1.1. Setup. In this talk k is a field of characteristic 0. Let (W,S) be a Coxeter group with |S| = n < ∞. Also we let V = ∑ s∈S kes be the reflection representation of W , and R = k[V ]. R is a graded algebra with V ∗ in degree 2. We abbreviate M ⊗R N = MN for a right R-module M and a left R-module N . We let {αs}s∈S be the dual basis of {es}s∈S , which can be considered as elements in V ∗ ⊂ R....

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 B=A?X|dX=t? be an extended DG algebra by the adjunction of a variable positive even degree n, and let N semi-free B-module that is assumed to bounded below as graded module. We prove in this paper liftable A if ExtBn+1(N,N)=0. Furthermore such lifting unique up isomorphisms ExtBn(N,N)=0.

Let G be a group with identity e. R G-graded commutative ring and M graded R-module. We introduce the concept of Ie-prime submodule as generalization prime for I =?g?G Ig fixed ideal R. give number results concerning this class submodules their homogeneous components. A proper N is said to if whenever rg ? h(R) mh h(M) rgmh IeN, then either (N :R M) or N.

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....

In this paper we study modules with periodic free resolutions (that is, periodic modules) over an exterior algebra. We show that any module with bounded Betti numbers (that is, a module whose syzygy modules have a bounded number of generators) must have periodic free resolution of period 2, and that for graded modules the period is 1. We also show that any module with a linear Tate resolution i...

We study the transfer of (dual) relative CS-Rickart properties via functors between abelian categories. consider fully faithful as well adjoint pairs functors. give several applications to Grothendieck categories and, in particular, (graded) module and comodule

L et I be an ideal, homogeneous with respect to the usual grading, in a polynomial ring R = k[x0, . . . , xn] in n+ 1 variables (over an algebraically closed field k). Denote the graded component of I of degree d by Id, and likewise the k-vector space of homogeneous forms of R of degree d by Rd. Since I is a graded R-module, we have k-linear maps μd,i : Id ⊗ Ri → Id+i given for each i and d by ...