The Albert-Brauer-Hasse-Noether Theorem and Global Dimension of Crossed Products

Aristotle University of Thessaloniki Faculty of Natural Sciences School of Mathematics

Palaisti, Marina

A thesis presented to The Department of Mathematics in partial fulﬁllment of the requirements for the degree of Master in Science in the subject of Pure Mathematics.

Thessaloniki, 2014 ii Aristotle University of Thessaloniki Faculty of Natural Sciences School of Mathematics

The Albert-Brauer-Hasse-Noether Theorem and Global Dimension of Crossed Products

Supervisor: Author: Prof. Palaisti, Marina Theohari-Apostolidi, Theodora

Committee Members: Prof. Theohari-Apostolidi, Theodora Prof. Haralambous, Hara Prof. Tzouvaras, Athanasios

Thessaloniki, 2014

Παλαιστή Μαρίνα, Πτυχιούχος Μαθηµατικός Α.Π.Θ.

Απαγορεύεται η αντιγραφή, αποθήκευση και διανοµή της παρούσης ερ- γασίας, εξ΄ ολοκλήρου ή τµήµατος αυτής, για εµπορικό σκοπό. Επιτρέπεται η ανατύπωση, αποθήκευση και διανοµή για σκοπό µη κερδοσκοπικό, εκ- παιδευτικής ή ερευνητικής ϕύσης, υπό την προϋπόθεση να αναφέρεται η πηγή προέλευσης και να διατηρείται το παρόν µήνυµα. Ερωτήµατα που αφορούν τη χρήση της εργασίας για κερδοσκοπικό σκοπό πρέπει να απευθύνονται στη συγγραφέα.

Οι απόψεις και τα συµπεράσµατα που περιέχονται σε αυτό το έγγραφο εκφράζουν τη συγγραφέα και δεν πρέπει να ερµηνευθεί ότι εκφράζουν το Α.Π.Θ.

iii iv "... he wrote to me that algebraic number theory was the most beautiful topic he had ever come across and that the sole consolation in misery was his lecturing on class ﬁeld theory... This was indeed the kind of mathematics he had admired most: the main results are of great scope, of great aesthetic beauty, but the proofs are technically extremely hard."

A.Borel about Harish-Chandra, 1995. vi Introduction

This thesis is concerned with the Albert-Brauer-Hasse-Noether Theorem, and also with global dimensions of crossed products, which are very important not only in the theory of structure of algebras, but also in algebraic number theory. The Albert-Brauer-Hasse-Noether Theorem was established in Novem- ber 9, 1931, by Abraham Adrian Albert(1905-1972), Richard Brauer(1901- 1977), Helmut Hasse(1898-1979) and Amalie Emmy Noether(1882-1935). It was published in 1932, and dedicated to Kurt Hensel(1861-1941), on the occasion of his 70th birthday. From the survey article of P. Roquette [Roq] we get the following historical elements of the collaboration of Albert, Brauer, Hasse and Noether, to prove their celebrating theorem. Brauer became interested in class field theory, because he believed that its results would be important when applied to representation theory of groups. Hasse became interested in the theory of algebras because he had noticed that class field theory and the local p-adic theory could be used there profitably. Finally, Noether, who brought Brauer and Hasse toghether, was motivated by her belief that the non-commutative theory of algebras should be used for a better understanding of class field theory. At that time the results of Dickson(1874-1954) and his disciples were noted with lots of interest by the mathematicians around Noether. There- fore, during the collaboration between Brauer, Hasse and Noether, Hasse wrote a letter to Dickson, describing his work in it, and asking about the existence of non-cyclic division algebras of index 4 over a number field. Dickson forwarded it to a disciple of him, Albert. The last replied to Hasse that he was very interested in it, and introduced himself, so they started communicating and exchanging ideas about the steps of the proof that were to handle. Albert had already in 1931 developed different tools from the other three, in order to approach a part of this theorem. Thus, in

