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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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References:
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