Let D be the sheaf of analytic differential operators of n variables x1, . . . , xn. We consider a left ideal I of D generated by {l1, . . . , lm} which are in the Weyl algebra D = C〈x1, . . . , xn, ∂1, . . . , ∂n〉. If no confusion arises, we also denote by I the left ideal D ·{l1, . . . , ln} in D. Assume that D/I is holonomic. It was proved by M.Kashiwara that the germs of the k-th extension ...

Introduction 1 0.1. Associated primes 2 0.2. Depth 2 1. Regular local rings and their cohomological properties 4 1.1. Notions of dimension 4 1.2. Dualizing functor 5 1.3. Local rings 6 1.4. Serre’s theorem on regular local rings 6 1.5. Cohomology of D when global dimension is finite 8 2. Depth 8 2.1. Property (Sk) 8 2.2. Serre’s criterion of normality 10 2.3. Cohen-Macaulay modules 11 3. Diviso...

It is a well established fact that the solutions of many problems involving families of vector bundles should essentially depend on good estimates for the dimension of the Hilbert schemes of coherent quotients of a given bundle. Deformation theory provides basic cohomological dimension bounds, but most of the time the cohomology groups involved are hard to estimate accurately and moreover do no...

Let D be the sheaf of analytic differential operators of n variables x1, . . . , xn. We consider a left ideal I of D generated by {l1, . . . , lm} which are in the Weyl algebra D = C〈x1, . . . , xn, ∂1, . . . , ∂n〉. If no confusion arises, we also denote by I the left ideal D ·{l1, . . . , ln} in D. Assume that D/I is holonomic. It was proved by M.Kashiwara that the germs of the k-th extension ...

Let Fn be the free group on n generators, and PΣn be the group of automorphisms of Fn which send each generator to a conjugate of itself. Let Kn be the kernel of the homomorphism PΣn → PΣn−1 induced by mapping one of the free group generators to the identity. We show that Kn has cohomological dimension (n − 1), and that Hi(Kn, Z) is infinitely generated for 2 ≤ i ≤ (n − 1).

Let A be a central simple algebra over a field F . Let k1, . . . , kr be cyclic extensions of F such that k1 ⊗F · · · ⊗F kr is a field. We investigate conditions under which A is a tensor product of symbol algebras where each ki is in a symbol F -algebra factor of the same degree as ki. As an application, we give an example of an indecomposable algebra of degree 8 and exponent 2 over a field of...