vii viii INTRODUCTION their paper, Brauer, Hasse and Noether inserted a footnote, aknowledging Albert and giving him "an independent share of the proof". Later, in 1932, in a joint paper between Albert and Hasse, the theorem had been proved again, using Albert’s methods this time. The Albert-Brauer-Hasse-Noether Theorem is one of the most profound results in the theory of central simple algebras. The paper starts with the following sentence. "At last our joint endeavours have finally been successful, to prove the following theorem, which is of fundamental importance for the structure theory of algebras, and also beyond..." The Albert-Brauer-Hasse-Noether Theorem is a very critical step in the view of algebraic number theory. It allows a complete classification of division algebras over a number field, by means of Hasse invariants. Therefore, one can determine the Brauer group of an algebraic number field, and also can describe the splitting fields of a division algebra, by describing their local behavior, which is extremely useful in representation theory. Fur- thermore, the Albert-Brauer-Hasse-Noether Theorem opened new paths, giving the ability to understand class field theory via the structure of algebras. In this thesis we discuss several topics of the theory of structure and properties of central simple algebras, from an algebraic number-theoretic point of view. We examine the Albert-Brauer-Hasse-Noether Theorem, and the required theories over algebraic number fields, and not exactly on global fields. In chapter 1 we introduce the central simple algebras over an algebraic number field K and their fundamental properties, and also we introduce the Brauer group of a field. Central simple K-algebras are really useful algebraic structures, not only because of their simplicity as algebras, but also due to the fact that their center is precisely the field K. An equivalence relation is defined on central simple algebras, and their classes form a group, the Brauer group. The elements of the last gives a classification of all division algebras over the field K. In chapter 2 we are concerned with the extremely important concept of crossed products algebras and the formalistic theory of group cohomology. Crossed products are a specific kind of central simple algebras, having especially beautiful properties, such as their correspondence to cocycles and their ability to preserve their natural properties. Furthermore, we discuss the basic stuff about cyclic algebras and we prove that a division ix algebra of degree 3 over an algebraic number field is cyclic. In chapter 3 we provide the needed machinery of the theory of valuations on algebraic number fields and the P-adic completions of them, in order to present and prove the main theorem of this thesis, the Albert- Brauer-Hasse-Noether Theorem. One of the main applications of the The- orem is that every central simple algebra is cyclic. Finally, in chapter 4 we discuss the generalization of our previous concepts. We give a little information about the cohomology modules, from an arbitrary algebra A to an (A, A)-bimodule M, and we compute the cohomology modules H 0,H 1 and H 2. Moreover, we introduce the construction of the general crossed products, and we view them from a homological point of view, giving relations between global and cohomological dimension for several crossed products. Finally, we concetrate to the case of zero global dimension, in which we prove that the group in this case is a torsion group, although it is not necessarily finite and we construct a crossed product which is a division ring. We refer to [Re] and [Pi] for the theory of central simple algebras and the cohomology theory, and we follow mainly [Re] for the proofs of the presented theorems. We refer to [AR] and [Yi] for the context of chapter 4. For many years, all division algebras were constructed as crossed products, starting in 1843 with Hamilton’s(1805-1865) real quaternions. Even- tually, Albert, Dickson and Wedderburn(1882-1948) proved that all division algebras of degree 2, 3, 4, 6 or 12 are cyclic, and hence crossed products, and for much of the 20th century, it was conjectured that all division algebras are crossed products. But, in 1972, Amitsur(1921-1994) pro- duced a counterexample. He constructed non-crossed product division algebras of any degree n divisible by 23 or p2, where p is a prime number, provided p is prime to the residue characteristic. This construction led to a number of questions about the properties of division algebras. One of the most important open problems regarding central simple K-algebras, and the motivation of the study of this thesis project, is to construct a non-cyclic division algebra of degree a prime p over a field K [ABGV, Problem 1.1]. It is trivial that a division algebra of degree 2 over K is cyclic. Also, a division algebra of degree 3 over K is cyclic, as shown by Wedderburn in 1921. For p > 3 this problem is completely open. If we specialize this problem to the case where K is a number field, then the answer is given by the Albert-Brauer-Hasse-Noether Theorem. This asserts that every division algebra of degree p over a number field is cyclic. x INTRODUCTION

From the Albert-Brauer-Hasse-Noether Theorem and some theorems of Frobenius(1849-1917), Hasse, Tsen(1898-1940) Wedderurn and Witt(1911-

1991), today we know that all division algebras over the fields K, KP ,KP ((t)), where K is any number field, Fq, Fq (t) , Fq ((t)) , R, R (t) , R ((t)) , C, C (t) , C ((t)) are crossed products [B]. I would wholeheartedly like to thank my supervisor, professor Theodora Theohari-Apostolidi, who has been an invaluable inspiration for me all these years, for the effort she put to develop this thesis, for the full understanding and endless patience, and for the detailed guidance in my every step. I would also like to thank the other members of my thesis committee, professor Hara Charalambous and professor Athanasios Tzouvaras, for their accurate and helpful comments, suggestions and corrections, which contributed to improve the present work. Εισαγωγή

