2019 Thematic Program (IV)

Derived loop spaces


October , 2020

W303  School of Mathematics, Sichuan University


Bertrand Toën (IMT)


Homological methods provide important information about the structure of associative algebras, revealing sometimes hidden connections amongst them. This course will be about an invariant preserved by derived equivalences: the Gerstenhaber bracket in the first Hochschild cohomology space of unital associative algebras over a field k. There has been a significant amount of effort expended by many authors in order to study this structure, especially in recent times. The first Hochschild cohomology space of an algebra A is the quotient of the k-linear derivations of A by the inner derivations, and the Gerstenhaber bracket provides it of a Lie algebra structure. Recent work in this area has been devoted to describe which Lie algebras appear in this way, and in particular to conditions on the algebra implying the solvability of HH^1(A) as Lie algebra. I will start by describing in detail HH^1(A), paying particular interest to its Lie structure, then treat some families of examples and finally give criteria on the algebra A to imply solvability of HH^1(A).

The purpose of the present series of lectures is to introduce notions from derived algebraic geometry in order to define the derived loop space of an algebraic variety (or a scheme or a stack). I will explain the relations between the derived loop spaces and classical notions such as de Rham cohomology and characteristic classes. One of the major result of this course is a highly structured version of the Hochschild-Kostant-Rosenberg theorem, and its applications to the construction of symplectic structures. In the final lecture, I will also present some new results concerning positive or mixed characteristic situations. 

• Lecture 1

Derived schemes I: simplicial commutative rings and their homotopy theory, examples.

• Lecture 2

Derived schemes II: derived schemes as ringed spaces, derived stacks as functors, cotangent complexes and formal comple- tion. Example : the derived moduli space of representations. 

• Lecture 3 

The derived loop space: definition and examples, the HKR theorem in characteristic zero, shifted symplectic structures. 

• Lecture 4 

The filtered loop space: the filtered circle and the HKR- filtration, applications to symplectic structures and the theory of sin- gular supports in non-zero characteristic. 


Bertrand Toën, Derived Algebraic Geometry. EMS Surv. Math. Sci. 1 (2014), no. 2, 153-245.