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

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
Zbl 0292.18004
Full Text:
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
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