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We show, in full generality, that Lusztig’s a-function describes the projective dimension of both indecomposable tilting modules and indecomposable injective modules in the regular block of the BGG category O, proving a conjecture from the first paper. On the way we show that the images of simple modules under projective functors can be represented in the derived category by linear complexes of...

We introduce the notion of residual intersections modules and prove their existence. show that projective dimension one have Cohen-Macaulay intersections, namely they satisfy relevant Artin-Nagata property. then establish a formula for core orientable satisfying certain homological conditions, extending previous results Corso, Polini, Ulrich on modules. Finally, we provide examples classes our ...

let $r=oplus_{nin bbb n_0}r_n$ be a noetherian homogeneous ring with local base ring $(r_0,frak{m}_0)$, $m$ and $n$ two finitely generated graded $r$-modules. let $t$ be the least integer such that $h^t_{r_+}(m,n)$ is not minimax. we prove that $h^j_{frak{m}_0r}(h^t_{r_+}(m,n))$ is artinian for $j=0,1$. also, we show that if ${rm cd}(r_{+},m,n)=2$ and $tin bbb n_0$, then $h^t_{frak{m}_0r}(h^2_{...

We classify contravariant pairings between standard Whittaker modules and Verma over a complex semisimple Lie algebra. These are useful in extending several classical techniques for category O to the Miličić–Soergel N. introduce class of costandard which generalize dual modules, describe canonical maps from terms pairings. show that have unique irreducible submodules share same composition fact...

Modules over monads (or: actions of monads on endofunctors) are structures in which a monad interacts with an endofunctor, composed either on the left or on the right. Although usually not explicitly identified as such, modules appear in many contexts in programming and semantics. In this paper, we investigate the elementary theory of modules. In particular, we identify the monad freely generat...

We explore the combinatorial properties of the extension monoid product of preinjective Kronecker modules. We state a theorem which characterizes the extension monoid product of preinjective (and dually preprojective) Kronecker modules in the most general case, over an arbitrary base field. As corollaries, we give another proof of an interesting theorem from [7] and restate the main theorem fro...

Morphisms and representations of a class of Banach C*-modules, called CQ*algebras, are considered. Together with a general method for constructing CQ*-algebras, two different ways of extending the GNS-representation are presented.

In this paper, we introduce a dimension, called the cotorsion dimension, for modules and rings. The relations between the cotorsion dimension and other dimensions are discussed. Various results are developed, some extending known results.