Semi-topological K-theory $K_*^{st}$ is a homological invariant of complex noncommutative spaces that interpolates between algebraic K-theory and topological K-theory. It is defined by starting with the presheaf defined by nonconnective algebraic K-theory, taking the associated etale sheaf, making it $\mathbf{A}^1$-homotopy invariant, and finally taking the topological realization of the resulting presheaf. One recovers topological K-theory from this by inverting the Bott element.
Semi-topological K-theory of complex varieties is related to morphic cohomology? in the same way that algebraic K-theory is related to motivic cohomology and topological K-theory is related to singular cohomology. In fact there is an analogue of the Atiyah-Hirzebruch spectral sequence.
Some examples suggest that semi-topological K-theory may be a better suited invariant for complex varieties than either algebraic or topological K-theory.
(topological realization of presheaves). Taking underlying topological spaces of complex points induces a functor of infinity-categories
from complex affine schemes to homotopy types. By left Kan extension it extends to a colimit-preserving functor
on the infinity-category of infinity-presheaves. This further extends to a colimit-preserving functor
on the infinity-category of presheaves of spectra, by taking the left Kan extension of the composite
along the canonical functor $P(Aff_\mathbf{C}) \to P^{Spt}(Aff_\mathbf{C})$ induced by taking suspension spectra objectwise.
Let $T$ be a dg-category over $\mathbf{C}$. Let $K(T)$ denote the nonconnective Waldhausen K-theory of $T$. This defines a presheaf of spectra
which sends $Spec(A)$ to $K(T \otimes_{\mathbf{C}} A)$.
The semi-topological K-theory of $T$ is the spectrum defined by the formula
where $L^{et}$ and $L^{A^1}$ are the etale and $\mathbf{A}^1$-homotopy invariant localization endofunctors, respectively.
The semi-topological K-theory of a complex scheme $X$ is the spectrum
where $Perf(X)$ is the dg-category of perfect complexes on $X$.
Let $T$ be a dg-category over $\mathbf{C}$ and $\mathcal{M}_T$ the moduli space of pseudo-perfect dg-modules over $T$. This is a derived stack, hence an infinity-stack on the infinity-category of simplicial commutative rings. Applying the topological realization (which extends to presheaves on simplicial commutative rings in the obvious way), one gets a spectrum $|\mathcal{M}_T|$.
$|\mathcal{M}_T|$ is an infinite loop space, and there is a canonical identification
where the left hand side is a connective version of semi-topological K-theory.
See (Blanc 13, 4.3), (Toen 10), (Kaledin 10, section 8).
(lattice conjecture). For every smooth proper dg-category $T$ over $\mathbf{C}$, the canonical morphism
induced by the noncommutative Chern character is an equivalence of spectra.
(move this to topological K-theory of a dg-category?…)
On a point, semi-topological K-theory coincides with topological K-theory:
while the homotopy groups of algebraic K-theory are uncountable in degree $i \gt 0$.
With finite coefficients, semi-topological K-theory coincides with algebraic K-theory:
while the topological K-theory only sees the homotopy type of the variety and not the finer algebraic structure.
Rationally, semi-topological K-theory contains information about the cycles on $X$, and conjecturally, the rational Hodge filtration on singular cohomology.
$K_0^{st}(X)$ coincides with the Grothendieck group of vector bundles up to algebraic equivalence?. Rationally, an analogue of the Chern character map induces canonical isomorphisms
where on the right hand side is the group of algebraic cycles modulo algebraic equivalence?.
A first definition of semi-topological K-theory for complex varieties was given in
An analogue of the Atiyah-Hirzebruch spectral sequence, and computations of semi-topological K-theory for a large class of varieties, are in
A survey is
Semi-topological K-theory for dg-categories was introduced by Bertrand Toen in lecture III of
Some discussion and interesting conjectures are in the last section of
A state of the art treatment via dg-categories, from the point of view of derived noncommutative algebraic geometry, is in
Last revised on May 28, 2017 at 09:37:30. See the history of this page for a list of all contributions to it.