Koszul Algebras and the Frobenius Automorphism

LET %’ be a highest weight category [CPSl] with finitely many simple objects and such that every object has finite length. Then Ce = mod S for a finite dimensional quasi-hereditary algebra S. In a series of papers, [CPS4,5,6,7], we have studied various Kazhdan-Lusztig theories for %. This concept arises naturally in, for example, the modular representation theory of semisimple algebraic groups. As shown in [CPS4; § 41, the existence of a Kazhdan-Lusztig theory in this case is equivalent to the validity of the Lusztig conjecture (and leads to a considerable simplification of that conjecture). Also, the theory permits a combinatorial calculation of the Yoneda extension groups for irreducible representations with regular high weight in the Jantzen region. More generally, as shown in [CPS7; 5 21, if (e has a Kazhdan-Lusztig theory, its homological dual is also a highest weight category. When the algebra S is (positively) graded, the associated category ?& of finitely generated, graded S-modules has the structure of a graded highest weight category. (For a discussion of this notion, see [CPS7; $ 11.) By analogy with the ungraded theory, one can consider the notion of a graded Kazhdan-Lusztig theory. This concept has been developed in detail in [CPS4,7]. The property of having a graded Kazhdan-Lusztig theory is closely related to another property of graded algebras: a positively graded algebra S is called Koszul provided each irreducible S-module L, when regarded as a graded S-module concentrated in degree 0, has a projective resolution * * * --+ P-‘-+ PO+ L + 0 in %& in which each P-” is generated by its term in grade n. (see [P], [BGS].) A main result in [CPS’I; (3.9)] establishes that if %& has a graded KazhdanLusztig theory, the associated algebra S is Koszul. The converse also holds in the presence of an (ungraded) Kazhdan-Lusztig theory on (e. We begin in Q 1 by formulating a new criterion guaranteeing that a (finite dimensional) k-algebra S is Koszul. Our result states essentially that, given an algebra S and an automorphism (7, if the Ext-algebra associated to S can be given its natural graded structure by means of the