DG indschemes (Q2875942)

The notion of differential graded scheme (DG scheme) was introduced by M. Kontsevich about twenty years ago, in a first approach to what is now called derived algebraic geometry. By definition, a DG scheme is a usual scheme \((X,\mathcal{O}_X)\) together with a sheaf \(\mathcal{O}_X^\bullet\) of differential graded \(\mathcal{O}_X\)-algebras such that the natural map \(\mathcal{O}_X\to H^0(X,\mathcal{O}_X^\bullet)\) is surjective. The theory of DG schemes was further developed by M. Kapranov, I. Ciocan-Fontanine, K. Behrend, and others, mainly in order to study derived moduli spaces and derived moduli stacks in algebraic geometry.NEWLINENEWLINENEWLINENEWLINEOn the other hand, there is the notion of indschem due to A. Beilinson and V. Drinfeld, which is defined as follows: An indscheme is a presheaf on the category of affine schemes that is representable as an inductive limit of closed embeddings of schemes. A prominent example of an indscheme is the infinite Grassmannian Gr\(_G\) corresponding to an algebraic group \(G\), which plays a crucial role in the geometric Langlands program. NEWLINENEWLINENEWLINEThe main motivation for the paper under review is to develop a suitable framework for the study of the category QCoh(Gr\(_G\)) of quasi-coherent sheaves on Gr\(_G\), and that in the context of \textit{D. Gaitsgory}'s multi-volume series [``Notes on geometric Langlands'', \url{http://www.math.harvard.edu/~gaitsgde/GL/}]. To this end, the authors introduce the conceptual framework of so-called DG indschemes, in which the afore-mentioned theories of DG schemes and indschemes are combined to create a new and powerful abstract machinery in derived algebraic geometry. As the entire approach is heavily based on the language, the methods, and the results of D. Gaitsgory's ``Notes on geometric Langlands'' ([loc. cit.] as well as of \textit{J. Lurie}'s large preprint series [``Derived algebraic geometry'', \url{http://www.math.harvard.edu/~lurie}], we here confine ourselves to briefly indicate the topics covered in the ten sections, instead of undertaking the hopeless attempt to explain any of the countless technical details. After a comprehensive introduction to the motivation, the structure, and the main results of the present paper, Section 1 is devoted to the definition of DG indschemes and their basic properties. Section 2 provides a detailed study of quasi-coherent sheaves and ind-coherent sheaves on DG indschemes, respectively, together with their fundamental functorial properties. Closed embeddings into a DG indscheme and push-outs are described in Section 3, whereas Sections 4 and 5 elaborate a characterization of DG indschemes via deformation theory. Thereafter, formal completions of DG indschemes are analyzed in Section 6, and the study of quasi-coherent and ind-coherent sheaves on formal completions of DG indschemes follows in Section 7. The notion of formal smoothness of DG indschemes is introduced in Section 8, and a characterization of formally smooth DG indschemes via deformation theory is provided there, too. Section 9 is devoted to a comparison between classical and derived formal smoothness of DG indschemes, with an outlook to applications with respect to loop groups and infinite Grassmannians, thereby coming back to the motivation for entire paper as indicated in the introduction above. Finally, Section 10 establishes a functorial equivalence between the categories of quasi-coherent sheaves and ind-coherent sheaves on a DG indscheme, respectively. As the authors point out, this main result of theirs was also proved by J. Lurie using a different method.NEWLINENEWLINEFor the entire collection see [Zbl 1285.00037].