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Higher \(K\)-theory of the category of weakly equivariant \({\mathcal D}\)- modules. (English) Zbl 0802.19002

Let \(X\) be a complex smooth quasiprojective algebraic variety and \(G\) a complex linear algebraic group acting on \(X\). This paper deals with equivariant versions of some of D. Quillen’s results on \(K\)-theory [in: Algebraic \(K\)-theory I, Lect. Notes Math. 341, 85-147 (1973; Zbl 0292.18004)] – namely, localization and Mayer-Vietoris sequences for the higher \(K\)-theory of coherent weakly \(G\)-equivariant \({\mathcal D}_ X\)- modules [see another paper by the author, Math. Z. 206, No. 1, 131-144 (1991; Zbl 0696.14008)], and higher \(K\)-theory of \(G\)-equivariant graded and filtered rings, in the case where the group \(G\) is a complex diagonalisable group acting linearly on \(X\). For the localisation and Mayer-Vietoris sequences, the key point is that every weakly \(G\)- equivariant quasi-coherent \({\mathcal D}_ X\)-module is the filtered colimit of its weakly \(G\)-equivariant coherent \({\mathcal D}_ X\)-submodules. As a consequence of the results on graded and filtered rings, the author obtains the equality between the higher \(K\)-theory of coherent \(G\)- equivariant \({\mathcal D}_ X\)-modules and coherent weakly \(G\)-equivariant \({\mathcal D}_ X\)-modules. This has been used [“Riemann-Roch for weakly equivariant \({\mathcal D}\)-modules. II”, Math. Z. 214, No. 2, 297-324 (1993)] to get a Riemann-Roch theorem at the level of the higher \(K\)- groups. The author also proves a generalization of the last result to the respective derived categories. For this, he uses Thomason’s approximation theorem [R. W. Thomason and T. Trobaugh in: The Grothendieck Festschrift, Vol. III, Progr. Math. 88, 247-435 (1990; Zbl 0731.14001)].

MSC:

19E08 \(K\)-theory of schemes
14C35 Applications of methods of algebraic \(K\)-theory in algebraic geometry
19L47 Equivariant \(K\)-theory
19L10 Riemann-Roch theorems, Chern characters
18E30 Derived categories, triangulated categories (MSC2010)
19D99 Higher algebraic \(K\)-theory
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References:

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