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Derived categories, quasi-hereditary algebras, and algebraic groups. (English) Zbl 0711.18002
Algebra, Proc. Workshop, Ottawa/Can. and Moosonee/Can. 1987, Math. Lect. Note Ser., Expo. Math., CRAF, Carleton Univ. 3, 105 p. (1988).
[For the entire collection see Zbl 0695.00008.]
Section one contains a fairly long survey of triangulated and derived categories and presents an alternative formulation of the octahedral axiom due to V. Dlab.
In the second section the notion of a triangulated category $${\mathcal D}$$ admitting ‘recollement’ (glueing or patching) relative to triangulated categories $${\mathcal D}'$$ and $${\mathcal D}''$$ is introduced along the lines of the basic example due to Beilinson, Bernstein and Deligne. This example says that, for a topological space X with i: $$F\subset X$$ a closed subspace and j: $$U\subset X$$ the open complement, the triangulated category $${\mathcal D}=D^+({\mathcal O}_ X$$-mod) admits recollement relative to the triangulated categories $${\mathcal D}'=D^+({\mathcal O}_ F$$-mod) and $${\mathcal D}''=D^+({\mathcal O}_ U$$-mod) by means of the induced morphisms $$i_*,i^*,i_ !,i^ !,...,j^ !$$. These induced morphisms satisfy several conditions, the glueing condition being that any element $$K\in Ob({\mathcal D})$$ determines distinguished triangles $i_ !i^ !K\to K\to j_*j^*K\to^{[1]}\text{ and } j_ !j^ !K\to K\to i_*i^*K\to^{[1]}.$ A further example of recollement is provided as follows: Let B be a finite dimensional algebra over a field k, let $$\Delta$$ be a finite dimensional division algebra over k, and also let M be a finite dimensional ($$\Delta$$,B)-bimodule. Form the algebra $$A=\left( \begin{matrix} B\quad 0\\ M\quad \Delta \end{matrix} \right)$$, the $$D^ b(mod$$-A) admits recollement relative to $$D^ b(mod$$-B) and $$D^ b(mod$$- $$\Delta$$). This fact may be used to prove that A is hereditary iff B is hereditary and M is a projective B-module. The definitions of t- structure, t-exact functor, the heart of a triangulated category with t- structure, perversity and the category of perverse sheaves on a topological space are recalled.
The third section treats Morita-theory for derived categories. The notions of a tilting A-module for a ring A and of a tilting complex are introduced, and a result of J. Rickard giving criteria for the equivalence of the bounded derived categories $$D^ b(mod$$-A) and $$D^ b(mod$$-B) for two rings A and B, one of these saying that $$D^ b(mod$$-A) is equivalent to $$D^ b(mod-B)\;\text{iff}\;B\cong\text{End}_{D(mod- A)}(T^.)$$ for a tilting complex $$T^.$$, is given. Filtered derived categories are discussed and a theorem on the equivalence between the derived category of bounded complexes of finitely generated right B- modules and the full, strict, triangulated subcategory of a triangulated category $${\mathcal D}$$, fully embeddable as a subcategory of $$D^-({\mathcal A})$$, $${\mathcal A}$$ abelian with enough projectives, and generated by direct summands of finite direct sums of copies of an object $$T^.\in Ob({\mathcal D})$$ with $$B=End_{{\mathcal D}}(T^.)$$, is proved.
In section four quasi-hereditary algebras are treated and a recollement theorem for $${\mathcal D}=D^ b(mod$$-A), A quasi-hereditary over a field k, and $${\mathcal D}'=D^ b(A/J)$$, J a so-called ideal of definition of A, and $${\mathcal D}''=D^ b(\Delta)$$, $$\Delta$$ a division algebra over k realized as $$\Delta\cong eAe$$ for a suitable primitive idempotent $$e\in A$$, is proved.
The next section introduces the notion of highest weight categories as locally artinian categories with simple object indexed by a poset and such that the injective envelope of a simple object has a ‘good filtration’. For a finite dimensional quasi-hereditary algebra A over a field k, the category of right A-modules is a highest weight category. This also holds for the category of finitely generated A-modules and the converse is also true. Any triangulated category $${\mathcal D}$$ over an algebraically closed field, with a set of objects indexed by a finite poset (‘Weyl modules’) and satisfying suitable conditions on the set of morphisms, determines a unique strict, full subcategory $${\mathcal C}$$ that is a full subcategory of a highest weight category $$\bar {\mathcal C}$$ such that distinguished triangles in $${\mathcal C}$$ become exact sequences in $$\bar {\mathcal C}$$. In case $${\mathcal D}$$ is a full triangulated subcategory of $$D^-({\mathcal A})$$, $${\mathcal A}$$ abelian with enough projectives, $$\bar {\mathcal C}$$ may be identified with a full, abelian subcategory of $${\mathcal D}$$ containing $${\mathcal C}$$. The theory may be applied to a topological space X with stratification $${\mathcal S}$$, perversity p: $${\mathcal S}\to {\mathbb{Z}}$$, and category of p-perverse sheaves $$\tilde {\mathcal C}=\tilde {\mathcal C}(X)$$. A criterion for $$\tilde {\mathcal C}$$ to be a highest weight category (with weight poset $${\mathcal S})$$ is given.
In section six numerous examples of highest weight categories from representation theory of semisimple groups and Lie-algebras, hereditary algebras, poset algebras and perverse sheaves are discussed.
The last section concerns an attempt to attack the Lusztig Conjecture. The Kazhdan-Lusztig conjecture is discussed and some history of quasi- hereditary algebras is given.
Reviewer: W.W.J.Hulsbergen

##### MSC:
 1.8e+31 Derived categories, triangulated categories (MSC2010)