Sheaves of E-infinity algebras and applications to algebraic varieties and singular spaces

Abstract

We describe the E-infinity algebra structure on the complex of singular cochains of a topological space, in the context of sheaf theory. As a first application, for any algebraic variety we define a weight filtration compatible with its E-infinity structure. This naturally extends the theory of mixed Hodge structures in rational homotopy to p-adic homotopy theory. The spectral sequence associated to the weight filtration gives a new family of multiplicative algebraic invariants of the varieties for any coefficient ring, carrying Steenrod operations. As a second application, we promote Deligne's intersection complex computing intersection cohomology, to a sheaf carrying E-infinity structures. This allows for a natural interpretation of the Steenrod operations defined on the intersection cohomology of any topological pseudomanifold.

To appear in Transactions of the American Mathematical Society

References in corpus (3)

• Topology of singular algebraic varieties
• Godement resolutions and sheaf homotopy theory
• The twisting procedure

Cited by in corpus (1)

• Prismatic cohomology and $p$-adic homotopy theory