Overconvergent Witt vectors (Q2904008)

Let \(A\) be a finitely generated algebra over a field \(K\) of positive characteristic. This paper constructs a subring \(W^\dagger(A)\) of the ring of Witt vectors \(W(A)\), called the ring of overconvergent Witt vectors. The expected application of this study is to construct the overconvergent de Rham-Witt complex, as completed in [\textit{C. Davis}, \textit{A. Langer} and \textit{T. Zink}, Ann. Sci. Éc. Norm. Supér. (4) 44, No. 2, 197--262 (2011; Zbl 1236.14025)] (although the latter paper appears in the literature before this paper).NEWLINENEWLINEFor the polynomial ring \(A = K[T_1, \dots, T_n]\), a Witt vector \((f_0,f_1, f_2, \dots) \in W(A)\) is called \textit{overconvergent} if there is a real number \(\epsilon>0\) and a real number \(C\) such that NEWLINE\[NEWLINE m-\epsilon p^{-m}\cdot \deg f_m \geq C \text{ for all } m \geq 0. NEWLINE\]NEWLINE The first result of this paper is that the subset of overconvergent Witt vectors form a subring \(W^\dagger(A)\) of \(W(A)\). For general finitely generated \(K\)-algebra \(A\), we first write it as a quotient of \(B = K[T_1, \dots, T_n]\). The ring of overconvergent Witt vectors \(W^\dagger(A)\) is defined to be the image of \(W^\dagger(B)\) under the natural map \(W(B) \twoheadrightarrow W(A)\). The authors check that the definition of \(W^\dagger(A)\) does not depend on the choice of the presentation of \(A\) as a quotient of a polynomial algebra.NEWLINENEWLINEThree important properties of the overconvergent Witt vectors are proved in this paper:NEWLINENEWLINE\quad (1) for \(A \hookrightarrow B\) a \textit{smooth} injective homomorphism of finitely generated \(K\)-algebras, \(W^\dagger(A) = W^\dagger(B) \cap W(A)\); a corollary of this fact is that the functor \(A \mapsto W^\dagger(A)\) is a sheaf for smooth topology (and hence for Zariski topology);NEWLINENEWLINE\quad (2) for \(A \to B\) a finite étale homomorphism of finitely generated \(K\)-algebra, the induced map \(W^\dagger(A) \to W^\dagger(B)\) is also finite and étale;NEWLINENEWLINE\quad (3) the map \(W^\dagger(A) \to A\) satisfies Hensel's lemma.NEWLINENEWLINEThe paper also includes a discussion on the relation with the work of de Jong and Kedlaya who studied the slope filtration of isocrystals over overconvergent Witt vectors of perfect fields with a valuation.NEWLINENEWLINEThe paper is well-written and is self-contained.