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On the $$K$$-theory of generalized fibre bundles and some of their twisted forms. (English) Zbl 0824.19001
Bokut’, L. A. (ed.) et al., Third Siberian school on algebra and analysis. Proceedings of the third Siberian school, Irkutsk State University, Irkutsk, Russia, August 30-September 4, 1989. Providence, RI: American Mathematical Society. Transl., Ser. 2, Am. Math. Soc. 163, 143-154 (1995).
A new $$K$$-theory for algebraic varieties is developed, first in the “twisted” and then in the “non-twisted” case. To show that this approach is meaningful, results of Quillen are reproved. Thus, for a locally free sheaf $$\mathcal E$$ on a quasi-compact scheme $$Y$$, the natural homomorphism $K_ 0({\mathcal P}_ Y({\mathcal E})) \bigotimes_{K_ 0(Y)} K_ *(Y)\to K_ *({\mathcal P}_ Y({\mathcal E}))$ is seen to be an isomorphism. Also, for a quasi-compact scheme $$Y$$, a Severi-Brauer scheme $$X$$ of relative dimension $$n- 1$$ over $$Y$$ and the associated Azumaya algebra $$A$$, the author shows that there is an isomorphism $\bigoplus^{n- 1}_{i= 0} K_ *(Y, A^{\otimes i})\to K_ *(X).$ Earlier results provable with similar methods are stated.
For the entire collection see [Zbl 0816.00016].
##### MSC:
 19D10 Algebraic $$K$$-theory of spaces 19E08 $$K$$-theory of schemes 14C35 Applications of methods of algebraic $$K$$-theory in algebraic geometry 14D15 Formal methods and deformations in algebraic geometry