Algebraic K-theory of Grassmann varieties and their convoluted forms. (English. Russian original) Zbl 0708.14033

Panin, I. A.

Funct. Anal. Appl. 23, No. 2, 143-144 (1989); translation from Funkts. Anal. Prilozh. 23, No. 2, 71-72 (1989).

Let \(Y\) be a Noetherian scheme over a field of characteristic zero, \(\mathcal E\) a vector bundle of rank \(n\) over \(Y\), and \(G=\text{Gr}(k,\mathcal E)\xrightarrow {p} Y\) the bundle of Grassmannians associated with \(\mathcal E\). The author shows that the natural homomorphism \(K_0(G)\otimes K_i(Y)\to K_i(G)\) is an isomorphism, and describes two bases of \(K_0(G)\) over \(K_0(Y)\). A similar result is obtained for the bundle of Grassmannians \(p: G\to Y\) determined by an element \(\phi \in H^1_{\text{et}}(Y, \mathrm{PGL}(n))\).
These results represent a generalization of results by D. Quillen [in: Algebraic K-theory I, Proc. Conf. Battelle Inst. 1972, Lecture Notes Math. 341, 85–147 (1973; Zbl 0292.18004)] for \(k=1\). The methods used by the author differ substantially from those of D. Quillen.

Reviewer: Jiří Vanžura (Brno)

MSC:

14M15	Grassmannians, Schubert varieties, flag manifolds
14C35	Applications of methods of algebraic \(K\)-theory in algebraic geometry
19E99	\(K\)-theory in geometry

Keywords:

\(K_0\); bundle of Grassmannians

Citations:

Zbl 0292.18004

References:

[1]	A. A. Beilinson, Funkts. Anal. Prilozhen.,12, No. 3, 68-69 (1978).
[2]	I. N. Bernshtein, I. M. Gel’fand, and S. I. Gel’fand, Funkts. Anal. Prilozhen.,12, No. 3, 66-67 (1978).
[3]	M. M. Kapranov, Funkts. Anal. Prilozhen.,17, No. 2, 78-79 (1985).
[4]	D. Mumford, Abelian Varieties [Russian translation], Mir, Moscow (1971). · Zbl 0222.14023
[5]	A. Grothendieck, North-Holland (1968).
[6]	D. Quillen, Lect. Notes Math.,341, 85-147 (1972). · Zbl 0292.18004