Extensions of the Frobenius to ring of differential operators on polynomial algebra in prime characteristic

2008/04/07 by Bavula, V. V.
#13A35 #13N10 #16S32 #16W20 #16W22 #Algebraic Geometry (math.AG) #Commutative Algebra (math.AC) #FOS: Mathematics #Rings and Algebras (math.RA)

paper · doi:10.48550/arxiv.0804.1091

Abstract

Let K be a field of characteristic p>0. It is proved that each automorphism \s ∈ \Aut_K(\CDPn) of the ring \CDPn of differential operators on a polynomial algebra P_n= K[x₁, ..., x_n] is \em uniquely determined by the elements \s (x₁), ... ,\s (x_n), and the set \Frob (\CDPn) of all the extensions of the Frobenius from certain maximal commutative polynomial subalgebras of \CDPn, like P_n, is equal to \Aut_K(\CDPn) ⋅ \CF where \CF is the set of all the extensions of the Frobenius from P_n to \CDPn that leave invariant the subalgebra of scalar differential operators. The set \CF is found explicitly, it is large (a typical extension depends on \em countably many independent parameters).

Related

• The group of order preserving automorphisms of the ring of differential operators on Laurent polynomial algebra in prime characteristic
• The inversion formulae for automorphisms of polynomial algebras and differential operators in prime characteristic
• Isomorphism problems and groups of automorphisms for Ore extensions K[x][y; f(d)/(dx) ] (prime characteristic)
• Generators and defining relations for ring of differential operators on smooth affine algebraic variety in prime characteristic
• Differential operators on classical invariant rings do not lift modulo p