[Paper Review] Categorical formal punctured neighborhood of infinity, I

Alexander I. Efimov|arXiv (Cornell University)|Nov 2, 2017

Algebraic structures and combinatorial models13 references3 citations

TL;DR

This paper introduces a DG categorical analogue of the formal punctured neighborhood of infinity, constructing a category $Πerf_{\text{top}}(\widehat{\mathcal{B}}_{\infty})$ for any smooth DG category $\mathcal{B}$, which recovers the algebro-geometric category $\operatorname{Perf}(\widehat{X}_{\infty})$ when $\mathcal{B} = \operatorname{Perf}(X)$. The construction generalizes the Calkin algebra and is independent of compactification choices, providing a quasi-invariant categorical framework for studying infinity neighborhoods.

ABSTRACT

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}.

Motivation & Objective

• To define a purely DG categorical version of the formal punctured neighborhood of infinity, independent of algebro-geometric compactifications.
• To generalize the Calkin algebra construction to DG categories, especially for infinite-dimensional endomorphism spaces modulo finite rank operators.
• To show that for $\mathcal{B} = \operatorname{Perf}(X)$, the construction recovers $\operatorname{Perf}(\widehat{X}_{\infty})$, the category of perfect complexes on the formal punctured neighborhood of infinity.
• To establish invariance under choice of compactification, proving the construction is well-defined for smooth varieties.
• To explore connections with existing frameworks, including Fukaya categories, Tate's work, and singularities via Orlov's category $D_{\text{sg}}(Y)^{\text{op}}$.

Proposed method

• Define the DG category $\operatorname{Perf}_{\text{top}}(\widehat{\mathcal{B}}_{\infty})$ as a quotient of $\operatorname{Perf}(\mathcal{B})$ by a suitable ideal, generalizing the Calkin algebra construction.
• Construct $\widehat{\mathcal{B}}_{\infty}$ as a DG algebra with non-negative cohomology, where $H^0(\widehat{\mathcal{B}}_{\infty}) \cong \operatorname{End}_k(B)/B^* \otimes B$ for an associative algebra $B$, modulo finite rank endomorphisms.
• Use the DG quotient $\operatorname{Calk}_k := \operatorname{Mod}_k / \operatorname{Perf}(k)$ as a foundational model, generalizing the algebraic Calkin algebra.
• Apply homological algebra techniques, including compact approximation and pseudo-perfect modules, to ensure well-behaved derived categories.
• Leverage the equivalence $D(X) \simeq D(A_X)$ for a generator $\mathcal{E}$ of $\operatorname{Perf}(X)$, with $A_X = \operatorname{End}(\mathcal{E})$, to relate geometric and categorical constructions.
• Prove that $\widehat{\mathcal{B}}_{\infty}$ is well-defined for any small DG category $\mathcal{B}$, not just smooth ones, via boundedness and compact approximation properties.

Experimental results

Research questions

• RQ1 Can the formal punctured neighborhood of infinity in algebraic geometry be reconstructed purely in terms of DG categories, without reference to compactifications?
• RQ2 How does the DG categorical construction relate to the Calkin algebra, particularly in the case of infinite-dimensional endomorphisms modulo finite rank operators?
• RQ3 Is the resulting category $\operatorname{Perf}_{\text{top}}(\widehat{\mathcal{B}}_{\infty})$ independent of the choice of smooth compactification when $\mathcal{B} = \operatorname{Perf}(X)$?
• RQ4 What is the categorical interpretation of the adele ring $\mathbb{A}_C$ for a smooth projective curve $C$, in terms of $\widehat{\mathcal{B}}_{\infty}$?
• RQ5 Does the construction recover known invariants such as Orlov’s category of singularities $D_{\text{sg}}(Y)^{\text{op}}$ for a proper singular scheme $Y$?

Key findings

• For any smooth algebraic variety $X$ over a field of characteristic zero, the construction yields a category $\operatorname{Perf}(\widehat{X}_{\infty})$ that is independent of the choice of smooth compactification $\overline{X} \supset X$, up to quasi-equivalence.
• When $\mathcal{B} = \operatorname{Perf}(X)$, the DG categorical construction recovers exactly $\operatorname{Perf}(\widehat{X}_{\infty})$, establishing a categorical analogue of the algebro-geometric construction.
• For the algebra $B = \mathrm{k}[t]$ of polynomials, the zeroth cohomology $H^0(\widehat{B}_{\infty})$ is isomorphic to $\mathrm{k}((t^{-1}))$, the field of Laurent series in $t^{-1}$, confirming the construction's consistency with known examples.
• For the algebra of rational functions on a smooth complete curve $C$, the construction yields the ring of adeles $\mathbb{A}_C$, demonstrating its relevance to global fields.
• For $\mathcal{B} = D^b_{\text{coh}}(Y)$ with $Y$ a proper singular scheme, the construction produces the opposite category of Orlov’s category of singularities, $D_{\text{sg}}(Y)^{\text{op}}$, linking it to singularity theory.
• The construction generalizes the Calkin algebra: $H^0(\widehat{B}_{\infty}) \cong \operatorname{End}_k(B)/B^* \otimes B$, where $B^* \otimes B$ corresponds to finite rank endomorphisms, and the map $b \mapsto \overline{L_b}$ gives the natural inclusion of $B$ into the quotient.