Η παρούσα εργασιά ασχολείται µε το Θεώρηµα των Albert-Brauer-Hasse- Noether, καθώς και µε τις ολικές διαστάσεις σταυρωτών γινοµένων, τα οποία είναι πολύ σηµαντικά όχι µόνο στη ϑεωρία δοµής των αλγεβρών, µα επιπλέον στην αλγεβρική ϑεωρία αριθµών. Το Θεώρηµα των Albert-Brauer-Hasse-Noether ϐρέθηκε στις 9 Νοεµβρίου 1931, από τους µαθηµατικούς Abraham Adrian Albert(1905-1972), Richard Brauer(1901-1977), Helmut Hasse(1898-1979) και Amalie Emmy Noether (1861-1941), όπως δηλώνει άλλωστε και το όνοµα του, και αφιερώθηκε στον Kurt Hensel(1861-1941), για την επέτειο των 70στών γενεθλίων του. Από το άρθρο του P. Roquette [Roq] αντλούµε τα ακόλουθα ιστορικά στοιχεία για την συνεργασία των τεσσάρων αυτών µεγάλων µαθηµατικών που οδήγησε στην απόδειξη του ϑεωρήµατός τους. Ο Brauer έδειξε ενδιαφέρον στην ϑεωρία κλάσεων σωµάτων, επειδή πίστευε πως αν τα αποτελέσµατα της ϑεωρίας αυτής εφαρµοστούν στη ϑεωρία ανα- παραστάσεων οµάδων ϑα οδηγούσαν σε σηµαντικά αποτελέσµατα. Ο Hasse στράφηκε προς τη ϑεωρία των αλγεβρών, διότι είχε παρατηρήσει πως η ϑεωρία κλάσεων σωµάτων, όπως και η ϑεωρία των p-αδικών αριθµών ϑα µπορούσαν να χρησιµοποιηθούν εκεί, αποφέροντας µεγάλα οφέλη. Τέλος, η Noether, η οποία ήταν ο συνδετικός κρίκος στην επικοινωνία των Brauer και Hasse, καθοδηγήθηκε από την ιδέα της πως η µη-µεταθετική ϑεωρία αλγεβρών ϑα έπρεπε να χρησιµοποιηθεί για µία καλύτερη κατανόηση της ϑεωρίας κλάσεων σωµάτων. Την εποχή εκείνη, οι µαθηµατικοί που περιστοίχιζαν τη Noether µελετού- σαν µε ενδιαφέρον τις εργασίες του Dickson(1874-1954) και των µαθητών του. ΄Ετσι, κατα τη διάρκεια της συνεργασίας µεταξύ των Brauer, Hasse και Noether, ο Hasse έγραψε µία επιστολή στον Dickson, µέσα στην οποία παρουσίαζε τη δουλειά του και αναρωτιόταν σχετικά µε την ύπαρξη µη- κυκλικών αλγεβρών δείκτη 4, υπεράνω ενός αλγεβρικού σώµατος αριθµών. Ο

xi xii ΕΙΣΑΓΩΓ΄Η

Dickson, τότε, την προώθησε στον Albert, που εκείνη την εποχή ήταν µαθητής του. Ο τελευταίος απάντησε στο γράµµα του Albert πως προσπαθούσε κι ο ίδιος να απαντήσει στην ίδια ερώτηση, κι έπειτα συστήθηκε. Κάπως έτσι, αυτοί οι δύο ξεκίνησαν να επικοινωνούν, καθώς και να ανταλλάζουν ιδέες γύρω από τα στάδια της απόδειξης. Ο Albert είχε, ήδη το 1931, δηµοσιεύσει εργασίες στις οποίες ανέπτυσσε διαφορετικά εργαλεία από τους υπόλοιπους τρεις, µε τα οποία προσεγγιζόταν το µεγαλύτερο µέρος της απόδειξης. ΄Ετσι, στην ερ- γασία τους, οι Brauer, Hasse και Noether πρόσθεσαν µία υποσηµείωση, αναγνωρίζοντας το έργο του Albert, καθώς και δίνοντάς του ένα ¨ξεχωριστό µερίδιο της απόδειξης¨. Αργότερα, το 1932, οι Albert και Hasse συνεργάστη- καν και απέδειξαν ξανά το Θεώρηµα, χρησιµοποιώντας τις µεθόδους του Al- bert αυτή τη ϕορά. Το Θεώρηµα των Albert-Brauer-Hasse-Noether αποτελεί ένα από τα πιο εµβριθή και ϐαθυστόχαστα ϑεωρήµατα στη ϑεωρία των απλών κεντρικών αλ- γεβρών. Η ίδια η εργασία ξεκινάει µε την ακόλουθη πρόταση. ¨Επιτέλους οι κοινές µας προσπάθειες στέφθηκαν µε επιτυχία, να αποδείξουµε το ακόλουθο ϑεώρηµα, το οποίο είναι ϑεµελιώδους σηµασίας για τη ϑεωρία δοµής αλγεβρών, καθώς επίσης και πέρα από αυτήν.¨ Το Θεώρηµα των Albert-Brauer-Hasse-Noether είναι ένα πολύ κρίσιµο ϐήµα στη ϑεώρηση της αλγεβρικής ϑεωρίας αριθµών. Επιτρέπει µια πλήρη ταξινό- µηση των αλγεβρών µε διαίρεση υπεράνω ενός αλγεβρικού σώµατος αριθµών, µέσω των αναλλοίωτων του Hasse. Ως εκ τούτου, επιτρέπει τον προσδιορισµό της οµάδας του Brauer ενός αλγεβρικού σώµατος αριθµών, καθώς επίσης και την περιγραφή των σωµάτων διάσπασης µίας άλγεβρας µε διαίρεση, περιγράφοντας την τοπική συµπεριφορά τους, µία ιδεα εξαιρετικά χρήσιµη στην ϑεωρία αναπαραστάσεων. Επιπλέον, το Θεώρηµα των Albert-Brauer- Hasse-Noether άνοιξε νέα µονοπάτια, δίνοντας τη δυνατότητα να κατανοήσου- µε τη ϑεωρία κλάσεων σωµάτων µέσω της ϑεωρίας δοµής των αλγεβρών. Σε αυτή την εργασία ϑα συζητήσουµε διάφορα ϑέµατα από τη ϑεωρία σχετικά µε τη δοµή καθώς και τις ιδιότητες των απλών κεντρικών αλγεβρών από τη σκοπιά της αλγεβρικής ϑεωρίας αριθµών. Εξετάζουµε το Θεώρηµα των Albert-Brauer-Hasse-Noether, καθώς και τις απαιτούµενες ϑεωρίες υπεράνω ενός αλγεβρικού σώµατος αριθµών, και όχι πάνω από ένα ολικό σώµα. Στο 1ο κεφάλαιο εισάγουµε την έννοια των απλών κεντρικών αλγεβρών υπεράνω ενός αλγεβρικού σώµατος αριθµών K, καθώς και τις ϑεµελιώδεις ιδιότητές τους. Επίσης, ορίζουµε την οµάδα του Brauer ενός σώµατος. Οι απλές κεντρικές K-άλγεβρες είναι ιδιαίτερα χρήσιµες αλγεβρικές δοµές, όχι µόνο εξαιτίας της ιδιότητάς τους να είναι απλές, µα επιπλέον εξαιτίας του xiii

