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

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.

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 |