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$$K$$-theory of twisted differential operators. (English) Zbl 0811.16020
This interesting article begins the study of $$K$$-groups of rings of differential operators with coefficients in vector bundles over homogeneous spaces. For example, denoting the line bundles over $$\mathbb{P}^ n$$ by $${\mathcal O} (k)$$, for $$k$$ an integer, and writing $$D_ k = \Gamma(\mathbb{P}^ n, {\mathcal D}_ k)$$ for the ring of globally defined differential operators with coefficients in $${\mathcal O}(k)$$, it is proved that $$G_ 0(D_{-1}) \cong \mathbb{Z}^ n \oplus \mathbb{Z}_{n + 1}$$ and $$K_ 0(D_{-1}) \cong \mathbb{Z}^ n$$. For $$n = 1$$, $$D_{-1}$$ is the unique non- regular primitive factor of $$U(\text{sl}_ 2)$$. The ring theoretical underpinnings of these results come from a study of the $$K$$-theory of endomorphism rings of finitely generated projective modules which occupies the first half of the article. The main result here is a generalization of the Bass Cancellation Theorem, called Replacement Theorem by the authors, which leads to an injectivity criterion for the Cartan map.
The work on the $$K$$-theory of endomorphism rings has subsequently been pushed further by M. Holland in the article “$$K$$-theory of endomorphism rings and of rings of invariants” (preprint, Univ. Sheffield, 1994), and the $$K$$-theory of twisted differential operators is taken up again in “$$K$$-theory of twisted differential operators on flag varieties” by M. Holland and P. Polo (preprint, Univ. Sheffield, 1994).

##### MSC:
 16S32 Rings of differential operators (associative algebraic aspects) 16E20 Grothendieck groups, $$K$$-theory, etc. 14M17 Homogeneous spaces and generalizations 19A49 $$K_0$$ of other rings
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