On the modules of \(m\)-integrable derivations in non-zero characteristic
Let \(k\) be a commutative ring, let \(A\) be a commutative \(k\)-algebra, and let \(\delta:A\to A\) be a \(k\)-linear derivation of \(A\). It can be identified with the \(k\)-algebra homomorphism \(\bar\delta: A\to A[\varepsilon]\), \(a\mapsto a+\delta(a)\cdot \varepsilon\), where \(A[\varepsilon]=At/(t^2)\) and \(\varepsilon = t+(t^2)\) is the infinitesimal parameter.NEWLINENEWLINELet \(m\) be a positive integer. The derivation \(\delta\) is called \(m\)-integrable if there is a \(k\)-algebra homomorphism \(\Phi: A\to At/(t^{m+1})\) that lifts the homomorphism \(\bar \delta\). Let \(\bar t\) denote the class of \(t\) in \(At/(t^{m+1})\), then NEWLINE\[NEWLINE \Phi(a)=a+\bar \delta\cdot \bar t+ D_1(a)\cdot \bar {t}^2+\dots+D_{m}\cdot \bar{t}^m, NEWLINE\]NEWLINE where \(D_i: A\to A\) are \(k\)-linear maps satisfying \(D_0=\mathrm{id}_A\), \(D_1=\delta\), and the Leibniz equality NEWLINE\[NEWLINED_{i}(ab)=\sum\limits_{r+s=i}D_r(a)D_s(b).NEWLINE\]NEWLINE Such a sequence \(D_0,\dots D_m\) of \(k\)-linear maps is called a Hasse-Schmidt derivation of length \(m\). The set \(\mathrm{Ider}_k(A; m)\) of all \(m\)-integrable \(k\)-derivations of \(A\) is an \(A\)-submodule of \(\mathrm{Der}_k(A)\).NEWLINENEWLINEThe derivation \(\delta\) is called (\(\infty\)-)integrable if \(\bar \delta\) can be lifted to a \(k\)-algebra homomorphism \(A\to At\), \(a\mapsto a+\delta(a)t+D_2(a)t^2+\dots\), and the sequence of \(k\)-linear operators NEWLINE\[NEWLINE D_0=\mathrm{id}_A,\quad D_1=\delta,\quad D_2, \dots NEWLINE\]NEWLINE satisfies the Leibniz equality and is called a Hasse-Schmidt derivation of infinite length. The set \(\mathrm{Ider}_k(A)=\mathrm{Ider}_k(A; \infty)\) of all integrable \(k\)-derivations of \(A\) is an \(A\)-submodule of \(\mathrm{Der}_k(A)\).NEWLINENEWLINEThere is a sequence of \(A\)-submodules NEWLINE\[NEWLINE \mathrm{Der}_k(A)=\mathrm{Ider}_k(A; 1)\supset \mathrm{Ider}_k(A; 2)\supset \dots \supset \mathrm{Ider}_k(A; \infty)=\mathrm{Ider}_k(A). NEWLINE\]NEWLINENEWLINENEWLINEThere are two main results of the paper.NEWLINENEWLINE(I) Let \(A\) be a finitely presented \(k\)-algebra and let \(m\) be a positive integer. Then the property of being \(m\)-integrable is a local property, i.~e., a \(k\)-derivation \(\delta: A\to A\) is \(m\)-integrable if and only if the induced \(k\)-derivations \(A_{\mathfrak p}\to A_{\mathfrak p}\) are \(m\)-integrable for all prime ideals \(\mathfrak p\subset A\).NEWLINENEWLINEOne can reformulate this as follows. Let \(f: X\to S\) be a locally finitely presented morphism of schemes and let \(m\) be a positive integer. Consider the \(\mathcal O_X\)-module \({Der}_S(\mathcal O_X)\) of \(\mathcal O_S\)-derivations of the structure sheaf \(\mathcal O_X\). Then there exists a quasi-coherent sub-module \(\mathrm{Ider}_S(\mathcal O_X; m)\subset {Der}_S(\mathcal O_X)\) such that for every affine open sets \(U=\mathrm{Spec} (A)\subset X\) and \(V=\mathrm{Spec}(k)\subset S\) with \(f(U)\subset V\) it holds \(\Gamma(U, \mathrm{Ider}_S(\mathcal O_X; m))=\mathrm{Ider}_k(A; m)\) and \(\mathrm{Ider}_S(\mathcal O_X; m)_x=\mathrm{Ider}_{\mathcal O_{S, f(x)}}(\mathcal O_{X, x}; m)\) for all \(x\in X\).NEWLINENEWLINE(II) For a given \(k\)-algebra \(A\) and a positive integer \(m\), there is an algorithm deciding whether \(\mathrm{Ider}_k(A; m)=\mathrm{Der}_k(A)\). The complexity of the algorithm is exponential with respect to \(m\).NEWLINENEWLINEThe paper under review consists of an introduction and 3 sections. In introduction an overview of the content of the paper is presented, main results of the paper are stated here as well. In Section~1, the notations used in the paper and basic facts about Hasse-Schmidt derivations are given. Section~2 deals with integrability questions, the main results of the paper are proven here. Section~3 provides some examples and open questions.
scientific article; zbMATH DE number 6023751
scientific article