γεγονότος ότι το κέντρο τους είναι ακριβώς το σώµα K. Στις απλές κεντρικές άλγεβρες ορίζεται µία σχέση ισοδυναµίας. ΄Ετσι, οι κλάσεις τους δοµούν µία οµάδα, τη λεγόµενη οµάδα του Brauer, τα στοιχεία της οποίας ταξινοµούν όλες τις άλγεβρες µε διαίρεση πάνω από το K. Στο 2ο κεφάλαιο ασχολούµαστε µε την ιδιαίτερα σηµαντική έννοια για τη µελέτη µας, αυτή των αλγεβρών-σταυρωτά γινόµενα, όπως επίσης και µε τη ϑεωρία συνοµολογίας οµάδων. Τα σταυρωτά γινόµενα είναι µία ειδική κατηγορία απλών κεντρικών αλγεβρών και παρουσιάζουν εξαιρετικά όµορφες ιδιότητες, όπως το ότι ϐρίσκονται σε ένα προς ένα αντιστοιχία µε συν-κύκλους, και το ότι έχουν την ικανότητα να διατηρούν τα ϕυσικά τους χαρακτηρισ- τικά. Περαιτέρω, συζητούµε τα ϐασικά γύρω από τη ϑεωρία των κυκλικών αλ- γεβρών, και αποδεικνύουµε πως µία άλγεβρα µε διαίρεση ϐαθµού 3 υπεράνω του αλγεβρικού σώµατος αριθµών K είναι κυκλική. Στο 3ο κεφάλαιο παρέχουµε τον ϐασικό εξοπλισµό από τη ϑεωρία ε- κτιµήσεων σε αλγεβρικά σώµατα αριθµών, όπως και από τη ϑεωρία των P- αδικών πληρώσεων αυτών. Σκοπός αυτού του κεφαλαίου είναι να παρουσιά- σουµε και να αποδείξουµε το κεντρικό ϑεώρηµα αυτής της εργασίας, το Θεώρηµα των Albert-Brauer-Hasse-Noether. Μία από τις εφαρµογές του ϑεωρήµατος αυτού είναι πως κάθε απλή κεντρική άλγεβρα είναι κυκλική. Τέλος, στο 4ο κεφάλαιο συζητούµε τη γενικέυση των παραπάνω εννοιών. ∆ίνουµε πληροφορίες για τα modules συνοµολογίας H 0,H 1 και H 2. Επιπλέον, εισάγουµε την κατασκευή της γενικής περίπτωσης των σταυρωτών γινοµένων, και τα εξετάζουµε αυτή τη ϕορά από οµολογικό πρίσµα, δίνοντας σχέσεις ανάµεσα στην ολική και την συνοµολογική διάσταση σε διάφορες δοµές αλ- γεβρών σταυρωτών γινοµένων. Εν συνεχεία, επικεντρωνόµαστε στην περίπτωση που η ολική διάσταση είναι µηδέν. Στην περίπτωση αυτή αποδεικνύουµε πως η οµάδα έχει στρέψη, µολονότι δεν είναι απαραίτητα πεπερασµένη. Τέλος, κατασκευάζουµε µία άλγεβρα σταυρωτό γινόµενο, η οποία είναι και άλγεβρα µε διαίρεση. Για τη ϑεωρία των απλών κεντρικών αλγεβρών, όπως και για τη ϑεωρία συνοµολογίας παραπέµπουµε στα [Re] και [Pi], και ακολουθούµε κυρίως τις µεθόδους που παρουσιάζονται στο [Re] για ϑεωρήµατα που παρουσιάζονται γύρω από αυτά τα ϑέµατα της ϑεωρίας. Επίσης, παραπέµπουµε στα [AR] και [Yi] για το περιεχόµενο του 4ου κεφαλαίου. Για πολλά χρόνια, όλες οι άλγεβρες µε διαίρεση κατασκευάζονταν ως άλγεβρες-σταυρωτά γινόµενα, ξεκινώντας µε τα quaternions του Hamilton (1805-1865), το 1843. Επιπλέον, οι Albert, Dickson και Wedderburn(1882- 1948) απέδειξαν πως όλες οι άλγεβρες µε διαίρεση ϐαθµού 2,3,4, 6 ή 12 είναι xiv ΕΙΣΑΓΩΓ΄Η

