×

A theory of Galois descent for finite inseparable extensions. (English) Zbl 1377.12002

Let \(L/k\) be a field extensions. If \(V\) is a \(K\)-vector space and \(W\) is a sub-vector space of \(V_L \simeq V\otimes_K L\), it is natural to ask whether \(W\) is defined over \(K\), namely whether there exists \(W_0\subseteq V\) a \(K\)-sub-vector space such that \(W \simeq W_0 \otimes_K L\). If \(L/k\) is finite Galois extension with Galois group \(G\), according to U. Görtz and T. Wedhorn [Algebraic geometry I. Schemes. With examples and exercises. Wiesbaden: Vieweg+Teubner (2010; Zbl 1213.14001), Theorem 14.83], the functor (\(k\)-vector spaces) \(\rightarrow\) (\(L\)-vector spaces with \(G\)-action over \(L\)), \(V \mapsto V \otimes_k L\), is an equivalence of categories with quasi-inverse \(V' \mapsto (V')^G\), the \(k\)-vector space of invariants of \(V'\) under \(G\), and therefore an algebraic objects over \(L\) with a suitable \(G\)-action are exactly those defined over \(K\). The goal of this article is to extend this result to finite modular normal (and a fortiori normal) extension \(L/k\) of exponent \(e\). It should be noted that the Heerema-Galois group \(HG(L/K) = \operatorname{Aut}(L[\bar{X}]/K[\bar{X}])\) where \(L[\bar{X}] = L[X]/(X^{p^e})\) (respectively, \(k[\bar{X}] = k[X]/(X^{p^e})\)) and [H. P. Allen and M. E. Sweedler, J. Algebra 12, 242–294 (1969; Zbl 0257.16024)] have proved very useful to the author to obtain descent conditions on \(L\)-vector spaces and consequently on algebras and more in general separated schemes of finite type, notably an algebraic object over \(L\) is defined over \(K\) if and only if its base change to \(L[X]\) admits a suitable \(HG(L/K)\)-action. The author then gives small generalization of these results in the fourth section. Finally, in order to fully extend the classical Galois descent theory, the author proposes the following open questions
Question 1. If \(L/K\) is a modular normal field extension of infinite exponent, is it possible to define the Heerema-Galois group of such extension?
Question 2. Which topology should then be given on \(HG(L/K)\) in order to extend the Krull topology and to get descent for objects endowed with a continuous action of \(HG(L/K)\)?

MSC:

12F15 Inseparable field extensions
14G17 Positive characteristic ground fields in algebraic geometry
14A15 Schemes and morphisms
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Allen, Harry Prince; Sweedler, Moss Eisenberg, A theory of linear descent based upon Hopf algebraic techniques, J. Algebra, 12, 242-294 (1969) · Zbl 0257.16024 · doi:10.1016/0021-8693(69)90051-9
[2] B\'egueri, Lucile, Sch\'ema d’automorphisms. Application \`“a l”\'etude d’extensions finies radicielles, Bull. Sci. Math. (2), 93, 89-111 (1969) · Zbl 0186.35502
[3] Chase, Stephen U., On the automorphism scheme of a purely inseparable field extension. Ring theory, Proc. Conf., Park City, Utah, 1971, 75-106 (1972), Academic Press, New York · Zbl 0262.12101
[4] Davis, R. L., A Galois theory for a class of inseparable field extensions, Trans. Amer. Math. Soc., 213, 195-203 (1975) · Zbl 0317.12106 · doi:10.2307/1998043
[5] Deveney, James K.; Mordeson, John N., Higher derivation Galois theory of inseparable field extensions. Handbook of algebra, Vol.1, Handb. Algebr. 1, 187-220 (1996), Elsevier/North-Holland, Amsterdam · Zbl 0868.12004 · doi:10.1016/S1570-7954(96)80010-6
[6] Grothendieck, Alexander, Technique de descente et th\'eor\`“emes d”existence en g\'eometrie alg\'ebrique. I. G\'en\'eralit\'es. Descente par morphismes fid\`“element plats. S\'”eminaire Bourbaki, Vol.5, 299-327 (1995, Exp.No.190), Soc. Math. France, Paris
[7] G\"ortz, Ulrich; Wedhorn, Torsten, Algebraic geometry I, Advanced Lectures in Mathematics, viii+615 pp. (2010), Vieweg + Teubner, Wiesbaden · Zbl 1213.14001 · doi:10.1007/978-3-8348-9722-0
[8] Heerema, Nickolas, A Galois theory for inseparable field extensions, Trans. Amer. Math. Soc., 154, 193-200 (1971) · Zbl 0211.37004 · doi:10.2307/1995437
[9] Mil James S. Milne, Algebraic geometry, http://www.jmilne.org/math/CourseNotes/ag.html.
[10] Mordeson, John N., Splitting of field extensions, Arch. Math. (Basel), 26, 6, 606-610 (1975) · Zbl 0319.12103 · doi:10.1007/BF01229788
[11] Pillay, Anand, Remarks on Galois cohomology and definability, J. Symbolic Logic, 62, 2, 487-492 (1997) · Zbl 0947.03054 · doi:10.2307/2275542
[12] Poizat, Bruno, Une th\'eorie de Galois imaginaire, J. Symbolic Logic, 48, 4, 1151-1170 (1984) (1983) · Zbl 0537.03023 · doi:10.2307/2273680
[13] Sancho de Salas, Pedro J., Automorphism scheme of a finite field extension, Trans. Amer. Math. Soc., 352, 2, 595-608 (2000) · Zbl 0933.14028 · doi:10.1090/S0002-9947-99-02361-2
[14] Sweedler, Moss Eisenberg, Structure of inseparable extensions, Ann. of Math. (2), 87, 401-410 (1968) · Zbl 0168.29203 · doi:10.2307/1970711
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.