Toward the compactification of the stack of Lie(G)-forms using perfect complexes

Abstract

To establish geometric properties of an algebraic stack, one can find a compactification. This method has been successfully employed to find irreducible components for example of the moduli stack of curves [DM69], vector bundles on a surface [O'G96], and Azumaya algebras on a surface [Lie09]. The latter two are moduli stacks of torsors, but these two classify locally free sheaves with possibly additional algebraic structure. For an arbitrary algebraic group G, one can study the stack of Lie(G)-forms, and try to find a compactification via degenerating the underlying locally free sheaves to perfect complexes. In order to avoid having to truncate the stack, we define a Lie infinity-operad, and Lie algebra objects in the additive symmetric monoidal infinity-category of perfect complexes. In the special case Lie(G)=sl_n, to get a candidate for a compactification as its essential image, we construct a functor mapping a perfect totally supported sheaf of rank n to the Lie algebra object of the traceless part of its derived endomorphism complex, in the setting of higher algebra.