Abstract
We study the action of substitution maps between power series rings as an additional algebraic structure on the groups of Hasse–Schmidt derivations. This structure appears as a counterpart of the module structure on classical derivations.
Dedicated to Antonio Campillo on the ocassion of his 65th birthday
Partially supported by MTM2016-75027-P, P12-FQM-2696 and FEDER.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
Pay attention that (φ + φ′)(r) ≠ φ(r) + φ′(r) for arbitrary r ∈ A[[s]]∇.
- 2.
Let us notice that there are canonical continuous isomorphisms of A-algebras \(A[[{\mathbf {s}}\sqcup {\mathbf {u}}]]_{\nabla \times \nabla '} \simeq A[[{\mathbf {s}}]]_\nabla \widehat {\otimes }_A A[[{\mathbf {u}}]]_{\nabla '}\), \(A[[{\mathbf {s}}\sqcup {\mathbf {u}}]]_{\varDelta \times \varDelta '} \simeq A[[{\mathbf {s}}]]_\varDelta \widehat {\otimes }_A A[[{\mathbf {u}}]]_{\varDelta '}\).
- 3.
This terminology is used for instance in [8].
- 4.
These HS-derivations are called of length m in [10].
- 5.
Actually, here an equality holds since the 0-term of E (as a series) is 1.
- 6.
Let us notice that \(\{E\in \operatorname {\mathrm {HS}}^{\mathbf {s}}_k(A;\varDelta )\ |\ \ell (E) > r\} = \ker \tau _{\varDelta ,\varDelta _r}\).
- 7.
The map π can be also understood as the truncation τ Δ,{0} : A[[t]]Δ → A[[t]]{0} = A.
References
Brown, W.C.: On the embedding of derivations of finite rank into derivations of infinite rank. Osaka J. Math. 15, 381–389 (1978).
Bourbaki, N.: Elements of Mathematics. Algebra II, chaps. 4–7. Springer, Berlin (2003)
Fernández-Lebrón, M., Narváez-Macarro, L.: Hasse-Schmidt derivations and coeflcient fields in positive characteristics. J. Algebra 265(1), 200–210 (2003). arXiv:math/0206261
Hasse, H., Schmidt, F.K.: https://doi.org/Noch eine Begründung der Theorie der höheren Differrentialquotienten in einem algebraischen Funktionenkörper einer Unbestimmten. J. Reine U. Angew. Math. 177, 223–239 (1937)
Hoffmann, D., Kowalski, P.: Integrating Hasse–Schmidt derivations. J. Pure Appl. Algebra 219, 875–896 (2015)
Hoffmann, D., Kowalski, P.: https://doi.org/Existentially closed fields with G-derivations. J. Lond. Math. Soc. (2) 93(3), 590–618 (2016)
Matsumura, H.: Integrable derivations. Nagoya Math. J. 87, 227–245 (1982)
Matsumura, H.: Commutative Ring Theory. Cambridge Studies in Advanced Mathematics, vol. 8. Cambridge University Press, Cambridge (1986)
Narváez Macarro, L.: https://doi.org/Hasse–Schmidt derivations, divided powers and differential smoothness. Ann. Inst. Fourier (Grenoble) 59(7), 2979–3014 (2009). arXiv:0903.0246
Narváez Macarro, L.: On the modules of m-integrable derivations in non-zero characteristic. Adv. Math. 229(5), 2712–2740 (2012). arXiv:1106.1391
Narváez Macarro, L.: Differential Structures in Commutative Algebra. Mini-course at the XXIII Brazilian Algebra Meeting, 27 July–1 August 2014, Maringá, Brazil
Narváez Macarro, L.: Rings of differential operators as enveloping algebras of Hasse–Schmidt derivations. arXiv:1807.10193
Vojta, P.: Jets via Hasse–Schmidt derivations. In: Diophantine geometry. CRM Series, vol. 4, pp. 335–361. Edizioni della Normale, Pisa (2007). arXiv:math/1201.3594
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2018 Springer Nature Switzerland AG
About this chapter
Cite this chapter
Narváez Macarro, L. (2018). On Hasse–Schmidt Derivations: The Action of Substitution Maps. In: Greuel, GM., Narváez Macarro, L., Xambó-Descamps, S. (eds) Singularities, Algebraic Geometry, Commutative Algebra, and Related Topics. Springer, Cham. https://doi.org/10.1007/978-3-319-96827-8_10
Download citation
DOI: https://doi.org/10.1007/978-3-319-96827-8_10
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-96826-1
Online ISBN: 978-3-319-96827-8
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)