An Introduction to Galois Module Structure. the Case O

GALOIS AND HOPF GALOIS: TOPICS

ELENA CAMPEDEL, ILARIA DEL CORSO, FABIO FERRI, AND LORENZO STEFANELLO

Abstract. Here we list the possible topics for the study group in Galois module theory and Hopf–Galois theory.

1. Part 1: Galois module theory Lecture 1: An introduction to Galois module structure. The case of tame extensions An introduction to the concepts of normal integral bases, tame extensions, Noether’s theorem, lattices, Hilbert–Speiser theorem and beyond. References: [Joh11], [FT93] Speaker: Ilaria Del Corso Date: 25th March 2021 Lecture 2: More tame Galois module structure and an introduction to wild Galois module structure Higher ramification groups. Definition of locally free class groups and easy consequences (e.g. if it is trivial and there is cancellation then we have NIB). Fröhlich’s conjecture at least in its consequences concerning tame abelian, dihedral, metacyclic, alternating and symmet- ric extensions of Q, which have normal integral basis. Introduction to associated order, first properties, towers and extensions. Sketch of partial proof of Leopoldt’s theorem. References: [Joh11],[Tay81],[Leo59],[Ser79] Speaker: Alessandro Cobbe Date: 8th April 2021 Lecture 3: Results on Galois module structure of wild extensions and local freeness over the associated order Overview on results about freeness of the ring of integers over the associated order in both local and global (local or global freeness) extensions: absolutely abelian, dihedral, metacyclic, Q8, weakly ramified extensions. More insights on local freeness: if a Galois extension of number fields is wildly ramified then we cannot always deduce that the ring of integers is locally free over the associated order if the ring of integers of the completion is free over the associated order. References: [Let98],[Ber72],[Mar72], [Jau81],[Joh15],[Ber79] 1 2ELENA CAMPEDEL, ILARIA DEL CORSO, FABIO FERRI, AND LORENZO STEFANELLO

Speaker: Fabio Ferri Date: 15th April 2021 Lecture 4: Explicit Galois module structure of weakly ramified extensions of local fields Let L/K be a finite Galois extension of complete local fields with finite residue fields and let G = Gal(L/K). Let G1 and G2 be the first and second ramification groups. Thus L/K is tamely ramified when G1 is trivial and we say that L/K is weakly ramified when G2 is trivial. Let be the valuation ring of L and let P be its maximal OL L ideal. We show that if L/K is weakly ramified and n 1 mod G ≡ | 1| then Pn is free over the group ring [G], and we construct an L OK explicit generating element. Under the additional assumption that L/K is wildly ramified, we then show that every free generator of PL over [G] is also a free generator of over its associated order in OK OL the group algebra K[G]. References: [Joh15] Speaker: Henri Johnston Date: 22nd April 2021 Lecture 5: Almost-maximal ramification, dihedral extensions, A4, S4, A5-extensions A deep study of local freeness allows us to approach the results of Bergé on dihedral extensions and new results on A4, A5 and S4- extensions. This is also linked to the property of a Galois extension to be almost-maximally ramified, which also turns out to be useful in extensions with cyclic inertia group. References: [Ber79],[Ber78], upcoming article of the speaker Speaker: Fabio Ferri Date: 29th April 2021 Lecture 6: Galois module theory of absolutely abelian extensions of p-adic fields From a result of Lettl we know that in an absolutely abelian extension of p-adic fields the ring of integers is free over the associated order, and important consequences on the structure of the associated order (for instance in some cases the associated order is maximal). Moreover we can deduce that in an absolutely abelian extension of number fields the ring of integers is locally free over the associated order. References: [Let98] Speaker: Francesco Campagna Date: 6th May 2021 Lecture 7: Cyclic extensions of degree p Thanks to the works of Bertrandias and Ferton we have a complete understanding on when in a cyclic extension of degree p of p-adic fields the ring of integers is free over the associated order, which GALOIS AND HOPF GALOIS: TOPICS 3

involves ramification jumps, almost-maximal ramification and con- tinued fractions. The same problem may be approached by studying the minimal index of the ring of integers over a free submodule over the integral group ring. If there is time, more insights on this latter work. References: [BF72],[DCFL20] Speaker: Davide Lombardo Date: 13th May 2021 Lecture 8: On K-theory, realisable classes, and Hilbert–Speiser fields Deeper insights on relation between locally free class groups, K- theory, Mayer–Vietoris sequences and Swan modules. This leads to the result by Greither, Replogle, Rubin and Srivastav which tells us that Q is the only Hilbert–Speiser field, that is, there is probably no hope to have a theory similar to Fröhlich’s conjecture when the base field is not Q. An important step is McCulloh’s theory of realisable classes, which we will only mention. If there is time we will see a recent article on weak normal integral bases. References: [McC83],[GRRS99],[FG19] Speaker: Alessandro Cobbe Date: 20th May 2021 4ELENA CAMPEDEL, ILARIA DEL CORSO, FABIO FERRI, AND LORENZO STEFANELLO

