Higher derivations of modules and the Hasse-Schmidt module (Q6164750)

Higher derivations of modules were introduced in [\textit{P. Ribenboim}, Port. Math. 39, 381--397 (1985; Zbl 0532.16024)] in analogy to higher derivations of rings to provide a similar notion of a map carrying ``infinitesimal information''. It was shown that there exists a universal object parametrizing such higher derivations of modules, which is called the Hasse-Schmidt module. This construction was used in [\textit{T. de Fernex} and \textit{R. Docampo}, Duke Math. J. 169, No. 2, 353--396 (2020; Zbl 1465.14019)] where the following statement was proved: Let \(A\) be a \(k\)-algebra (\(k\) is a commutative ring) and for any \(n\in\mathbb{N}\cup\{\infty\}\), let \(\mathbb{H}\mathbb{S}^{n}_{k}(A)\) denote the \(n\)th Hasse-Schmidt algebra of \(A\). Then there exists an \(A\)-module \(Q_{n}\) such that \(\Omega_{\mathbb{H\mathbb{S}}^{n}_{k}(A)/k} = \Omega_{A/k}\bigotimes_{A}Q_{n}\). Since \(\mathbb{H}\mathbb{S}^{n}_{k}(A)\) parametrizes infinitesimal data on \(A\) up to order \(n\) (i.e. \(n\)-jets on \(A\)), the last formula suggests that tangents (i.e. infinitesimal data up to order 1) of \(n\)-jets on \(A\) can be recovered from some higher order operation on tangents on \(A\). The paper under review makes this idea precise. The authors extend known properties of higher derivations to graded higher derivations and prove that if \(A = \bigoplus_{i\in\mathbb{N}}A_{n}\) is a graded \(k\)-algebra, then for all \(m, n\in\mathbb{N}\cup\{\infty\}\), there exist natural isomorphisms of graded \(A_{n}\)-algebras \(\mathbb{H}\mathbb{S}^{n}_{k}(\mathbb{H}\mathbb{S}^{m}_{k}(A))\simeq \mathbb{H}\mathbb{S}^{m}_{k}(\mathbb{H}\mathbb{S}^{n}_{k}(A))\) (the left-hand side is considered with its structural grading and the right-hand side is considered with its induced grading). Furthermore, as the final ingredient, the authors prove the following theorem, which connects the Hasse-Schmidt algebra to the Hasse-Schmidt module by means of the symmetric algebra: Let \(A\) be a \(k\)-algebra and \(M\) an \(A\)-module. Then for each \(n\in\mathbb{N}\cup\{\infty\}\), there are isomorphisms of \(\mathbb{N}\)-graded \(A_{n}\)-algebras Sym\(_{A_{n}}(\mathbb{H}\mathbb{S}^{n}_{A/k}(M))\simeq \mathbb{H}\mathbb{S}^{n}_{k}(\mathrm{Sym}_{A}(M))\) which are natural in \(M\). (Here \(\mathbb{H}\mathbb{S}^{n}_{A/k}(M)\) is the \(n\)th Hasse-Schmidt module of \(M\). The left-hand side is considered with the natural grading on the symmetric algebra, whereas the right-hand side is considered with the induced grading of \(\mathbb{H}\mathbb{S}^{n}_{k}(\mathrm{Sym}_{A}(M))\).) The last part of the paper discusses globalizing of the authors' construction to obtain sheaves on jet and arc spaces.