Zoran Skoda Zbl076835085

DE076835085 Zbl76835085

Porta, Mauro; Sala, Francesco, Two-dimensional categorified Hall algebras, J. Eur. Math. Soc. (JEMS) 25, 1113–1205 (2023) doi

Classification: 14A20 17B37 55P99

Keywords: Hall algebras; Higgs bundles; flat bundles; local systems; categorification; stable $\infty$-categories

Review

For a small Abelian category $\mathcal{A}$ of finite cohomological dimension and with finite $\mathrm{Hom}$- and $\mathrm{Ext}^i$-groups, one classically defines its Hall algebra as an associative algebra whose vector space basis is given by isomorphism classes of objects and equipped with structure constants defined in terms of counting extensions. More flexible approaches and generalizations are in place with important examples including quantum groups, Yangians, algebras of BPS-states in physics, cohomological Hall algebras of quivers etc. A cohomological Hall algebra attached to $\mathcal{A}$ is defined via a diagram $\mathcal{M}_{\mathcal{A}}\times\mathcal{M}_{\mathcal{A}}\stackrel{p}\leftarrow\mathcal{M}_{\mathcal{A}}^{\mathrm{ext}}\stackrel{q}\rightarrow\mathcal{M}_{\mathcal{A}}$ where $\mathcal{M}_{\mathcal{A}}$ and $\mathcal{M}_{\mathcal{A}}^{\mathrm{ext}}$ are moduli stacks of objects and of extensions in $\mathcal{A}$ and $p,q$ are natural projections. The multiplication is then defined on Borel-Moore homology classes as a pull-push (convolution) product $q_*\circ p^*\colon H_*^{\mathrm{BM}}(\mathcal{M}_{\mathcal{A}})\otimes H_*^{\mathrm{BM}}(\mathcal{M}_{\mathcal{A}})\to H_*^{\mathrm{BM}}(\mathcal{M}_{\mathcal{A}})$. This construction works under some regularity assumptions on $p$ which hold if $\mathcal{A}$ has cohomological dimension $1$ but may fail if $\mathcal{A}$ has cohomological dimension $2$. A uniform satisfactory approach in dimension $2$ to the construction of Hall multiplication is proposed in the article under review by replacing moduli stacks $\mathcal{M}_{\mathcal{A}}, \mathcal{M}_{\mathcal{A}}^{\mathrm{ext}}$ by certain derived enhancements $\mathbb{R}\mathcal{M}_{\mathcal{A}}, \mathbb{R}\mathcal{M}_{\mathcal{A}}^{\mathrm{ext}}$. In fact, a categorification of Hall multiplication is constructed: instead of working with homological or K-theoretic classes, the convolution formula gives a tensor product $q_*\circ p^*\colon\mathrm{Coh}^{\mathrm{b}}(\mathbb{R}\mathcal{M}_{\mathcal{A}})\otimes\mathrm{Coh}^{\mathrm{b}}(\mathbb{R}\mathcal{M}_{\mathcal{A}})\to\mathrm{Coh}^{\mathrm{b}}(\mathbb{R}\mathcal{M}_{\mathcal{A}})$ of an $\mathbb{E}_1$-monoidal structure on the dg-category of coherent sheaves on $\mathbb{R}\mathcal{M}_{\mathcal{A}}$ with bounded cohomology. Coherences for this $\mathbb{E}_1$-monoidal structure would be probably too hard to construct using deformation theory in terms of triangulated categories; throughout, the setup of stable $\infty$-categories is used instead.

Given a smooth and proper scheme $S$, authors construct a derived moduli stack $\mathbf{Coh}(S)$ of coherent sheaves on $S$ and its categorical Waldhausen S-construction $\mathcal{S}_\bullet\mathbf{Coh}(S)$, a simplicial object in stable $\infty$-category of derived stacks satisfying 2-Segal condition, where $\mathcal{S}_1\mathbf{Coh}(S)=\mathbf{Coh}(S)$ and $\mathcal{S}_2\mathbf{Coh}(S)=\mathbf{Coh}^{\mathrm{ext}}(S)$ is the derived moduli stack of extensions of coherent sheaves. The corresponding convolution diagram is $\mathbf{Coh}(S)\times\mathbf{Coh}(S)\stackrel{(\partial_o,\partial_1)}\longleftarrow\mathbf{Coh}^{\mathrm{ext}}(S)\stackrel{\partial_1}\rightarrow\mathbf{Coh}(S)$. The construction in the article builds on an insight of Dyckerhoff and Kapranov how 2-Segal simplicial objects induce Hall type structures [Dyckerhoff, T., Kapranov, M.: Higher Segal Spaces. Lecture Notes in Math. 2244, Springer, Cham (2019) Zbl1459.18001 MR3970975].

Hall-type $\mathbb{E}_1$-monoidal structure on the stable $\infty$-category $\mathrm{Coh}^{\mathrm{b}}_{\mathrm{pro}}(\mathbf{Coh}(X))$ is constructed if $X$ is a complex scheme of dimension $1$ or $2$ or the Betti, de Rham or Dolbeault stack of a smooth projective curve. Derived stacks of coherent sheaves on the latter stacks (Simpson’s shapes) enhance classical stacks of local systems, flat vector bundles and Higgs sheaves on $X$, respectively. This enables introducing the Hall monoidal products for the latter. An analytic version of the formalism is developed as well and a cohomological Hall algebra version of the derived Riemann-Hilbert correspondence is proven as an equivalence of stable $\mathbf{E}_1$-monoidal $\infty$-categories, relating the de Rham and Betti side in analytic setup. A version of non-abelian Hodge correspondence at the categorified Hall algebra level is demonstrated as well, in terms of Deligne shape interpolating between the de Rham and Dolbeault shape. These constructions are introduced with rich motivation from previously known cohomological Hall algebra structures in dimensions $1$ and $2$. Paths to decategorifications like $K$-theoretic Hall algebras are studied to make connections to the earlier known Hall algebras and missing conjectured cases. One of the motivations was to relate different categorifications of quantum groups, which have Hall algebra interpretations.

The technical part of the paper are numerous auxiliary results in derived algebraic geometry, including methods of deformation theory in construction of derived moduli stacks. These are refining and adapting known techniques (including some from earlier papers of the authors) to the present needs. Such needs are concisely but explicitly articulated. The wealth of new and background material is presented precisely, clearly and sufficiently motivated even for non-specialists.

Literature

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