nLab
semi-topological K-theory

Contents

Contents

Idea

Semi-topological K-theory K * stK_*^{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 A 1\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 CSpc |-| : Aff_C \longrightarrow Spc

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

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

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

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

along the canonical functor P(Aff C)P Spt(Aff C)P(Aff_\mathbf{C}) \to P^{Spt}(Aff_\mathbf{C}) induced by taking suspension spectra objectwise.

Let TT be a dg-category over C\mathbf{C}. Let K(T)K(T) denote the nonconnective Waldhausen K-theory of TT. This defines a presheaf of spectra

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

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

Definition

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

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

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

Definition

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

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

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

Properties

Let TT be a dg-category over C\mathbf{C} and T\mathcal{M}_T the moduli space of pseudo-perfect dg-modules over TT. 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 | T||\mathcal{M}_T|.

Proposition

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

K˜ st(T)| T| \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 TT over C\mathbf{C}, the canonical morphism

ch top SHC:K top(T) SHCHP(T) 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

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

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

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.