In mathematics, the Verschiebung or Verschiebung operatorV is a homomorphism between affine commutative group schemes over a field of nonzero characteristic p. For finite group schemes it is the Cartier dual of the Frobenius homomorphism. It was introduced by Witt (1937) as the shift operator on Witt vectors taking (a0, a1, a2, ...) to (0, a0, a1, ...). ("Verschiebung" is German for "shift", but the term "Verschiebung" is often used for this operator even in other languages.)
The Verschiebung operator V and the Frobenius operator F are related by FV = VF = [p], where [p] is the pth power homomorphism of an abelian group scheme.
Examplesedit
If G is the discrete group with n elements over the finite field Fp of order p, then the Frobenius homomorphism F is the identity homomorphism and the Verschiebung V is the homomorphism [p] (multiplication by p in the group). Its dual is the group scheme of nth roots of unity, whose Frobenius homomorphism is [p] and whose Verschiebung is the identity homomorphism.
For Witt vectors the Verschiebung takes (a0, a1, a2, ...) to (0, a0, a1, ...).
On the Hopf algebra of symmetric functions the Verschiebung Vn is the algebra endomorphism that takes the complete symmetric function hr to hr/n if n divides r and to 0 otherwise.
Witt, Ernst (1937), "Zyklische Körper und Algebren der Characteristik p vom Grad pn. Struktur diskret bewerteter perfekter Körper mit vollkommenem Restklassenkörper der Charakteristik pn", Journal für die Reine und Angewandte Mathematik (in German), 176: 126–140, doi:10.1515/crll.1937.176.126
January 01, 1970
verschiebung, operator, mathematics, verschiebung, homomorphism, between, affine, commutative, group, schemes, over, field, nonzero, characteristic, finite, group, schemes, cartier, dual, frobenius, homomorphism, introduced, witt, 1937, shift, operator, witt, . In mathematics the Verschiebung or Verschiebung operator V is a homomorphism between affine commutative group schemes over a field of nonzero characteristic p For finite group schemes it is the Cartier dual of the Frobenius homomorphism It was introduced by Witt 1937 as the shift operator on Witt vectors taking a0 a1 a2 to 0 a0 a1 Verschiebung is German for shift but the term Verschiebung is often used for this operator even in other languages The Verschiebung operator V and the Frobenius operator F are related by FV VF p where p is the pth power homomorphism of an abelian group scheme Examples editIf G is the discrete group with n elements over the finite field Fp of order p then the Frobenius homomorphism F is the identity homomorphism and the Verschiebung V is the homomorphism p multiplication by p in the group Its dual is the group scheme of nth roots of unity whose Frobenius homomorphism is p and whose Verschiebung is the identity homomorphism For Witt vectors the Verschiebung takes a0 a1 a2 to 0 a0 a1 On the Hopf algebra of symmetric functions the Verschiebung Vn is the algebra endomorphism that takes the complete symmetric function hr to hr n if n divides r and to 0 otherwise See also editDieudonne moduleReferences editDemazure Michel 1972 Lectures on p divisible groups Lecture Notes in Mathematics vol 302 Berlin New York Springer Verlag doi 10 1007 BFb0060741 ISBN 978 3 540 06092 5 MR 0344261 Witt Ernst 1937 Zyklische Korper und Algebren der Characteristik p vom Grad pn Struktur diskret bewerteter perfekter Korper mit vollkommenem Restklassenkorper der Charakteristik pn Journal fur die Reine und Angewandte Mathematik in German 176 126 140 doi 10 1515 crll 1937 176 126 Retrieved from https en wikipedia org w index php title Verschiebung operator amp oldid 1183925713, wikipedia, wiki, book, books, library,
