Steenrod operations on the Chow groups modulo a prime number p are not available when the characteristic of the base field is equal to p. We build operations on the restriction to a splitting field of the Chow group of a smooth projective homogeneous variety under a semi-simple linear algebraic group. These operations respect rationality of cycles provided that the base field admits a form of resolution of singularities, which is given by a result of Gabber when the base field has a characteristic different from p. Therefore we recover a weak form of Steenrod operations, in the cases when they are already constructed, using a very different approach. We show that the first Steenrod square (p = 2) can be constructed without using resolution of singularities. As a consequence we prove a theorem on the parity of the Witt index of a quadratic form. Another part of this work consists of proving directly that Chow motives of smooth projective quadrics decompose in the same way when the coefficients are either Z or Z/2.

(2009). Steenrod operations and quadratic forms. (Tesi di dottorato, Università degli Studi di Milano-Bicocca, 2009).

Steenrod operations and quadratic forms

HAUTION, OLIVIER JEAN-LAURENT
2009

Abstract

Steenrod operations on the Chow groups modulo a prime number p are not available when the characteristic of the base field is equal to p. We build operations on the restriction to a splitting field of the Chow group of a smooth projective homogeneous variety under a semi-simple linear algebraic group. These operations respect rationality of cycles provided that the base field admits a form of resolution of singularities, which is given by a result of Gabber when the base field has a characteristic different from p. Therefore we recover a weak form of Steenrod operations, in the cases when they are already constructed, using a very different approach. We show that the first Steenrod square (p = 2) can be constructed without using resolution of singularities. As a consequence we prove a theorem on the parity of the Witt index of a quadratic form. Another part of this work consists of proving directly that Chow motives of smooth projective quadrics decompose in the same way when the coefficients are either Z or Z/2.
Karpenko, Nikita
Chow groups, Grothendieck groups, projective homogeneous varieties, Steenrod operations, Adams operations, Riemann-Roch theorems, Witt indices, motivic decompositions
English
9-dic-2009
École Doctorale Paris Centre
2010
N/A
Università degli Studi di Milano-Bicocca
(2009). Steenrod operations and quadratic forms. (Tesi di dottorato, Università degli Studi di Milano-Bicocca, 2009).
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Tipologia di allegato: Doctoral thesis
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/10281/449099
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