Geometric models for algebraic \(K\)-theory. (English) Zbl 0977.19002

Grayson, Daniel R.; Walker, Mark E.

\(K\)-Theory 20, No. 4, 311-330 (2000).

Suppose that \(X\) and \(Y\) are quasi-projective varieties over a field \(k\). Let \(\mathcal{P}^{d}(X,Y)\) denote the full subcategory of coherent sheaves on \(X \times Y\) consisting of those coherent sheaves \(\mathcal{F}\) such that \(\mathcal{F}\) is flat over \(X\) and the support of \(\mathcal{F}\) maps properly to \(X\) with fibres at most \(d\). The category \(\mathcal{P}^{d}(X,Y)\) is exact, and the authors write \(K^{d}(X,Y)\) for the associated \(K\)-theory space. They also write \(K^{d}(X \times \Delta^{\bullet},Y)\) for the geometric realization of the simplicial space arising from the standard cosimplicial scheme \(\Delta^{\bullet}\) of affine spaces.
The main result of this paper is that \(K^{d}(X \times \Delta^{\bullet},Y)\) is homotopy equivalent to the group completion of a \(\Gamma\)-space \(\hom(X \times \Delta^{\bullet},K_{Y,d})\), where \(K_{Y,d}\) is a \(\Gamma\)-object in the category of ind-schemes associated to generating families of coherent sheaves as above which are generated by global sections. The object \(K_{Y,d}\) can be regarded as a family of generalized Grassmanians. The bivariant \(K\)-theory functor described here specializes in several interesting ways (e.g. to ordinary \(K\)-theory of regular schemes, \(K'\)-theory, Suslin \(K\)-homology), and the main result of this paper implies representability results in those cases.

Reviewer: Rick Jardine (London /Ontario)

MSC:

19E08	\(K\)-theory of schemes
19D10	Algebraic \(K\)-theory of spaces
19E15	Algebraic cycles and motivic cohomology (\(K\)-theoretic aspects)
14C35	Applications of methods of algebraic \(K\)-theory in algebraic geometry

Keywords:

category of ind-schemes; \(\Gamma\)-spaces; representability; bivariant \(K\)-theory functor