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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).

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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