bf 27} (1994), 63-102.

bibitem{K3} Bernhard Keller, {em Derived categories and their uses}, Handbook of Algebra, vol. I, North-Hollad (1996), 671-701.

bibitem{K2} Bernhard Keller, {em On differential graded categories}, In:

International Congress of Mathematics, vol. II. Eur. Math. Soc. Zurich (2006), 151-190.

bibitem{Kl-R}

Mark M. Kleiner and Andrei V. Rojter, {em Representations of differential graded categories}, in Proceed. 1st International Conference on Representations of Algebras, Ottawa 1974, Springer Lect. Notes Math. {bf 488} (1975), 316-339.

bibitem{M}

Barry Mitchell, {em Rings with several objects}, Adv. Math. {bf 8} (1972), 1-161

bibitem{NVO}

Constantin Nastasescu and Freddy Van Oystaeyen, {sc Graded Ring Theory}, North-Holland (1982).

bibitem{Neeman}

Amnon Neeman, {sc Triangulated Categories}, Princeton University Press 2001.

bibitem{NS-Japan}

Pedro Nicol'as and Manuel Saor'in, {em Classical derived functors as fully

faithful embeddings}. Proc. 46th Japan Symp. Ring Theory and Repres. Theory

(edited by I. Kikumasa). Yamaguchi University (2014), 137-187.

bibitem{NS-GTT}

Pedro Nicol'as and Manuel Saor'in, {em Generalized tilting theory}, Preprint available at https://arxiv.org/abs/1208.2803

bibitem{O}

Serge Ovsienko, {em Bimodule and matrix problems}, in 'Computational Methods for Representations of Groups and Algebras', Proceed. Euroconfer. Essen 1977, P. Dr"axler, C.M. Ringel and G.O. Michler (edts.), Birkh"auser Progress in Maths. {bf 173} (1999), 325-357.

bibitem{SZ2}

Manuel Saor'i n and Alexander Zimmermann, {em Symmetry of the definition of degeneration in triangulated categories}, Preprint

bibitem{Ta}

Goncalo Tabuada, {em Une structure de cat'egorie de mod`eles de Quillen sur la cat'egorie des dg-cat'egories}, Compt. Rend. Acad. Sci. Paris, s'er {bf I 340} (2005), 15-19.

bibitem{T}

Bertrand To"en, {em The homotopy theory of dg-categories and derived Morita theory}, Invent. Math. {bf 167} (2007), 615-667.

bibitem{Verdier}

Jean-Louis Verdier, {em Des cat'egories d'eriv'ees des cat'egories abeliennes}.

Ast'erisque {bf 239}, Soc. Math. France (1996).

bibitem{Wis}

Robert Wisbauer, {sc Foundations of Module and Ring Theory}, Gordon and Breach Science Publishers (1991).

bibitem{reptheobuch}

Alexander Zimmermann, {sc Representation Theory; A homological algebra point of view},

Springer Verlag (2014).

end{thebibliography}


Refbacks

  • There are currently no refbacks.