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Derived categories of modules and coherent sheaves. (English) Zbl 1113.16022
Lossen, Christoph (ed.) et al., Singularities and computer algebra. Selected papers of the conference, Kaiserslautern, Germany, October 18–20, 2004 on the occasion of Gert-Martin Greuel’s 60th birthday. Cambridge: Cambridge University Press (ISBN 0-521-68309-2/pbk). London Mathematical Society Lecture Note Series 324, 79-128 (2006).
The author surveys some of his recent results, obtained in colaboration with V. I. Bekkert and I. I. Burban, on the structure of the derived category of the category of modules over an algebra and of the category of coherent sheaves on projective curves.
Let $$A$$ be a finite dimensional algebra over an algebraically closed field $$k$$ and let mod-$$A$$ denote the category of finitely generated left $$A$$-modules (or finite dimensional representations of $$A$$). A fundamental result of Yu. A. Drozd [Representations and quadratic forms, Collect. sci. Works, Kiev 1979, 39-74 (1979; Zbl 0454.16014)] asserts that $$A$$ is either ‘tame’, i.e., for any $$r$$ there are $$A$$-$$k[T]$$-bimodules $$M_1,\dots,M_N$$, finitely generated and free over $$k[T]$$, such that any indecomposable $$A$$-module of dimension $$\leq r$$ is isomorphic to some $$M_i\otimes_{k[T]}k[T]/(T-\lambda)$$ or ‘wild’, i.e., there is an $$A$$-$$k\langle X,Y\rangle$$-bimodule $$M$$, f.g. and free over the free algebra $$k\langle X,Y\rangle$$, such that the functor $$M\otimes_{k\langle X,Y\rangle}-$$ sends non-isomorphic finite dimensional $$k\langle X,Y\rangle$$-modules to non-isomorphic $$A$$-modules.
The first main result of the survey under review is due to V. I. Bekkert and Yu. A. Drozd [Tame-wild dichotomy for derived categories. (preprint), arXiv:math.RT/0310352] and asserts that $$A$$ is either ‘derived tame’ or ‘derived wild’. The definition of derived tame and derived wild is similar to the above definition of tame and wild: one replaces mod-$$A$$ by its derived category and the bimodules by bounded complexes of projective bimodules. Before stating the result, the author recalls a result from the Appendix of the paper of I. Burban and Yu. Drozd [J. Algebra 272, No. 1, 46-94 (2004; Zbl 1047.16005)] asserting that if $$\Lambda$$ is a finite dimensional algebra over a complete local Noetherian ring then every object of the derived category $$\text{D}^-(\text{mod-}\Lambda)$$ decomposes (uniquely) into an at most countable direct sum of objects with local endomorphism rings.
As in the case of the above mentioned theorem of Yu. A. Drozd, the proof of the derived tame-wild dichotomy uses the technique of ‘matrix problems’ plus some ideas from the work of B. Huisgen-Zimmermann and M. Saorín [Trans. Am. Math. Soc. 353, No. 12, 4757-4777 (2001; Zbl 1036.16003)]. As an application, one gets that for a family $$\mathcal A$$ of algebras over a variety $$X$$ the set $$\{x\in X\mid{\mathcal A}(x)$$ is derived tame} is a countable intersection of open subsets and the author conjectures that, in fact, this set is open (although even the analogue of this conjecture for usual tame algebras has not yet been proved).
Then the author presents the main result of his joint paper with I. Burban quoted above which consists in a description of the indecomposable objects of the derived category $$\text{D}^-(\text{mod-}A)$$, for $$A$$ a ‘nodal ring’ (a not necessarily commutative generalization of the completion of the local ring of the simple double point of an algebraic curve, i.e., of $$k[[X,Y]]/(XY)$$). Finally, the author explains the analogous result for the derived category $$\text{D}^-(\text{Coh\,}X)$$, where $$X$$ is a projective curve whose singular points are nodes, whose intersection graph has a special form, and whose normalization is a disjoint union of rational curves. This result is due to I. Burban and Yu. Drozd [Duke Math. J. 121, No. 2, 189-229 (2004; Zbl 1065.18009)]. In both cases the description reduces to a special class of matrix problems (“bunches of chains” or “clans”). The paper ends with an application of the last result in the theory of Cohen-Macaulay modules over surface singularities.
For the entire collection see [Zbl 1086.14001].
##### MSC:
 16G60 Representation type (finite, tame, wild, etc.) of associative algebras 18E30 Derived categories, triangulated categories (MSC2010) 15A21 Canonical forms, reductions, classification 16G20 Representations of quivers and partially ordered sets 16G30 Representations of orders, lattices, algebras over commutative rings 14H60 Vector bundles on curves and their moduli