Quadratic forms and Clifford algebras on derived stacks (Q305710)

The paper under review develops the generalisation of quadratic forms and Clifford algebras to derived algebraic geometry. It is the quadratic analogue of [\textit{T. Pantev} et al., Publ. Math., Inst. Hautes Étud. Sci. 117, 271--328 (2013; Zbl 1328.14027)], which introduced shifted symplectic structures. It lays the foundations for the theory, explains how shifted quadratic forms are related to shifted symplectic forms, introduces the analogues of Clifford algebras in derived algebraic geometry, and discusses several open questions.NEWLINENEWLINERecall that over a field \(k\) (not of characteristic two) a quadratic form on a vectorspace \(V\) can be seen as a symmetric bilinear form, i.e.\ as a map of vectorspaces \(\mathrm{Sym}^2V\to k\). The paper under review replaces this by a map in the derived category of vector spaces, whose codomain can moreover be shifted to \(k[n]\).NEWLINENEWLINEThe paper explains how one can in complete generality define quadratic forms in modules over a derived stack. As a special case one can consider quadratic forms for the tangent complex, in which case the structure is called a \textit{shifted quadratic stack}.NEWLINENEWLINEThe first main result of the paper is an existence theorem for shifted quadratic stacks, similar to that obtained in [loc. cit.] for shifted symplectic structures. Observe that the results of op. cit. imply these existence results, but because there is no need for closedness data in a shifted quadratic structure it is possible to prove this directly.NEWLINENEWLINEThe paper also defines the analogue of the Grothendieck-Witt ring, which is only an abelian group now (unless one simultaneously takes all shifts into account).NEWLINENEWLINEThe second main result of the paper is the notion of \textit{derived Clifford algebra}. This is a derived version of the usual Clifford algebra, in the sense that its zeroth cohomology is the usual one. It is explained that already for the zero quadratic form on a one-dimensional vectorspace this derived Clifford algebra is not discrete. Classically one has that a Clifford algebra is an Azumaya algebra, but in the derived setting this is less clear, and the paper raises several interesting questions in this regard.NEWLINENEWLINEIn Sections 1 and 2 the (derived) affine situation is discussed: first derived quadratic complexes are introduced, then it is shown how one can associate a derived Clifford algebra to this (provided the shift is even). Section 3 is the global version of Section 1, and Section 4 contains the existence theorems for the objects introduced in the previous section. Section 5 is the global analogue of Section 2. The paper also contains an appendix on \({\mathbb Z}/2{\mathbb Z}\)-weights, necessary for derived Clifford algebras.