Generators for Hall algebras of surfaces (Q5918580)

Letting \(S\) to be a smooth surface over \(\mathbb{C}\), let \(\mathfrak{M}_\beta\) be the derived moduli stack of coherent sheaves on \(S\) with support class \(\beta\) for an algebraic class \(\beta\in H^2(S,\mathbb{Z})\oplus H^4(S,\mathbb{Z})\). For classes \(\beta\) and \(\gamma\), there are maps \(\mathfrak{M}_{\beta}\times \mathfrak{M}_{\gamma}\xleftarrow{q_{\beta, \gamma}}\mathfrak{M}_{\beta, \gamma}\xrightarrow{p_{\beta,\gamma}}\mathfrak{M}_{\beta+\gamma}\), where \(\mathfrak{M}_{\beta, \gamma}\) is the corresponding stack of extensions. The map \(q_{\beta,\gamma}\) is quasi-smooth and \(p_{\beta,\gamma}\) is proper, so they induce functors \(m_{\beta, \gamma} := p_{\beta,\gamma*}q^*_{\beta,\gamma}: D^b\left(\mathfrak{M}_{\beta}\right)\otimes D^b\left(\mathfrak{M}_{\gamma}\right)\to D^b\left(\mathfrak{M}_{\beta+\gamma}\right)\). The category \(\bigoplus_{\beta} D^b\left(\mathfrak{M}_{\beta}\right)\) is monoidal with respect to the functors \(m_{\beta,\gamma}\). Taking the Grothendieck group of this category, one obtains the \(K\)-theoretic Hall algebra of a surface for sheaves of dimension zero. Categorical (\(K\)-theoretic) Hall algebras for quivers with potential are local version of these categories (algebras). Particular cases of equivariant \(K\)-theoretic Hall algebras of quivers with potentials, namely preprojective ones, are expected to be positive parts of quantum affine groups. T. Pădurariu constructs semi-orthogonal decompositions for categorical Hall algebras of points on \(S\). He refines these decompositions in \(K\)-theory for a topological \(K\)-theoretic Hall algebra.