2. Part 2: Hopf–Galois theory Lecture 9: An introduction to Hopf algebras. After a brief description of the motivations behind this theory, we will introduce the protagonists: Hopf algebras. In particular, we will see definition, examples, and main properties. We will discuss the dual of a Hopf algebra, the description of the grouplike elements, and the module and comodule algebras. References: [Chi00, §1-2], [CS69], [Und15] Speaker: Lorenzo Stefanello Date: 3rd June 2021 Lecture 10: Hopf–Galois extensions We will introduce Hopf Galois extensions of commutative rings. In order to do this, we will need to define module and comodule algebras, and generalise the classical definition of Galois extensions also for commutative rings. We will also discuss about ways to start from an Hopf Galois extension and get others, over different rings: base change and Galois descent. References: [Chi00, §2], [CS69] Speaker: Date: 10th June 2021 Lecture 11: Greither–Pareigis theorem and Byott’s transla- tion The problem of knowing how many different Hopf–Galois structures can a given Hopf Galois extension have, and how do those structures look like, can be shifted to a group theory matter, thanks to two fundamental results due to Greither and Pareigis, and Byott. We will present these two theorems together with some applications. References: [Chi00, §6-7], [GP87], [Byo96b], [Byo96a], more to be added Speaker: Elena Campedel Date: 17th June 2021 Lecture 12: On the algebra structure of Hopf algebras occur- ring in Hopf–Galois theory In Hopf–Galois theory one frequently encounters Hopf algebras H which are “forms” of a group ring. This means that if H is base- changed from the base field K to a suitable extension field L, it becomes a group ring. If is very well understood under which con- ditions such forms are “trivial” in the Hopf sense, that is, already isomorphic to a group ring over K. Roughly speaking, this happens very rarely. It came as a little surprise that a nontrivial form (in the Hopf sense) can very well be isomorphic to a group ring, as a K-algebra. This talk tries to explain why this often happens, and that it will not happen all the time. Before coming to this, we will review the theory (and practice) of Galois descent, a technique which GALOIS AND HOPF GALOIS: TOPICS 5

is highly relevant in many places of Hopf–Galois theory. References: [Gre21], [KO74]. Speaker: Cornelius Greither Date: 25th June 2021 Lecture 13: Induced Hopf–Galois structures We consider Hopf–Galois structures of a Galois extension L/K whose Galois group G is a semidirect product. If G = J ⋊ G′ with J a normal subgroup of G and E = LG′ , F = LJ , induced Hopf– Galois structures are built from a Hopf–Galois structure on E/K and a Hopf–Galois structure on L/E. They can be described either in terms of the corresponding Hopf algebra and Hopf action or in terms of the attached regular subgroup under the Greither–Pareigis theorem. The type of an induced Hopf Galois structure is the direct product of the types of the corresponding Hopf Galois structures of L/E and E/K. We will see that this gives a generalisation of some results for classical Galois extension whose Galois group is a direct product of subgroups. In particular, assuming that inducing subextensions are arithmetically disjoint, we have that the associated order decom- poses as tensor product as the associated orders of the corresponding factors. References: [Byo13], [CRV16a], [CRV16b], [GMR20] Speaker: Anna Rio Date: 1st July 2021 Lecture 14: Hopf–Galois module theory We apply the perspective provided by Hopf–Galois theory to some classical notions of Galois module theory. The notions of integrals and Hopf orders are introduced, as well as the notion of associated order in an arbitrary Hopf–Galois structure of a finite extension of fields. For a commutative ring R, we shall study a generalisation of the tame condition for an R-algebra S which is an H-module algebra for some finite cocommutative R-Hopf algebra H. Finally, we prove different results concerning logical relations between different properties of S involving H, for instance H-free, H-tame and H- Galois. References: [Chi00, §5, Chapter 3] Speaker: Daniel Gil-Muñoz Date: 8th July 2021 Lecture 15: Hopf orders as associated orders We will look at when the associated order is a Hopf order, in the con- text of totally ramified Galois p-extensions L/K of p-adic fields. We will see that this strongly restricts the ramification breaks of L/K. We will then see how formal groups can be used to construct Hopf orders, and use this to obtain an extension which behaves differently in a non-classical Hopf–Galois structure than in the classical one. 6ELENA CAMPEDEL, ILARIA DEL CORSO, FABIO FERRI, AND LORENZO STEFANELLO

