nLab semi-topological K-theory

Idea

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.

Definition

Definition

(topological realization of presheaves). Taking underlying topological spaces of complex points induces a functor of infinity-categories

$$|-| : Aff_C \longrightarrow Spc$$

from complex affine schemes to homotopy types. By left Kan extension it extends to a colimit-preserving functor

$$|-| : P(Aff_\mathbf{C}) \longrightarrow Spc$$

on the infinity-category of infinity-presheaves. This further extends to a colimit-preserving functor

$$|-| : P^{Spt}(Aff_\mathbf{C}) \longrightarrow Spt$$

on the infinity-category of presheaves of spectra, by taking the left Kan extension of the composite

$$P(Aff_\mathbf{C}) \stackrel{|-|}{\longrightarrow} Spc \stackrel{\Sigma^{\infty}}{\longrightarrow} Spt$$

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

$$K(T \otimes_{\mathbf{C}} -) : Aff_C^{op} \longrightarrow Spt$$

which sends $Spec(A)$ to $K(T \otimes_{\mathbf{C}} A)$.

Definition

The semi-topological K-theory of $T$ is the spectrum defined by the formula

$$K^{st}(T) = |L^{A^1}(L^{et}(K(T \otimes_{\mathbf{C}} -)))|$$

where $L^{et}$ and $L^{A^1}$ are the etale and $\mathbf{A}^1$-homotopy invariant localization endofunctors, respectively.

Definition

The semi-topological K-theory of a complex scheme $X$ is the spectrum

$$K^{st}(T) = K^{st}(Perf(X))$$

where $Perf(X)$ is the dg-category of perfect complexes on $X$.

Properties

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 $|\mathcal{M}_T|$.

Proposition

$|\mathcal{M}_T|$ is an infinite loop space, and there is a canonical identification

$$\tilde{K}^{st}(T) \simeq |\mathcal{M}_T|$$

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).

Conjectures

Conjecture

(lattice conjecture). For every smooth proper dg-category $T$ over $\mathbf{C}$, the canonical morphism

$$ch^{top} \wedge_{\mathbf{S}} H \mathbf{C} : K^{top}(T) \wedge_{\mathbf{S}} H\mathbf{C} \longrightarrow HP(T)$$

induced by the noncommutative Chern character is an equivalence of spectra.

(move this to topological K-theory of a dg-category?…)

Examples

• On a point, semi-topological K-theory coincides with topological K-theory:

  $$K^{st}(\Spec(\mathbf{C})) \simeq K^{top}(\Spec(\mathbf{C})),$$

  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:

  $$K_*^{st}(X, \mathbf{Z}/n) \simeq K_*^{alg}(X, \mathbf{Z}/n)$$

  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

  $$K_0^{st}(X, \mathbf{Q}) \stackrel{\sim}{\longrightarrow} CH^*_alg(X, \mathbf{Q}),$$

  where on the right hand side is the group of algebraic cycles modulo algebraic equivalence?.

References

A first definition of semi-topological K-theory for complex varieties was given in

• Eric M. Friedlander and Mark E. Walker?, Semi-topological K-theory using function complexes, Topology, 41(3):591–644, 2002.

An analogue of the Atiyah-Hirzebruch spectral sequence, and computations of semi-topological K-theory for a large class of varieties, are in

• Eric M. Friedlander, Christian Haesemeyer?, and Mark E. Walker?. Techniques, computations, and conjectures for semi-topological K-theory, Math. Ann., 330(4):759–807, 2004, web.

A survey is

• Eric M. Friedlander, Mark E. Walker?, Semi-topological K-theory, in Handbook of K-theory, Springer, 2005, web.

Semi-topological K-theory for dg-categories was introduced by Bertrand Toen in lecture III of

• Bertrand Toen, Saturated dg-categories, lectures at Workshop on Homological Mirror Symmetry and Related Topics, January 2010, University of Miami, notes.

Some discussion and interesting conjectures are in the last section of

• D. Kaledin, Motivic structures in non-commutative geometry, arXiv:1003.3210.

A state of the art treatment via dg-categories, from the point of view of derived noncommutative algebraic geometry, is in

• Anthony Blanc, Topological K-theory of complex noncommutative spaces, Ph.D. thesis, 2013, arXiv:1211.7360.