κυκλικές, και κατά συνέπεια σταυρωτά γινόµενα, και για ένα µεγάλο µέρος του 20ού αιώνα εικαζόταν πώς όλες οι άλγεβρες µε διαίρεση είναι σταυρωτά γινόµενα. Αλλά, το 1972, ο Amitsur(1921-1994) παρήγαγε το πρώτο α- ντιπαράδειγµα. Κατασκεύασε µη-σταυρωτά γινόµενα άλγεβρες µε διαίρεση µε ϐαθµό n, για κάθε n, υπό την προϋπόθεση ότι ο n διαιρείται είτε από τον αριθµό 23 είτε από τον p2, όπου p είναι ένας πρώτος άριθµός, ο οποίος είναι πρώτος και µε τη χαρακτηριστική του σώµατος κλασµάτων. Η κατασκευή αυτή οδήγησε σε µία πληθώρα ερωτήσεων γύρω από τις ιδιότητες µίας άλγε- ϐρας µε διαίρεση. ΄Ενα από τα σηµαντικότερα ανοιχτά προβλήµατα που αφορούν τις απλές κεντρικές K-άλγεβρες, και το οποίο αποτέλεσε κίνητρο για την εκπόνηση αυτής της εργασίας, είναι η κατασκευή µίας µη-κυκλικής άλγεβρας µε ϐαθµό p, όπου p είναι πρώτος αριθµός, υπεράνω ενός σώµατος K, και είναι καταχωρη- µένο ως Πρόβληµα 1.1 στην εργασία [ABGV]. Είναι προφανές πως µία άλγε- ϐρα µε διαίρεση ϐαθµού 2 πάνω από το K είναι κυκλική. Επιπρόσθετα, µία άλγεβρα µε διαίρεση ϐαθµού 3 πάνω από το K είναι κυκλική, όπως απεδείχθη από τον Wedderburn το 1921. Στην περίπτωση που p > 3 το πρόβληµα είναι τελείως ανοιχτό. Αν περιορίσουµε, όµως, το πρόβληµα στην περίπτωση που το σώµα K είναι αλγεβρικό σώµα αριθµών, τότε η απάντηση δίνεται από το Θεώρηµα των Albert-Brauer-Hasse-Noether, το οποίο ακριβώς ισχυρίζεται πως κάθε άλγεβρα µε διαίρεση ϐαθµού p υπεράνω ενός αλγεβρικού σώµατος αριθµών είναι κυκλική. Από το Θεώρηµα των Albert-Brauer-Hasse-Noether, και µαζί µε ϑεωρήµα- τα των Frobenius(1849-1917), Hasse, Tsen(1898-1940) Wedderburn και Witt (1911-1991), σήµερα µας είναι γνωστό ότι όλες οι άλγεβρες µε διαίρεση

υπεράνω των σωµάτων K, KP ,KP ((t)), όπου K είναι τυχαίο σώµα αριθµών,

Fq, Fq (t) , Fq ((t)) , R, R (t) , R ((t)) , C, C (t) , C ((t)) είναι άλγεβρες-σταυρωτά γινόµενα [B]. Θα ήθελα να ευχαριστήσω ολόψυχα την επιβλέπουσα καθηγήτριά µου, Καθηγήτρια Θεοδώρα Θεοχάρη-Αποστολίδη, η οποία υπήρξε ανεκτίµητη πηγή έµπνευσης για µένα όλα αυτά τα χρόνια, για την προσπάθεια που κατέβαλλε µαζί µου στην ανάπτυξη της παρούσας εργασίας, την αµέριστη κατανόηση σε όλα τα -µαθηµατικά και µη- ϑέµατα που παρουσιάστηκαν, την ατέλειωτη υποµονή, και τέλος, για τη λεπτοµερή καθοδήγηση σε κάθε µου ϐήµα. Θα ήθελα επίσης να ευχαριστήσω τα υπόλοιπα µέλη της τριµελούς εξεταστικής επιτροπής, καθηγήτρια Χαρά Χαραλάµπους και καθηγητή Αθανάσιο Τζουβά- ϱα, για τις ακριβείς και χρήσιµες παρατηρήσεις, τις προτάσεις και τις διορθώσεις τους, οι οποίες συνέβαλαν στη ϐελτίωση της παρούσας εργασίας. Contents