References: [Chi00, Chapters 6 and 11], [Byo97] Speaker: Nigel Byott Date: 15th July 2021 Lecture 16: An assortment of associated orders in Hopf–Galois extensions We will survey a range of topics in Hopf–Galois module theory. In particular, we will look at tools for studying integral Hopf–Galois module structure when the associated order is not a Hopf order, Hopf–Galois module structure for “opposite” Hopf–Galois structures, the case of tamely ramiﬁed extensions, and extensions of number ﬁelds. References: [Byo02], [Tru11], [Tru12], [Tru13], [Tru18a], [Tru18b], [Tru20] Speaker: Paul Truman Date: 22nd July 2021

References [Ber72] Anne-Marie Bergé, Sur l’arithmétique d’une extension diédrale, Ann. Inst. Fourier (Grenoble) 22 (1972), no. 2, 31–59. MR 371857 [Ber78] , Arithmétique d’une extension galoisienne à groupe d’inertie cyclique, Ann. Inst. Fourier (Grenoble) 28 (1978), no. 4, 17–44, ix. [Ber79] , Projectivite des anneaux d’entiers sur leurs ordres associes., Astérisque 61 (1979), 15–28 (French). [BF72] Françoise Bertrandias and Marie-Josée Ferton, Sur l’anneau des entiers d’une extension cyclique de degré premier d’un corps local, C. R. Acad. Sci. Paris Sér. A-B 274 (1972), A1330–A1333. MR 296047 [Byo96a] Nigel P. Byott, Corrigendum: “Uniqueness of Hopf Galois structure for separable field extensions”, Comm. Algebra 24 (1996), no. 11, 3705. MR 1405283 [Byo96b] , Uniqueness of Hopf Galois structure for separable field extensions, Comm. Algebra 24 (1996), no. 10, 3217–3228. MR 1402555 [Byo97] , Galois structure of ideals in wildly ramified abelian p-extensions of a p-adic field, and some applications, J. Théor. Nombres Bordeaux 9 (1997), no. 1, 201–219. MR 1469668 [Byo02] , Integral Hopf-Galois structures on degree p2 extensions of p-adic fields, J. Algebra 248 (2002), no. 1, 334–365. MR 1879021 [Byo13] , Nilpotent and abelian Hopf-Galois structures on field extensions, J. Algebra 381 (2013), 131–139. MR 3030514 [Chi00] Lindsay N. Childs, Taming wild extensions: Hopf algebras and local Galois module theory, Mathematical Surveys and Monographs, vol. 80, American Mathematical Society, Providence, RI, 2000. MR 1767499 [CRV16a] Teresa Crespo, Anna Rio, and Montserrat Vela, Induced Hopf Galois structures, J. Algebra 457 (2016), 312–322. MR 3490084 [CRV16b] , On the Galois correspondence theorem in separable Hopf Galois theory, Publ. Mat. 60 (2016), no. 1, 221–234. MR 3447739 [CS69] Stephen U. Chase and Moss E. Sweedler, Hopf algebras and Galois theory, Lec- ture Notes in Mathematics, Vol. 97, Springer-Verlag, Berlin-New York, 1969. MR 0260724 [DCFL20] Ilaria Del Corso, Fabio Ferri, and Davide Lombardo, How far is an extension of p-adic fields from having a normal integral basis?, arXiv preprint arXiv:2005.07932 (2020). GALOIS AND HOPF GALOIS: TOPICS 7

