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.

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:

18E30 | Derived categories, triangulated categories (MSC2010) |