Nested Witt vectors and their \(q\)-deformation
Let \(N\subset {\mathbb N}\) be a truncation set, i. e. a subset of \({\mathbb N}\) containing every divisor of each of its elements. Let \(Ring\) be the category of commutative rings not necessarily unital and \(gh_N:Ring\rightarrow Ring\) the \(N\)-nested ghost ring functor associating to each ring \(A\) the ring whose underlying set equals \(A^N\) and whose operation are defined componentwise. If \(f:A\rightarrow B\) is a morphism of Ring then \(gh_N(f)\) is given by \((x_n)_{n\in N}\rightarrow (f(x_n))_{n\in N}\). The ring of \(N\)-nested Witt vectors \(W_N:Ring\rightarrow Ring\) is a functor characterized by the following properties: i) \(W_N(A)=A^N\) as a set, ii) for a morphism \(f:A\rightarrow B\) of Ring, \(W_N(f):W_N(A)\rightarrow W_N(B)\) is given by \((x_n)_{n\in N}\rightarrow (f(x_n))_{n\in N}\), iii) the map \(W_N(A)\rightarrow gh_N(A)\) given by \((x_n)_{n\in N}\rightarrow (\Sigma_{d|n} dx_d^{n/d})_{n\in N}\) is a ring morphism [see \textit{R. Auer}, J. Algebra 252, 293--299 (2002; Zbl 1011.13013)]. The \(q\)-deformation of \(N\)-nested Witt vector \(W_N^q:Ring\rightarrow Ring\) is a functor characterized by the similar conditions of i),ii) and by iii') the map \(W_N^q(A)\rightarrow gh_N(A)\) given by \((x_n)_{n\in N}\rightarrow (\Sigma_{d|n} dq^{(n/d) -1}x_d^{n/d})_{n\in N}\) is a ring morphism. Here it gives necessary and sufficient conditions to have \(W_N^q(A)\) strictly isomorphic with \(W_N^r(A)\) for some \(q,r\in {\mathbb N}\). The \(q\)-deformed \(N\)-nested necklace ring is a functor \(Nr_N^q:Ring\rightarrow Ring\) characterized by the similar conditions of i),ii) and by iii'') the map \(Nr_N^q(A)\rightarrow gh_N(A)\) given by \((x_n)_{n\in N}\rightarrow (\Sigma_{d|n} dq^{(n/d) -1}x_d)_{n\in N}\) is a ring morphism. If \(N,M\) are coprime truncated sets then \(NM\) is still a truncated set and there exists a functorial isomorphism \(W_N^q\cdot W_M^q\cong W_{NM}^q\), \(Nr_N^q\cdot Nr_M^q\cong Nr_{NM}^q\).
scientific article; zbMATH DE number 5141738
scientific article