Introduction vii

Εισαγωγή xi

1 Central Simple Algebras and the Brauer Group 1 1.1 Central Simple Algebras ...... 1 1.2 The Brauer Group ...... 9

2 Galois Cohomology 17 2.1 Crossed-product algebras ...... 17 2.2 Galois Cohomology ...... 20 2.3 Cyclic Algebras and Division Algebras of Small Degrees . . . 37

3 The Albert-Brauer-Hasse-Noether Theorem 45 3.1 Valuations and Completions ...... 45 3.2 The Albert-Brauer-Hasse-Noether Theorem ...... 47

4 The General Case 55 4.1 The General Crossed Products ...... 56 4.2 Global Dimensions of Crossed Products ...... 61 4.3 Global Dimension 0 ...... 67

Bibliography 71

xv xvi CONTENTS Chapter 1

Central Simple Algebras and the Brauer Group

The aim of this chapter is to study the central simple algebras over an algebraic number field. By K we denote an algebraic number field, that is, a finite extension of the field Q of the rational numbers. Also, we introduce the reader to the Brauer group, a very critical concept in our study. For further information about this chapter, we refer to [Re] and [Pi].

1.1 Central Simple Algebras

Deﬁnition 1 A central simple K-algebra A is a ring with center K with no nontrivial two-sided ideals, which is a ﬁnite dimensional K-vector space.

By Wedderburn’s Structure Theorem, a central simple K-algebra A is isomorphic to a matrix algebra over a division K-algebra. So

A Mr (D) , where D is a central division K-algebra for some number r. We refer to D as the division algebra part of A. By a division algebra we mean a ring whose nonzero elements form a multiplicative group, and it is of ﬁnite dimension over its center, say K.

The ﬁrst example of a division algebra is the Hamilton’s quaternion algebra H = {a + bi + cj + dk : a, b, c, d ∈ R, i2 = j2 = k2 = ijk = −1}.

1 2CHAPTER 1. CENTRAL SIMPLE ALGEBRAS AND THE BRAUER GROUP

Two central simple K-algebras A and B are called equivalent or Brauer equivalent, and we note them as A ∼ B, if their division algebra parts are K-isomorphic. Therefore the central simple K-algebras A and B are equivalent if there exists a ring isomorphism

A ⊗K Mr (K) B ⊗K Ms (K) , for some integers r, s. Another way of looking at this equivalence is the following one. Let

A Mρ (D1) and B Mλ (D2) ; then

A D1 ⊗K Mρ (K) and B D2 ⊗K Mλ (K) .

Therefore A ∼ B if and only if D1 ∼ D2. In other words, the equivalence relation deﬁned for central simple algebras over K is one really deﬁned for division algebras over K. Let [A] denote the equivalence class of A. Our aim is to give an algebraic structure on the set of equivalence classes of central simple algebras. For this reason we have to study the central simple algebras in more detail. Also, by Wedderburn’s Structure Theorem for central simple algebras, we get the next.

Theorem 1 A central simple K-algebra A is isomorphic to an algebra of r × r matrices over a division ring. The algebra A determines r uniquely, and determines the division ring D up to isomorphism. If V is any minimal ideal of A, then D EndA (V ) is a division ring, and

o A = EndD (V ) Mr (D ) , where r is the dimension of the left D-space V . Hence

(A : K) = r2 (D : K) .

In the sequel we prove some of the main properties of central simple algebras.

Theorem 2 Let A and B be two central simple K-algebras. Then A ⊗K B is also a central simple K-algebra. 1.1. CENTRAL SIMPLE ALGEBRAS 3

Proof. At ﬁrst we compute the center of A ⊗K B. Since B is a ﬁnite dimen- Ls { } sional K-algebra, we can write B = i=1 Kei , for some e1, . . . , es K-basis for B. Then Ms A ⊗K B = A ⊗K Kei . i=1 P Let x ∈ Z (A ⊗K B), then x = ai ⊗ ei . for some elements ai ∈ A, 1 ≤ i ≤ s. Moreover, x has to commute with the element a ⊗ 1 for all a ∈ A. Hence Xs 0 = (a ⊗ 1 − 1 ⊗ a) x = (aai − ai a) ⊗ ei = 0, i=1 and since ei are linearly independent over K, we get aai = aai for all a ∈ A.

