Earlier we defined the Proj of a graded ring. In these notes we introduce a relative version of this construction, which is the Proj of a sheaf of graded algebras S over a scheme X. This construction is useful in particular because it allows us to construct the projective space bundle associated to a locally free sheaf E , and it allows us to give a definition of blowing up with respect to an a...

Let M be a Riemannian manifold. For p ∈ M , the tensor algebra of the negative part of the (complex) affinization of the tangent space of M at p has a natural structure of a meromorphic open-string vertex algebra. These meromorphic open-string vertex algebras form a vector bundle over M with a connection. We construct a sheaf V of meromorphic open-string vertex algebras on the sheaf of parallel...

Monday December 9: We discussed the sheaf ΩX of regular differential forms on a variety X , which is sheaf of OX-modules, locally free IFF X is smooth, in which case its rank is equal to the dimension. We computed several examples: the sheaf on A is freely generated (on every open set) by dx1, . . . , dxn. On the parabola defined by y = x in A, the elements dx and dy generate on every open set,...

A notion of Poincaré series was introduced in [1]. It was developed in [2] for a multi-index filtration corresponding to the sequence of blow-ups. The present paper suggests the way to generalize the notion of Poincaré series to the case of arbitrary locally free sheaf on the modification of complex plane C 2. This series is expressed through the topological invariants of the sheaf. For the she...

We generalize the theory of sheaves to chamber systems. We prove that, given a chamber system C and a family R of proper residues of C containing all residues of rank c1, every sheaf defined over R admits a completion which extends C. We also prove that, under suitable hypotheses, a sheaf defined over a truncation of C can be extended to a sheaf for C. In the last section of this paper, we appl...

It is shown that every (small) topos is equivalent to the category of global sections of a sheaf of so-called hyperlocal topoi, improving on a result of Lambek & Moerdijk. It follows that every boolean topos is equivalent to the global sections of a sheaf of well-pointed topoi. Completeness theorems for higher-order logic result as corollaries. The main result of this paper is the following. Th...

After a brief survey of the primary ideas involved in the theory of connections on vector and principal sheaves (studied in [7], [8], [14], [15]), we examine the behaviour of connections under various types of morphisms between sheaves of the considered category. The results thus obtained are useful in the development of a non-smooth geometry in the aforementioned abstract framework and related...

We show that Boehmians defined over open sets of R constitute a sheaf. In particular, it is shown that such Boehmians satisfy the gluing property of sheaves over topological spaces. MSC: Primary 44A40, 46F99; Secondary 44A35, 18F20

HOMOTOPY THEORY AND GENERALIZED SHEAF COHOMOLOGY BY KENNETHS. BROWN0) ABSTRACT. Cohomology groups Ha(X, E) are defined, where X is a topological space and £ is a sheaf on X with values in Kan's category of spectra. These groups generalize the ordinary cohomology groups of X with coefficients in an abelian sheaf, as well as the generalized cohomology of X in the usual sense. The groups are defin...

We x a a prime number l. We denote by E a nite extension of Ql inside a chosen algebraic closure Ql of Ql, by O the ring of integers in E , by F its residue eld, and by F an algebraic closure of F . We take as coeÆcient eld A one of the elds on the following list: F , F , E , or Ql. We work over a eld k in which l is invertible. We are given a smooth connected k-scheme S=k, separated and of nit...