In mathematics, the Verschiebung or Verschiebung operator ''V'' is a
homomorphism
In algebra, a homomorphism is a structure-preserving map between two algebraic structures of the same type (such as two groups, two rings, or two vector spaces). The word ''homomorphism'' comes from the Ancient Greek language: () meaning "same" ...
between affine commutative
group schemes over a
field
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grass ...
of nonzero characteristic ''p''. For finite group schemes it is the
Cartier dual In mathematics,
Cartier duality is an analogue of Pontryagin duality for commutative group schemes. It was introduced by .
Definition using characters
Given any finite flat commutative group scheme ''G'' over ''S'', its Cartier dual is the group o ...
of the
Frobenius homomorphism. It was introduced by as the shift operator on Witt vectors taking (''a''
0, ''a''
1, ''a''
2, ...) to (0, ''a''
0, ''a''
1, ...). ("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 ''p''th power homomorphism of an abelian group scheme.
Examples
*If ''G'' is the discrete group with ''n'' elements over the finite field ''F''
''p'' 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 ''n''th roots of unity, whose Frobenius homomorphism is
'p''and whose Verschiebung is the identity homomorphism.
*For Witt vectors the Verschiebung takes (''a''
0, ''a''
1, ''a''
2, ...) to (0, ''a''
0, ''a''
1, ...).
*On the Hopf algebra of symmetric functions the Verschiebung ''V''
''n'' is the algebra endomorphism that takes the complete symmetric function ''h''
''r'' to ''h''
''r''/''n'' if ''n'' divides ''r'' and to 0 otherwise.
See also
*
Dieudonne module
References
*
* {{Citation , url=http://www.digizeitschriften.de/main/dms/img/?IDDOC=504725 , last1=Witt , first1=Ernst , author1-link = Ernst Witt , title=Zyklische Körper und Algebren der Characteristik p vom Grad p
n. Struktur diskret bewerteter perfekter Körper mit vollkommenem Restklassenkörper der Charakteristik p
n , language=German , year=1937 , journal=Journal für die Reine und Angewandte Mathematik , volume=176 , pages=126–140 , doi=10.1515/crll.1937.176.126
Algebraic groups