Derived algebraic geometry over En̳-rings

• Main Author: Francis, John (John Nathan Kirkpatrick)
• Other Authors: Michael Hopkins.
• Format: Others
• Language: English
• Published: Massachusetts Institute of Technology 2008
• Subjects: Mathematics.
• Online Access: http://hdl.handle.net/1721.1/43792

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2008. === In title on t.p., double underscored "n" appears as subscript. === Includes bibliographical references (p. 55-56). === We develop a theory of less commutative algebraic geometry where the role of commutative rings is assumed by En-rings, that is, rings with multiplication parametrized by configuration spaces of points in Rn. As n increases, these theories converge to the derived algebraic geometry of Tobn-Vezzosi and Lurie. The class of spaces obtained by gluing En-rings form a geometric counterpart to En-categories, which are higher topological variants of braided monoidal categories. These spaces further provide a geometric language for the deformation theory of general E, structures. A version of the cotangent complex governs such deformation theories, and we relate its values to E&-Hochschild cohomology. In the affine case, this establishes a claim made by Kontsevich. Other applications include a geometric description of higher Drinfeld centers of SE-categories, explored in work with Ben-Zvi and Nadler. === by John Francis. === Ph.D.