[FG19] Fabio Ferri and Cornelius Greither, Tame Galois module structure revisited, Ann. Mat. Pura Appl. (4) 198 (2019), no. 6, 2029–2042. MR 4031837 [FT93] A. Fröhlich and M. J. Taylor, Algebraic number theory, Cambridge Studies in Advanced Mathematics, vol. 27, Cambridge University Press, Cambridge, 1993. MR 1215934 [GMR20] Daniel Gil-Muñoz and Anna Rio, On induced hopf galois structures and their local hopf galois modules, arXiv preprint arXiv:1910.06083 (2020). [GP87] Cornelius Greither and Bodo Pareigis, Hopf Galois theory for separable field extensions, J. Algebra 106 (1987), no. 1, 239–258. MR 878476 [Gre21] Cornelius Greither, Descent theory and the structure of hopf algebras acting on separable field extensions, chapter 9 of: Hopf algebras and Galois Module theory (Lindsay N. Childs, Cornelius Greither, Kevin Keating, Alan Koch, Timothy Kohl, Paul J. Truman, and Robert Underwood, eds.), to appear as an AMS monograph, 2021. [GRRS99] Cornelius Greither, Daniel R. Replogle, Karl Rubin, and Anupam Srivastav, Swan modules and Hilbert-Speiser number fields, J. Number Theory 79 (1999), no. 1, 164–173. MR 1718724 [Jau81] Jean-François Jaulent, Sur la l-structure galoisienne des idéaux ambiges dans une extension métacyclique de degré nl sur le corps des rationnels, Number theory, 1979–1980 and 1980–1981, Publ. Math. Fac. Sci. Besançon, Univ. Franche- Comté, Besançon, 1981, pp. Exp. No. 3, 20. MR 748000 [Joh11] Henri Johnston, Notes on galois modules, Notes accompanying the course ‘Ga- lois Modules’ given in Cambridge (2011). [Joh15] , Explicit integral Galois module structure of weakly ramified extensions of local fields, Proc. Amer. Math. Soc. 143 (2015), no. 12, 5059–5071. MR 3411126 [KO74] Max-Albert Knus and Manuel Ojanguren, Théorie de la descente et algèbres d’Azumaya, Lecture Notes in Mathematics, Vol. 389, Springer-Verlag, Berlin- New York, 1974. MR 0417149 [Leo59] Heinrich-Wolfgang Leopoldt, Über die Hauptordnung der ganzen Elemente eines abelschen Zahlkörpers, J. Reine Angew. Math. 201 (1959), 119–149. MR 108479 [Let98] Günter Lettl, Relative Galois module structure of integers of local abelian fields, Acta Arith. 85 (1998), no. 3, 235–248. MR 1627831 [Mar72] Jacques Martinet, Sur les extensions à groupe de Galois quaternionien, C. R. Acad. Sci. Paris Sér. A-B 274 (1972), A933–A935. MR 299593 [McC83] Leon R. McCulloh, Galois module structure of elementary abelian extensions, J. Algebra 82 (1983), no. 1, 102–134. MR 701039 [Ser79] Jean-Pierre Serre, Local fields, Graduate Texts in Mathematics, vol. 67, Springer-Verlag, New York-Berlin, 1979, Translated from the French by Marvin Jay Greenberg. MR 554237 [Tay81] M. J. Taylor, On Fröhlich’s conjecture for rings of integers of tame extensions, Invent. Math. 63 (1981), no. 1, 41–79. MR 608528 [Tru11] Paul J. Truman, Towards a generalisation of Noether’s theorem to nonclassical Hopf-Galois structures, New York J. Math. 17 (2011), 799–810. MR 2862153 [Tru12] , Hopf-Galois module structure of tame biquadratic extensions, J. Théor. Nombres Bordeaux 24 (2012), no. 1, 173–199. MR 2914905 [Tru13] , Integral Hopf-Galois structures for tame extensions, New York J. Math. 19 (2013), 647–655. MR 3119101 [Tru18a] , Commutative Hopf-Galois module structure of tame extensions, J. Al- gebra 503 (2018), 389–408. MR 3780002 [Tru18b] , Commuting Hopf-Galois structures on a separable extension, Comm. Algebra 46 (2018), no. 4, 1420–1427. MR 3780516 8ELENA CAMPEDEL, ILARIA DEL CORSO, FABIO FERRI, AND LORENZO STEFANELLO

[Tru20] , Hopf-Galois module structure of tamely ramiﬁed radical extensions of prime degree, J. Pure Appl. Algebra 224 (2020), no. 5, 106231, 13. MR 4046237 [Und15] Robert G. Underwood, Fundamentals of Hopf algebras, Universitext, Springer, Cham, 2015. MR 3379140