Categorical formal punctured neighborhood of infinity, I

In this paper we introduce and study the formal punctured neighborhood of infinity, both in the algebro-geometric and in the DG categorical frameworks. For a smooth algebraic variety $X$ over a field of characteristic zero, one can take its smooth compactification $\bar{X}\supset X,$ and then take the DG category of perfect complexes on the formal punctured neighborhood of the infinity locus $\bar{X}-X.$ The result turns out to be independent of $\bar{X}$ (up to a quasi-equivalence) and we denote this DG category by $\operatorname{Perf}(\hat{X}_{\infty}).$ We show that this construction can be done purely DG categorically (hence of course also $A_{\infty}$-categorically). For any smooth DG category $\mathcal{B},$ we construct the DG category $\operatorname{Perf}_{top}(\hat{\mathcal{B}}_{\infty}),$ which we call the category of perfect complexes on the formal punctured neighborhood of infinity of $\mathcal{B}.$ The construction is closely related to the algebraic version of a Calkin algebra: endomorphisms of an infinite-dimensional vector space modulo endomorphisms of finite rank. We prove that the DG categorical construction is compatible with the algebro-geometric one. We study numerous examples. In particular, for the algebra of rational functions on a smooth complete connected curve $C$ we obtain the algebra of adeles $\mathbb{A}_C,$ and for $\mathcal{B}=D^b_{coh}(Y)$ for a proper singular scheme $Y$ we obtain the category $D_{sg}(Y)^{op}$ -- the opposite category of the Orlov's category of singularities. Among other things, we discuss the relation with the papers of Tate \cite{Ta} and Arbarello, de Concini, and Kac \cite{ACK}.