Let Y, Z be a pair of smooth coisotropic subvarieties in a smooth algebraic Poisson variety X. We show that any data of first order deformation of the structure sheaf OX to a sheaf of noncommutative algebras and of the sheaves OY and OZ to sheaves of right and left modules over the deformed algebra, respectively, gives rise to a Batalin-Vilkoviski algebra structure on the Tor-sheaf TorX q (OY ,...

Let Y, Z be a pair of smooth coisotropic subvarieties in a smooth algebraic Poisson variety X. We show that any data of first order deformation of the structure sheaf OX to a sheaf of noncommutative algebras and of the sheaves OY and OZ to sheaves of right and left modules over the deformed algebra, respectively, gives rise to a Batalin-Vilkoviski algebra structure on the Tor-sheaf TorX q (OY ,...

Let $\{\Lambda ^\infty _t\}$ be an isotopy of Legendrians (possibly singular) in a unit cosphere bundle $S^*M$ that arise as slices singular Legendrian $\Lambda _I^\infty \subset S^*M \times T^*I$ . $\mathcal {C}_t = Sh(M, \Lambda _t)$ the differential graded derived category constructible sheaves on $M$ with support at infinity contained _t$ We prove if embeds into Liouville hypersurfaces, the...

For a coherent filtered D-module we show that the dual of each graded piece over the structure sheaf is isomorphic to a certain graded piece of the ring-theoretic local cohomology complex of the graded quotient of the dual of the filtered D-module along the zero-section of the cotangent bundle. This follows from a similar assertion for coherent graded modules over a polynomial algebra over the ...

6 Introduction to Affine Schemes 2 6.1 Rings and Modules of Fractions . . . . . . . . . . . . . . . . . 2 6.2 The Spectrum of a Unital Commutative Ring . . . . . . . . . 5 6.3 The Spectrum of a Quotient Ring . . . . . . . . . . . . . . . . 7 6.4 The Spectrum of a Ring of Fractions . . . . . . . . . . . . . . 9 6.5 Intersections of Prime Ideals . . . . . . . . . . . . . . . . . . . 11 6.6 Topolo...

One feature which clearly distinguishes the theory of schemes from the older theory of varieties is the possibility of having nilpotent elements in the structure sheaf of a scheme. In particular, if Y is a closed subvariety of a variety X, defined by a sheaf of ideals I , then for any n ≥ 1 we can consider the closed subscheme Yn defined by the nth power I n of the sheaf of ideals I . For n ≥ 2...

For any smooth projective variety X of dimension n over an algebraically closed field k of characteristic p > 0 with μ(Ω X ) > 0. If T(Ω X ) (0 < l < n(p − 1)) are semi-stable, then the sheaf B X of exact 1-forms is stable. When X is a surface with μ(Ω X ) > 0 and Ω X is semi-stable, the sheaf B X of exact 2-forms is also stable. Moreover, under the same condition, the sheaf Z X of closed 1-for...

Sheaf theoretic notions and methods have entered model theory during recent years and important constructions like reduced powers and products, unions of chains, Boolean valued models and models for non-classical logics appear here in a unified setting. If one looks at a sheaf as a generalized structure, then forcing becomes the genuine extension of the validity relation between models and form...