The Hilbert functions and the regularity of the graded components of local cohomology of a bigraded algebra are considered. Explicit bounds for these invariants are obtained for bigraded hypersurface rings. Introduction In this paper we study algebraic properties of the graded components of local cohomology of a bigradedK-algebra. Let P0 be a Noetherian ring, P = P0[y1, . . . , yn] be the polyn...

Let k be a commutative ring, A a commutative k-algebra and D the filtered ring of k-linear differential operators of A. We prove that: (1) The graded ring grD admits a canonical embedding θ into the graded dual of the symmetric algebra of the module ΩA/k of differentials of A over k, which has a canonical divided power structure. (2) There is a canonical morphism θ from the divided power algebr...

Let S be a standard N k-graded polynomial ring over a field K, let I be a multigraded homogeneous ideal of S, and let M be a finitely generated Z k-graded S-module. We prove that the resolution regularity, a multigraded variant of Castelnuovo-Mumford regularity, of I n M is asymptotically a linear function. This shows that the well known Z-graded phenomenon carries to multigraded situation.

We identify the ring of odd symmetric functions introduced by Ellis and Khovanov as the space of skew polynomials fixed by a natural action of the Hecke algebra at q = −1. This allows us to define graded modules over the Hecke algebra at q = −1 that are ‘odd’ analogs of the cohomology of type A Springer varieties. The graded module associated to the full flag variety corresponds to the quotient...

We construct the H-von Neumann regular radical for H-module algebras and show that it is an H-radical property. We obtain that the Jacobson radical of twisted graded algebra is a graded ideal. For twisted H-module algebra R, we also show that rj(R#σH) = rHj(R)#σH and the Jacobson radical of R is stable, when k is an algebraically closed field or there exists an algebraic closure F of k such tha...

The idea of associating a free resolution to a finitely generated module was introduced in two famous papers by Hilbert in 1890 [Hi1] and 1893 [Hi2]. He proved Hilbert’s Syzygy Theorem 4.9, which says that the minimal free resolution of every finitely generated graded module over a polynomial ring is finite. Since then, there has been a lot of progress on the structure and properties of finite ...

Let S = k[x1, . . . , xn] be a polynomial ring over a field k with n variables x1, . . . , xn, m the irrelevant maximal ideal of S, I a monomial ideal in S and I ′ the polarization of I in the polynomial ring S′ with ρ variables. We show that each graded piece H i m (S/I)a, a ∈ Z , of the local cohomology module H i m (S/I) is isomorphic to a specific graded pieceH i+ρ−n m ′ (S′/I )α, α ∈ Z , o...

We introduce a notion of strongly C×-graded generalized g-twisted V -module for an automorphism g, not necessarily of finite order, of a vertex operator algebra. Let V = ∐ n∈Z V(n) be a vertex operator algebra such that V(0) = C1 and V(n) = 0 for n < 0 and let u be an element of V of weight 1 such that L(1)u = 0, ResxY (u, x) has only real eigenvalues, and the sizes of the Jordan blocks of Resx...