We show that there exists a natural non-degenerate pairing of the homomorphism space between two neighbor standard modules over a quasi-hereditary algebra with the first extension space between the corresponding costandard modules and vise versa. Investigation of this phenomenon leads to a family of pairings involving standard, costandard and tilting modules. In the graded case, under some ”Kos...

Let S = K[X 1 ,. .. , X n ] be the polynomial ring over a field K. For bounded below Z n-graded S-modules M and N we show that if Tor S p (M, N) = 0, then for 0 ≤ i ≤ p, the dimension of the K-vector space Tor S i (M, N) is at least p i. In particular, we get lower bounds for the total Betti numbers. These results are related to a conjecture of Buchsbaum and Eisenbud.

Given a module M over a ring R that has a grading by a semigroup Q, we present a spectral sequence that computes the local cohomology H I(M) at any graded ideal I in terms of Ext modules. We use this method to obtain finiteness results for the local cohomology of graded modules over semigroup rings. In particular we prove that for a semigroup Q whose saturation Q is simplicial, and a finitely g...

For a coherent filtered D-module we show that the dual of each graded piece over the structure sheaf is isomorphic to a certain graded piece of the ring-theoretic local cohomology complex of the graded quotient of the dual of the filtered D-module along the zero-section of the cotangent bundle. This follows from a similar assertion for coherent graded modules over a polynomial algebra over the ...

To frame and motivate the goals pursued in the present article we recall that, loosely speaking, the most common among the blowup algebras are the Rees algebra R[It] = ⊕∞ n=0 I ntn and the associated graded ring grI(R) = ⊕∞ n=0 I n/In+1 of an ideal I in a commutative Noetherian local ring (R,m). The three main clusters around which most of the current research on blowup algebras has been develo...

Let φ : Rm → Rd be a map of free modules over a commutative ring R. Fitting’s Lemma shows that the “Fitting ideal,” the ideal of d × d minors of φ, annihilates the cokernel of φ and is a good approximation to the whole annihilator in a certain sense. In characteristic 0 we define a Fitting ideal in the more general case of a map of graded free modules over a Z/2graded skew-commutative algebra a...

In the first part of this paper the projective dimension of the structural modules in the BGG category O is studied. This dimension is computed for simple, standard and costandard modules. For tilting and injective modules an explicit conjecture relating the result to Lusztig’s a-function is formulated (and proved for type A). The second part deals with the extension algebra of Verma modules. I...

We compute the Hilbert coefficients of a graded module with pure resolution and discuss lower and upper bounds for these coefficients for arbitrary graded modules.

For any finitely generated abelian group Q, we reduce the problem of classification of Q-graded simple Lie algebras over an algebraically closed field of “good” characteristic to the problem of classification of gradings on simple Lie algebras. In particular, we obtain the full classification of finite-dimensional Q-graded simple Lie algebras over any algebraically closed field of characteristi...