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Action of the Cremona group on a noncommutative ring. (English) Zbl 1250.14008
The author constructs a noncommutative algebra \(A\) on which the Cremona group acts by outer automorphisms. Let us recall that the Cremona group is the group of birational transformations of the projective plane, or equivalently the group of \(\mathbb{C}\)-linear automorphisms of the field \(\mathbb{C}(x,y)\). Two different constructions of the algebra \(A\) are provided: the first one proceeds by brute force using a noncommutative analog of the classical presentation of the Cremona group due to V. A. Iskovskikh [Russ. Math. Surv. 40, No. 5, 231–232 (1985; Zbl 0613.14012)]. The second one is more conceptual and uses derived categories of coherent sheaves.

MSC:
14E07 Birational automorphisms, Cremona group and generalizations
14A22 Noncommutative algebraic geometry
14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
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