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.File | Dimensione | Formato | |
---|---|---|---|
Haution-2009-Steenrod operations and quadratic forms-VoR.pdf
accesso aperto
Descrizione: Tesi di dottorato
Tipologia di allegato:
Doctoral thesis
Licenza:
Altro
Dimensione
1.19 MB
Formato
Adobe PDF
|
1.19 MB | Adobe PDF | Visualizza/Apri |
I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.