On the Grothendieck ring of varieties
2015 ◽
Vol 158
(3)
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pp. 477-486
Keyword(s):
AbstractLet K0(Vark) denote the Grothendieck ring of k-varieties over an algebraically closed field k. Larsen and Lunts asked if two k-varieties having the same class in K0(Vark) are piecewise isomorphic. Gromov asked if a birational self-map of a k-variety can be extended to a piecewise automorphism. We show that these two questions are equivalent over any algebraically closed field. If these two questions admit a positive answer, then we prove that its underlying abelian group is a free abelian group and that the associated graded ring of the Grothendieck ring is the monoid ring $\mathbb{Z}$[$\mathfrak{B}$] where $\mathfrak{B}$ denotes the multiplicative monoid of birational equivalence classes of irreducible k-varieties.
2012 ◽
Vol 55
(1)
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pp. 208-213
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2008 ◽
Vol 18
(01)
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pp. 165-180
2018 ◽
Vol 2018
(739)
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pp. 159-205
1975 ◽
Vol 78
(1)
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pp. 117-123
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2002 ◽
Vol 01
(01)
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pp. 107-112
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Keyword(s):
1959 ◽
Vol 14
◽
pp. 223-234
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Keyword(s):
2013 ◽
Vol 89
(2)
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pp. 234-242
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