Topological \(K\)-theory of complex noncommutative spaces (Q2805051)

The paper under review introduces a new localising invariant for dg categories, called \textit{topological \(K\)-theory}. One interpretation of noncommutative algebraic geometry consists of the study of (sufficiently nice) dg categories as being the central objects, where objects from algebraic geometry are considered via (dg enhancements of) their derived categories. Many invariants from algebra and geometry have a generalisation to one for arbitrary dg categories agreeing with the original invariant. Examples of these are Hochschild homology, various flavours of algebraic \(K\)-theory, various flavours of cyclic homology, \dots\ and these are the object of study in the setting of Tabuada's model structures on the category of dg categories.NEWLINENEWLINEFor schemes over the complex numbers it is possible to consider the \({\mathbb C}\)-valued points together with the Euclidean topology, and for this topological space one can consider (various flavours of) topological \(K\)-theory. Based on a suggestion from Toën as outlined in \S 2.2.6(b) in [\textit{L.\ Katzarkov} and \textit{M.\ Kontsevich} and \textit{T.\ Pantev}, Proceedings of Symposia in Pure Mathematics 78, 87--174 (2008, Zbl 1206.14009)], the paper under review constructs an invariant for dg categories that indeed generalises the usual topological \(K\)-theory, together with the various expected compatibilities such as the Chern character map. One motivation for having such an invariant for dg categories stems from the use of Hodge structures in homological mirror symmetry, where one wishes to use ``Betti data'' to define the \({\mathbb Q}\)-structure on period cyclic homology, i.e.\ find an integral lattice inside the period cyclic homology as the image of the Chern character map from topological \(K\)-theory.NEWLINENEWLINEThe construction starts by considering algebraic \(K\)-theory and turning it into a presheaf of spectra on the category of affine schemes. Using the left Kan extension of topological realisation one obtains the (connective) \textit{semitopological \(K\)-theory} as an intermediate invariant. It is shown that it has an appropriate module structure for the ring spectrum that corepresents topological \(K\)-theory, and by inverting the element in \(\pi_2\) associated to Bott periodicity one obtains the desired invariant. The construction is highly non-trivial, and depends on several results of independent interest.NEWLINENEWLINEThe paper contains three conjectures regarding topological \(K\)-theory. The first (and most important one) is the \textit{lattice conjecture} that says that for a smooth and proper dg category the Chern map indeed defines a lattice inside the period cyclic homology, hence the Betti data for a noncommutative Hodge structure. It is shown (by agreement with usual topological \(K\)-theory) to be true for dg categories of geometric origin. By the compatibility of the lattice conjecture with retracts, this means it is true for the class of geometric dg categories, and by Orlov's geometricity result for all finite-dimensional algebras of finite global dimension. The second conjecture states that semitopological \(K\)-theory vanishes in negative degree for smooth and proper dg categories, as it does for smooth commutative algebras of finite global dimension. The third conjecture states that modulo \(n\) the algebraic and topological \(K\)-theory for dg categories are equivalent via the obvious morphism.NEWLINENEWLINEThe author moreover defines an analogue of Deligne cohomology for dg categories, and expresses semitopological \(K\)-theory in terms of the group completion of the topological realisation of the derived moduli stack of perfect complexes. Finally it gives an independent proof of the lattice conjecture for a pseudoconnective version of topological \(K\)-theory of the lattice conjecture for finite-dimensional algebras of finite global dimension, not using Orlov's geometricity result.