Integrable derivations in the sense of Hasse-Schmidt for some binomial plane curves (Q2039364)

This paper is about the integrable Hasse-Schmidt derivations. In particular, the focus is on the integrable derivation for \(x^n-y^q\). Let \(A\) be a commutative algebra over a commutative ring \(k\). Then we say a sequence \(D=(D_0,D_1,\cdots,D_m)\) is a Hasse-Schmidt derivation (over \(k\)), if every \(D_i\), \(0\leq i\leq m\) is a \(k\)-linear map \(D_i:A\rightarrow A\) satisfying \[ D_0=ID_A, \quad D_i(xy)=\sum_{a+b=i}D_a(x)D_b(y), \] where \(x,y\in A\). Denote by \(HS_k(A;m)\) the set of Hasse-Schmidt derivations over \(k\) of length \(m\) (\(HS_k(A)\) if \(m=\infty\)). Definition [\textit{M. De La P. Tirado Hernández}, Rend. Semin. Mat. Univ. Padova 145, 53--71 (2021; Zbl 07367429)]. Let \(D\in HS_k(A;m)\), where \(m\in \bar{\mathbb{N}}\) and \(n\geq m\). Let \(I\) be an ideal of \(A\). 1. Let \(q\in \{1,\cdots,m\}\) be an integer. A truncation map, denote by \(\tau_{mq}(D)\) is defined by \(\tau_{mq}(D)=(Id, D_1,\cdots,D_q)\). 2. \(D\) is said to be \(I\)-logarithmic, if \(D_i(I)\subset I\) for every \(i\). Denote by \(HS_k(\log I;m)\) the set of all \(I\)-logarithmic Hasse-Schmidt derivations, \(HS_k(\log I):=HS_k(\log I; \infty)\), and \(Der_k(\log I):= HS_k(\log I;1)\). 3. \(D\) is said to be \(n\)-integrable if there exists \(E\in HS_k(A;n)\) such that \(\tau_{nm}(E)=D\). Any such \(E\) is called an \(n\)-integral of \(D\). If \(D\) is \(\infty\)-integrable, it is called integrable. If \(m=1\) then the set of all \(n\)-integrable derivation is denoted by \(IDer_k(A;n)\), and \(IDer_k(A):=IDer_k(A;\infty)\). 4. If \(D\in HS_k(\log I;m)\) then \(D\) is called \(I\)-logarithmically \(n\)-integrable if there exists \(E\in HS_k(\log I;n)\) such that \(E\) is an \(n\)-integral of\(D\). The set of \(I\)-logarithmically \(n\)-integrable derivations (i.e. for \(m=1\)) is denoted by \(IDer_k(\log I;n)\) and \(IDer_k(\log I):=IDer_k(\log I;\infty)\). 5. The \(k\)-algebra \(A\) has a leap at \(s>1\) if the inclusion \(IDer_k(A;s-1)\nsupseteq IDer_k(A;s)\) is proper. The set of leaps of \(A\) over \(k\) is denoted by \(Leaps_k(A)\). From now let \(k\) to be a unique factorization domain of characteristic \(p>0\). Let also \(R=k[x_1,\cdots,x_d]\). Proposition [\textit{M. De La P. Tirado Hernández}, Rend. Semin. Mat. Univ. Padova 145, 53--71 (2021; Zbl 07367429)]. If \(f,g\in R\) are coprime then for every \(n\in \bar{\mathbb{N}}\) we have \(HS_k(\log fg;n)=HS_k(\log f;n)\cap HS_k(\log g;n)\). Proposition [\textit{M. De La P. Tirado Hernández}, Rend. Semin. Mat. Univ. Padova 145, 53--71 (2021; Zbl 07367429)]. Let \(k\) be a UFD of characteristic \(p>0\) and let \(R= k[x_1,\cdots,x_d]\). Let \(h\) be a polynomial of \(R\). For every \(n\in \bar{\mathbb{N}}\), we have \(IDer_k(\log h;n)=IDer_k(\log h^p;np)\). The next proposition expresses a relation between the modules of \(i\)-integrable derivations of \(R/\langle h\rangle\) and \(R/\langle h^{p^\tau}\rangle\), where \(\tau\geq 1\). Proposition [\textit{M. De La P. Tirado Hernández}, Rend. Semin. Mat. Univ. Padova 145, 53--71 (2021; Zbl 07367429)]. Let \(k\) be a UFD of characteristic \(p>0\), \(R=k[x_1,\cdots,x_d]\), \(h\in R\) and \(\tau\geq 1\). Denote \(A:=R/\langle h^{p^\tau}\rangle\) and \(A':=R/\langle h\rangle\). Then, \[ Leaps_k(A)=\begin{cases} \{np^\tau | n \in Leaps_k(A')\} & \text{if} \, Der_k(\log h)=Der_k(R).\\ \{np^\tau |n \in Leaps_k(A')\}\cup \{p^\tau\} & \text{if}\, Der_k(\log h)\neq Der_k(R). \end{cases} \] Let \(q,n\geq 1\), \(R=k[x,y]\), where \(k\) is a reduced ring of characterstic \(p>0\). Using the last proposition, now for every \(i\), one can study the module of \(i\)-integrable derivations of \(A:=R/\langle h\rangle\), for \(h=x^n-y^q\in R\). Let \(s=n/p^\alpha\) where \(\alpha=val_p(n)\) is the \(p\)-adic valuation of \(n\).Let also \(m\) be the reminder of the division of \(q\) by \(p\), and \(\beta:=val_p(q-m)\). If \(B\) be a \(k\)-algebra and \(J\) be its ideal, then the following map is defined \[ \Pi: IDer_k(\log B;m)\rightarrow IDer_k(B/J;m), \] by \(\Pi(\delta)=\bar{\delta}\), where \(\bar{\delta}(x)=x+I\) for every \(\delta\in IDer_k(\log B;m)\) and \(x\in B\). Define \[ \gamma:=\min\{i|ip^\alpha\geq q-1\}=\lceil(q-1)/p^\alpha\rceil, \] where \(\lceil x\rceil\) is the least integer greater or equal to \(x\). Proposition [\textit{M. De La P. Tirado Hernández}, Rend. Semin. Mat. Univ. Padova 145, 53--71 (2021; Zbl 07367429)]. We have the following properties. 1. If \(n,q\not\equiv 0\) mod \(p\), then \(IDer_k(A)=Der_k(A)=\langle \overline{\delta_1},\overline{\delta_2}\rangle\) where \(\delta_1= qx\partial_x+ny\partial_y\) and \(\delta_2=qy^{q-1}\partial_x+nx^{n-1}\partial_y\). 2. If \(n\equiv 0\) mod \(p\) and \(q=1\), then \(IDer_k(A)=Der_k(A)=\langle \overline{\partial_x}\rangle\). 3. If \(\alpha,m\geq 1\) and \(q\geq 2\), then \qquad \qquad -if \(s=1, \alpha\leq \beta\), and \(m=1\), \[ \begin{cases} \langle \overline{\partial_x}\rangle & \text{if}\, 1\leq i<p^\alpha,\\ \langle \overline{x\partial_x},\overline{y^\gamma \partial_x} \rangle & \text{if} \, p^\alpha\leq i < p^{\alpha+\beta},\\ \langle \overline{x\partial_x},\overline{y^{\gamma+1}\partial_x} \rangle & if\, i\geq p^{\alpha+\beta}\, \text{or}\, i=\infty; \end{cases} \] \qquad \qquad -otherwise \[ \begin{cases} \langle \overline{\partial_x}\rangle & \text{if}\, 1\leq i<p^\alpha,\\ \langle \overline{x\partial_x},\overline{y^\gamma \partial_x} \rangle & \text{if} \, i\geq p^{\alpha}\, \text{or}\, i=\infty. \end{cases} \]