This means that ai ∈ Z (A) = K. Thus Xs Xs Xs x = ai ⊗ ei = 1 ⊗ ai ei = 1 ⊗ ai ei ; i=1 i=1 i=1 in other words x = 1 ⊗ b0, for some b0 ∈ B. But also x has to commute with the elements 1 ⊗ b, for all b ∈ B. Hence

x (1 ⊗ b) = (1 ⊗ b) x ⇒ 1 ⊗ b0b = 1 ⊗ bb0, for all b ∈ B, so b0 ∈ Z (B) = K. Therefore x ∈ K and so Z (A ⊗K B) ⊂ K. Now it is clear that K ⊂ Z (A ⊗K B). Hence Z (A ⊗K B) = K. The K-algebra

A ⊗K B is of ﬁnite dimension, since this happens for A and B, so it remains 0 to prove that A ⊗K B is simple. For this, let A = Mr (D) and B = Mt (D ), for central K-division algebras D and D0. Then

0 0 0 Mr (D) ⊗K Mt D Mr (K) ⊗K Mt (K) ⊗K D ⊗K D Mrt (K) ⊗K D ⊗K D ,

0 Thus in order to prove that A⊗K B is simple, it is enough to prove that D⊗K D Ln ∈ is simple. Let D = i=1 Kdi , for some number n and some elements di D, 1 ≤ i ≤ n. Then n 0 M 0 D ⊗K D = di ⊗K D . i=1 0 Pλ 0 Any element of D ⊗K D is of the form i=1 di ⊗ di , for some number λ and 0 0 some elements di ∈ D , 1 ≤ i ≤ λ. Let now X be a nonzero two-sided ideal 0 Pm 0 of D ⊗K D . We can choose an element 0 , x ∈ X such that x = i=1 di ⊗ di 4CHAPTER 1. CENTRAL SIMPLE ALGEBRAS AND THE BRAUER GROUP with m the smallest between the elements of X, that is, of shortest length. 0 −1 We get the element x ⊗ 1 ⊗ (di ) ∈ X. Then

X 0 x = d1 ⊗ 1 + di ⊗ di ∈ X, and the element 0 0−1 y = 1 ⊗ d x 1 ⊗ d ∈ X, for any element d0 ∈ D0. We see that

m X 0 0 0−1 y = d1 ⊗ 1 + di ⊗ d di d . i=1

Since x, y ∈ X, then x − y ∈ X. But the element x − y is shorter than 0 x, hence x − y = 0 and so x = y. This means that di , 1 ≤ i ≤ m, 0 0 0 commutes with every element d ∈ D , hence di ∈ K. Therefore x ∈ D 0 0 and as nonzero, x is a unit of D ⊗K F , hence X = D ⊗K D . So we conclude 0 0 that D ⊗K D has no nonzero proper two-sided ideal, i.e. D ⊗K D is simple.

A result that arises from the above theorem is the next.

Corollary 1 If A is a central simple K-algebra and L ⊃ K, then L ⊗K A is a central simple L-algebra.

A left A-module M, for a K-algebra A, is called faithful if the relation aM = 0, for a ∈ A, implies that a = 0. We denote by

AL := {aL ∈ EndZ (M) : aL (m) = am, ∀m ∈ M}.

It is clear that M is a faithful left A-module if and only if the ring homomorphism A −→ AL , a 7→ aL is a monomorphism.

Deﬁnition 2 Let M be a left A-module for a K-algebra A. The pair (A, M) has the double centralizer property if AL = EndD (M), where D := EndA (M).

We remark that the pair (A, AA) has the double centralizer property with

D = EndA (AM) which is the ring

AR = {ar ∈ EndZ (A) : aR (x) = xa, ∀x ∈ A} of right multiplications by elements of A. 1.1. CENTRAL SIMPLE ALGEBRAS 5

Theorem 3 Let A be a ﬁnite dimensional K-algebra.

(i) For each A-module N, the pair (A,A A ⊕ N) has the double centralizer property. (ii) Let M be an A-module and k be a positive integer. If A, M(k) has the double centralizer property, then so does (A, M).

Proof. For the proof see [Re, Theorem (7.9)].

Corollary 2 Let A be a central simple K-algebra and M a faithful left A- module of ﬁnite K-dimension. Then (A, M) has the double centralizer property.

Theorem 4 (Double Centralizer Theorem) Let A be a central simple K- algebra and B be a simple subring of A containing K. Then the centralizer

CA (B) = {x ∈ A : xb = bx, for all b ∈ B} of B in A is a simple ring, and B is its centralizer in A.

Proof. Let V be a simple left A-module, D = EndA (V ). By Theorem 1, we know that A = EndD (V ) and D is a division algebra with center K. Now, we may form aL , bL and dL , the left multiplications on V , for a ∈ A, b ∈ B and d ∈ D. We observe that V may be viewed as a left D ⊗K B-module, via the mapping

(d ⊗ b) v = dL bL v, v ∈ V.

