Let A be a graded C∗-algebra. We characterize Kasparov’s K-theory group K̂0(A) in terms of graded ∗-homomorphisms by proving a general converse to the functional calculus theorem for self-adjoint regular operators on graded Hilbert modules. An application to the index theory of elliptic differential operators on smooth closed manifolds and asymptotic morphisms is discussed.

We continue with [LY] to construct and classify graded simple twisted modules for the N-graded vertex algebras constructed by Gorbounov, Malikov and Schechtman from vertex algebroids. Meanwhile we determine the full automorphism groups of those N-graded vertex algebras in terms of the automorphism groups of the corresponding vertex algebroids.

We define the concept of regularity for bigraded modules over a bigraded polynomial ring. In this setting we prove analogs of some of the classical results on m-regularity for graded modules over polynomial algebras.

We describe the cone of Betti tables of all finitely generated graded modules over the homogeneous coordinate ring of three non-collinear points in the projective plane. We also describe the cone of Betti tables of all finite length modules.

In this paper we prove parts of a conjecture of Herzog giving lower bounds on the rank of the free modules appearing in the linear strand of a graded k-th syzygy module over the polynomial ring. If in addition the module is Z n-graded we show that the conjecture holds in full generality. Furthermore, we give lower and upper bounds for the graded Betti numbers of graded ideals with a linear reso...

In this paper, we introduce a family of indecomposable finitedimensional graded modules for the twisted current algebras. These modules are indexed by an |R+|-tuple of partitions ξ = (ξ)α∈R+ satisfying a natural compatibility condition. We give three equivalent presentations of these modules and show that for a particular choice of ξ these modules become isomorphic to Demazure modules in variou...