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...

The Hilbert function of a module over a positively graded algebra is of quasi-polynomial type (Hilbert–Serre). We derive an upper bound for its grade, i. e. the index from which on its coefficients are constant. As an application, we give a purely algebraic proof of an old combinatorial result (due to Ehrhart, McMullen and Stanley). 1. HILBERT QUASIPOLYNOMIALS Let K be a field, and R a positive...

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 cohomol-ogy. We establish the key properties of regularity: its connection with the ...

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 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...

Let A be a differential graded algebra with cohomology ring HA. A graded module over HA is called realisable if it is (up to direct summands) of the form HM for some differential graded A-module M . Benson, Krause and Schwede have stated a local and a global obstruction for realisability. The global obstruction is given by the Hochschild class determined by the secondary multiplication of the A...

The first Weyl algebra over k, A1 = k〈x, y〉/(xy− yx− 1) admits a natural Z-grading by letting deg x = 1 and deg y = −1. Paul Smith showed that gr -A1 is equivalent to the category of quasicoherent sheaves on a certain quotient stack. Using autoequivalences of gr -A1, Smith constructed a commutative ring C, graded by finite subsets of the integers. He then showed gr -A1 ≡ gr -(C,Zfin). In this p...