Articles written in Proceedings – Mathematical Sciences

• Aperiodic rings, necklace rings and Witt vectors—II

In an earlier paper the author and Dr K Wehrhahn have introduced the concept of the aperiodic ring Ap(A) of a commutative ringA. It is a commutative ring equipped with two families of operatorsVr: Ap(A)→Ap(A),Fr: Ap(A)→Ap(A) for every integerr≥1, called the Verschiebung and Frobenius operators. Let$$D(A) = \{ \sum\nolimits_{k \geqslant 1} {V_k \underline {S(a_k )} \left| {a_k \in A} \right.} \}$$, where for anya∈A,$$\underline {S(a)}$$ is the elementS(a, 1),S(a, 2),S(a, 3),...)of Ap (A). Let W(A) denote the ring of Witt vectors ofA. Let χ: W(A)→Ap(A) denote the map$$\sum\nolimits_{k \geqslant 1} {V_k \underline {S(a_k )} }$$. We prove that χ is a ring homomorphism preserving the Verschiebung and Frobenius operators with image χ=D(A). Moreover χ:W(A)→D(A) is an isomorphism if and only if the additive group ofA is torsion-free.

• Verschiebung and Frobenius operators

Metropolis and Rota introduced the concept of the necklace ring Nr(A) of a commutative ringA. WhenA contains Q as a subring there is a natural bijection γ:Nr(A→1+tA[]. Grothendieck has introduced a ring structure on 1+tA[t] while studyingK-theoretic Chern classes. Nr(A) comes equipped with two families of operatorsFr,Vr called the Frobenius and Verschiebung operators. Mathematicians studying formal group laws have introduced two families of operators,Fr, andVr on 1+tA[t]. Metropolis and Rota have not however tried to show that γ preserves, these operators. They transport the operators from Nr(A) to 1+tA[t] using γ. In our present paper we show that γ does preserve all these operators.

• Verschiebung and Frobenius operators

• # Editorial Note on Continuous Article Publication

Posted on July 25, 2019