Several kinds of quotient triangulated categories arising naturally in representations of algebras are studied; their relations with the stable categories of Frobenius exact categories are investigated; the derived categories of Gorenstein algebras are explicitly computed inside the stable categories of the graded module categories of the corresponding trivial extension algebras; new descriptio...

We describe a software package for constructing minimal free resolutions ofGLn(Q)-equivariant graded modules M over Q[x1, . . . , xn] such that for all i, the ith syzygy module of M is generated in a single degree. We do so by describing some algorithms for manipulating polynomial representations of the general linear group GLn(Q) following ideas of Olver and Eisenbud– Fløystad–Weyman.

The equivariant Kazhdan-Lusztig polynomial of a braid matroid may be interpreted as the intersection cohomology of a certain partial compactification of the configuration space of n distinct labeled points in C, regarded as a graded representation of the symmetric group Sn. We show that, in fixed cohomological degree, this sequence of representations of symmetric groups naturally admits the str...

Let R = ⊕ n≥0 Rn be a homogeneous Noetherian ring with local base ring (R0,m0) and let M be a finitely generated graded R-module. Let Hi R+ (M) be the i-th local cohomology module of M with respect to R+ := ⊕ n>0 Rn. If dimR0 ≤ 1, the R-modules Γm0R(H R+ (M)), (0 :Hi R+ (M) m0) and Hi R+ (M)/m0H i R+ (M) are Artinian for all i ∈ N0. As a consequence, much can be said on the asymptotic behaviour...

Let R = ⊕ g∈GRg be a G-graded ring. It is well known (see e.g. [D], [M1], [N], [NRV], [NV]) that in the study of the connections that may be established between the categories R-gr of graded R-modules and R1-mod (1 is the unit element of G), an important role is played by the following system of functors : (−)1 : R-gr → R1-mod given by M 7→ M1, where M = ⊕ g∈GMg is a graded left R-module, the i...

In 1982 Richard P. Stanley conjectured that any finitely generated Zn-graded module M over a finitely generated Nn-graded K-algebra R can be decomposed as a direct sum M = ⊕t i=1 νi Si of finitely many free modules νi Si which have to satisfy some additional conditions. Besides homogeneity conditions the most important restriction is that the Si have to be subalgebras of R of dimension at least...

We develop a multigraded variant of Castelnuovo-Mumford regularity. Motivated by toric geometry, we work with modules over a polynomial ring graded by a finitely generated abelian group. As in the standard graded case, our definition of multigraded regularity involves the vanishing of graded components of local cohomology. We establish the key properties of regularity: its connection with the m...

There has been much work on strong and weak Lefschetz conditions for graded Artinian algebras A, especially those that are Gorenstein. A more general invariant of an algebra or finite A-module M we consider here is the set Jordan types elements maximal ideal

let $r$ be a right artinian ring or a perfect commutative‎‎ring‎. ‎let $m$ be a noncosingular self-generator $sum$-lifting‎‎module‎. ‎then $m$ has a direct decomposition $m=oplus_{iin i} m_i$‎,‎where each $m_i$ is noetherian quasi-projective and each‎‎endomorphism ring $end(m_i)$ is local‎.