Frobenius endomorphism

Noun

Frobenius endomorphism (plural Frobenius endomorphisms)

1. (commutative algebra, field theory) Given a commutative ring R with prime characteristic p, the endomorphism that maps x → xᵖ for all x ∈ R.
   • Section 3 concerns what properties of the ring other than regularity are reflected by the homological properties of the Frobenius endomorphism.2003, Claudia Miller, “The Frobenius endomorphism and homological dimensions”, in Luchezar L. Avramov, Marc Chardin, Marcel Morales, Claudia Polini, editors, Commutative Algebra: Interactions with...
   • Let k=◌̅ 𝔽_𝕡, and let q be a power of p such that the group G is defined over 𝔽_𝕢. We then denote by F:G→G the corresponding Frobenius endomorphism. The Lie algebra 𝒢 and the adjoint action of G on 𝒢 are also defined over 𝔽_𝕢 and we still denote by F:𝒢→𝒢 the Frobenius endomorphism on 𝒢.
     […] Assume...2005, Emmanuel Letellier, Fourier Transforms of Invariant Functions on Finite Reductive Lie Algebras, Springer, Lecture Notes in Mathematics 1859, page 11

   Synonyms:

   Frobenius homomorphism