In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, the Verschiebung or Verschiebung operator ''V'' is a
homomorphism
In algebra, a homomorphism is a morphism, structure-preserving map (mathematics), map between two algebraic structures of the same type (such as two group (mathematics), groups, two ring (mathematics), rings, or two vector spaces). The word ''homo ...
between affine commutative
group scheme
In mathematics, a group scheme is a type of object from algebraic geometry equipped with a composition law. Group schemes arise naturally as symmetries of schemes, and they generalize algebraic groups, in the sense that all algebraic groups hav ...
s 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 a scheme ''S'', its Cartier dual is th ...
of the
Frobenius homomorphism
In commutative algebra and field theory, the Frobenius endomorphism (after Ferdinand Georg Frobenius) is a special endomorphism of commutative rings with prime characteristic , an important class that includes finite fields. The endomorphism m ...
. It was introduced by as the shift operator on
Witt vector
In mathematics, a Witt vector is an infinite sequence of elements of a commutative ring. Ernst Witt showed how to put a ring structure on the set of Witt vectors, in such a way that the ring of Witt vectors W(\mathbb_p) over the finite field o ...
s 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
In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field (mathematics), field that contains a finite number of Element (mathematics), elements. As with any field, a finite field is a Set (mathematics), s ...
''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
In mathematics, a root of unity is any complex number that yields 1 when raised to some positive integer power . Roots of unity are used in many branches of mathematics, and are especially important in number theory, the theory of group char ...
, 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
In mathematics, a Hopf algebra, named after Heinz Hopf, is a structure that is simultaneously a ( unital associative) algebra and a (counital coassociative) coalgebra, with these structures' compatibility making it a bialgebra, and that moreover ...
of symmetric functions, the Verschiebung ''V''
''n'' is the algebra
endomorphism
In mathematics, an endomorphism is a morphism from a mathematical object to itself. An endomorphism that is also an isomorphism is an automorphism. For example, an endomorphism of a vector space is a linear map , and an endomorphism of a g ...
that takes the complete symmetric function ''h''
''r'' to ''h''
''r''/''n'' if ''n'' divides ''r'' and to 0 otherwise.
See also
*
Dieudonné module
In mathematics, a Dieudonné module introduced by , is a module over the non-commutative Dieudonné ring, which is generated over the ring of Witt vectors by two special endomorphisms F and V called the Frobenius and Verschiebung operators. The ...
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, url-access=subscription
Algebraic groups