Seminars: I. Baskov, The de Rham cohomology of soft function algebras

SEMINARS


Joint Mathematical seminar of Saint Petersburg State University and Peking University December 26, 2024 16:00, St. Petersburg, online

The de Rham cohomology of soft function algebras I. Baskov St. Petersburg Department of Steklov Mathematical Institute of Russian Academy of Sciences
Abstract: For a commutative $k$-algebra $A$ one defines the de Rham cohomology $H^_{DR} (A)$. The Grothendieck's comparison theorem states that the de Rham cohomology groups of the algebra of the regular functions on a smooth affine algebraic variety $X$ are isomorphic to the singular cohomology groups of the analytification of $X$. If one considers an arbitrary soft function $R$-algebra $A$ on a space $X$ then there is no such isomorphism, but the singular cohomology groups canonically split off of the de Rham cohomology groups of $A$. The splitting is obtained via the canonical maps $\Lambda : H^_{sing} (X, R) \to H^_{DR} (A)$ and $\Psi : H^_{DR} (A) \to H^_{sing} (X, R)$ During the talk we will briefly discuss the construction of these maps. We will consider these maps for various function algebras: - an algebra of continuous functions on a topological space, - an algebra of smooth functions on a smooth manifold, - an algebra of piecewise polynomial functions on a polyhedron, - an algebra of polynomial functions on a polyhedron. We will also discuss when the maps $\Lambda$ and $\Psi$ are isomorphisms. The talk is based on the papers https://arxiv.org/abs/2208.11431 and https://arxiv.org/abs/2208.11431. Language:* English