In recent notes of complex geometry by Clausen–Scholze, they gave a theory of analytification of finite type $\mathbb C$-schemes. It seems to me that there is a non-commutative analogue which works for any DG-category over $\mathbb C$ (we take large categories, not just perfect objects there). For sake of simplicity, we ignore the size issues in the sequel.

$\DeclareMathOperator\Cond{Cond}\DeclareMathOperator\Liq{Liq}\DeclareMathOperator\an{an}$Let $\mathcal C$ be a DG-category over $\mathbb C$ (or more precisely, a presentable stable $\mathbb C$-linear $\infty$-category). Then $\Cond(\mathcal C)$ is tensored with $D(\Cond(\mathbb C))$, and via base change along $D(\Cond(\mathbb C))\to D(\Liq(\mathbb C))$, we get a $\Liq(\mathbb C)$-linear category $\mathcal C^{\Liq}$.

Recall that the analytification of $A:=\mathbb C[T]$ is obtained by killing the boundary idempotent algebra $A_\infty:=A(\{|T|\gg O(1)\})$ in $D(\Liq(\mathbb C[T]))$. The idempotent algebra $A_\infty$ gives rise to a split Verdier sequence (we adopt the definition in Calmès–Dotto–Harpez–Hebestreit–Land–Moi–Nardin–Nikolaus–Steimle, Hermitian $K$-theory for stable $\infty$-categories II: cobordism categories and additivity, including the splitness below) $$ D(A_\infty)\longrightarrow D(\Liq(\mathbb C[T]))\longrightarrow C(\mathbb C[T],\mathbb C[T]). $$ The analytification of every finitely generated $\mathbb C$-algebra $B$ can be understood as killing $A_\infty$ in $D(\Liq(B))$ for every map $A\to B$ of $\mathbb C$-algebras. This also works for $\mathcal C^{\Liq}$. We form the full subcategory $\mathcal C^{\an}\subseteq\mathcal C^{\Liq}$ spanned by objects $M\in\mathcal C^{\Liq}$ such that, for every $\Liq(\mathbb C)$-linear functor $F\colon D(\Liq(A))\to\mathcal C^{\Liq}$ with right adjoint $G\colon\mathcal C^{\Liq}\to D(\Liq(A))$, we have $G(M)\in C(\mathbb C[T],\mathbb C[T])$. Under finiteness conditions on $\mathcal C$ (a possible option is of finite type in Toën–Vaquié, Moduli of objects in DG-categories), it seems that this could be tested on finitely many $F$, and thus $\mathcal C^{\Liq}\to\mathcal C^{\an}$ is a split Verdier quotient.

Here are some mutually related questions:

  1. Can we compare this construction with any classical notion of analytification in noncommutative algebraic geometry?
  2. Can we apply any variant of Rosenberg's spectrum to this situation? This might be closely related to Soibelman, On non-commutative analytic spaces over non-archimedean fields. Rosenberg's spectrum usually only applies to non-derived settings, although he has also worked out some version which might be applicable to derived settings.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy