Quasicoherent sheaves on complex noncommutative two-tori (Q2472424)

The article under review extends the study of holomorphic vector bundles on noncommutative two-tori \(T_{\theta,\tau} = T\) (parametrized by \(\theta\in\mathbb R\) and complexified by a differential \(\delta_\tau,\tau\in\mathbb{H}\), which is an analogue of the \(\bar{\partial}\) operator) as \(C^\infty(T)\)-modules carrying connections compatible with \(\delta_\tau\) to the quasicoherent level. The idea behind this definition of the derived category of holomorphic vector bundles on \(T\) is the following fact: given a complex manifold \(X\), the category of finitely presented modules with compatible connections over the differential algebra \((C^\infty(X),\bar{\partial})\) is derived equivalent to \(D^b(X)\), the bounded derived category of analytic coherent sheaves over \(X\) (actually there is an exact equivalence at the level of abelian categories, see \textit{N. Pali} [Math. Ann. 336, No. 3, 571--615 (2006; Zbl 1110.32003)]. Building upon the work of \textit{A. Connes} and \textit{M. A. Rieffel} [Contemp. Math. 62, 237--266 (1987; Zbl 0633.46069)], the author had initiated the study of the category of holomorphic vector bundles on noncommutative two-tori along with \textit{A. Schwarz} [Commun. Math. Phys. 236, No. 1, 135--159 (2003; Zbl 1033.58009)] and subsequently shown that it is derived equivalent to \(D^b(\mathbb C/(\mathbb Z +\tau\mathbb Z))\) [see \textit{A. Polishchuk}, Doc. Math., J. DMV 9, 163--181 (2004; Zbl 1048.32012)]. We set \(A_\theta = C^\infty(T)\). Quite naturally, coherent sheaves are defined as finitely presented modules with connections over \((A_\theta,\delta_\tau)\) and quasicoherent sheaves are defined as inductive limits inside the category of all differentiable modules. The author lifts the above-mentioned derived equivalence to the quasicoherent level, i.e., \(D^b(\text{Qcoh}(T))\cong D^b(\text{Qcoh}(\mathbb C/(\mathbb Z+\tau\mathbb Z)))\) as a consequence of the following general result (also proven in the article), which might be interesting in its own right. Let \(\mathcal A\) be a noetherian abelian category of homological dimension \(\leqslant 1\) equipped with a cotilting torsion pair and let \(\mathcal A_t\) be the tilted heart. It is known that \(D^b(\mathcal A_t)\cong D^b(\mathcal A)\) since the torsion pair is assumed to be cotilting \textit{A. I. Bondal} and \textit{M. Van den Bergh} [Mosc. Math. J. 3, No.~1, 1--36 (2003; Zbl 1135.18302)]. The author proves that the torsion pair lifts to \(\text{Ind}(\mathcal A)\) (the inductive completion of \(\mathcal A\)) and there are equivalences of categories \(\text{Ind}(\mathcal A)_t\cong\text{Ind}(\mathcal A_t)\) (additive equivalence) and \(D^b(\text{Ind}(\mathcal A))\cong D^b(\text{Ind}(\mathcal A_t))\) (exact equivalence). The author extends the definition of rank to \(\text{Qcoh}(T)\), whose values exhaust \(\mathbb R_{\geqslant 0}\cup\{\infty\}\), and discusses some concepts like slope, stability, Harder-Narasimhan filtration, etc. Most of the results are in keeping with our intuition from the classical picture, provided one takes into account some finiteness conditions. Another interesting result is that the full subcategory consisting of finite rank objects of \(\text{Qcoh}(\eta_T):=\text{Qcoh}(T)/\text{Tors}\), where \(\text{Tors}\) is the Serre subcategory consisting of rank zero sheaves, is equivalent to the category of finitely presented right modules over a semihereditary algebra. The semihereditary algebra arises as \(\text{End}_{\text{Qcoh}(\eta_T)}(P)\), where \(P\) is any finite rank projective object in \(\text{Qcoh}(\eta_T)\), which is obtained as a filtering union of holomorphic vector bundles. Since the stalk of a torsion sheaf at a generic point vanishes in the classical setting, \(\text{Qcoh}(\eta_T)\) is a noncommutative analogue of the category of quasicoherent sheaves on a ``generic point'' of \(T\) and from this perspective the semihereditary algebra may be regarded as an analogue of the function field. The author also proves that the projective objects of finite rank are classified up to isomorphism by their ranks in \(\text{Qcoh}(\eta_T)\).