We also see that dL bL = bL dL , for b ∈ B and d ∈ D. Thus, the elements of DL and BL commute, and we also know that DL and BL are K-subalgebras of f EndK (V ). This means that the mapping D ⊗K B −→ DL BL is an isomorphism of K-algebras. Indeed,

Ker f = {d ⊗ b ∈ D ⊗K B : dL bL = 0} = {d ⊗ b ∈ D ⊗K B : dL = 0 or bL = 0} =

= {d ⊗ b ∈ D ⊗K B : d = 0 or b = 0} = {0 ⊗ b or d ⊗ 0} = {0}, thus f is a monomorphism, and also

Im f = {f (d ⊗ b) : d ⊗ b ∈ D ⊗K B} = {dL bL : d ∈ D, b ∈ B} = DL BL , which means that it also is an epimorphism. So by Theorem 2, D ⊗K B is a central simple K-algebra, which means that DL BL is simple too. Moreover, 6CHAPTER 1. CENTRAL SIMPLE ALGEBRAS AND THE BRAUER GROUP

V is a faithful DL BL -module, so the pair (DL BL ,V ) has the double centralizer property, by Corollary 2. Now, let φ ∈ EndDL BL (V ). Then φ ∈ EndDL (V ) = AL , so φ = aL for some a ∈ A which centralizes BL . But A and B act faithfully on V , thus a ∈ CA (B), and we obtain that (CA (B))L = EndDL BL (V ). And because of the fact that CA (B) (CA (B))L , CA (B) is simple. Now, (DL BL ,V ) has the double centralizer property, thus DL BL = End(CA(B))L (V ). We choose an element x ∈ A which centralizes CA (B). Then the element xL ∈ AL centralizes (CA (B))L , so xL ∈ DL BL . But since xL also belongs in AL , we obtain that xL centralizes DL . In the same manner as in the proof of Theorem 2, we have that the elements in D ⊗K B which centralize D ⊗ 1 are contained in

Z (D) ⊗K B, that is, in B itself, since Z (D) = K. Therefore xL must lie in BL , which means that x ∈ B. The latter gives us the inclusion CA (CA (B)) ⊆ B.

From the proof of Theorem 4, we get the next useful result.

o Corollary 3 Let A be a central simple K-algebra. Then A ⊗K A Mλ (K), where λ = (A : K) and Ao is the opposite ring of A.

Definition 3 Let A be a central simple K-algebra. A field extension L of K is called a splitting field of A (or L splits A) if L ⊗K A Ms (L), for some number s.

From the above definition it follows that any field extension E of a splitting field L of the central simple K-algebra A is also a splitting field of A, since

E ⊗K A E ⊗L (L ⊗K A) E ⊗L Ms (L) Ms (E) . Moreover, a splitting ﬁeld L of the central division algebra D is also a splitting ﬁeld of the algebra Mr (D), and the reverse also holds.

From the above, the importance of splitting fields of central simple algebras follows. Let Ke be an algebraic closure of the field K. Then Ke is a splitting field of the K-algebra Mr (D) for some central K-division algebra D. Indeed 0 Ke ⊗K Mr (D) Mr Ke ⊗K D Mλ D , for some Ke-central division algebra D0. We know that D0 = Ke, since Ke is the algebraic closure of K and hence every element of D0 is algebraic over Ke. Thus there always are splitting fields for central simple algebras. Our 1.1. CENTRAL SIMPLE ALGEBRAS 7 interest is to find a splitting field of a central simple K-algebra, which is a separable extension of K. We can do it due to the next result.

Theorem 5 (Noether- Kothe)¨ Let D be a central division K-algebra, where K is an algebraic number field. Then (i) Every maximal subfield E of D contains K, and is a splitting field for D. 2 Moreover, if (E : K) = m, then (D : K) = m , and E ⊗K D Mm (E). (ii) There exists a maximal subfield L of D which is separable over K.

Proof. (i) Since (D : K) is finite, D contains maximal subfields, so let E be a maximal subfield of D. Then K ⊂ E, otherwise E (K) is a larger subfield of

D. Now, we shall use the results of Theorem 4 using A = D, V = DD and B = E. We obtain that E ⊆ CD (E). Further, let x ∈ CD (E). If x < E, then E (x) is a subﬁeld of D, which contains E, which is a contradiction, since we chose E to be maximal. Thus E = CD (E). By Corollary 3 we have

D ⊗K E EndE (V ) Mm (E) ,

2 2 where m = (V : E) and (E : K) = (D : K). But m = (D ⊗K E : E) = (D : K) = (E : K)2, so m = (E : K). (ii) We observe that for every d ∈ D, K (d) /K is separable. Indeed, we have to show that for every extension F of K, the tensor product K (d) ⊗K F has no nonzero nilpotent elements. Let ((a + bd) ⊗ f )n = 0, for a, b ∈ K, f ∈ F. Then, (a + bd)n ⊗ f n = 0, and since char K = 0, we get that either a + bd = 0 ⇒ a = b = 0 or f = 0.

In both cases, (a + bd) ⊗ f = 0, therefore the tensor product K (d) ⊗K F has no nonzero nilpotent elements, and the extension K (d) /K is separable.

We remark, also, that if A is a central simple K-algebra, then

2 2 2 (A : K) = (Mr (D): K) = r (D : K) = r m , where m = ind D.

The main result that leads to the deﬁnition of the crossed product is Skolem-Noether Theorem, which is very important in general in ring theory. 8CHAPTER 1. CENTRAL SIMPLE ALGEBRAS AND THE BRAUER GROUP

Theorem 6 (Skolem-Noether) Let A be a central simple K-algebra. Then every K-isomorphism φ : B −→ Be between two simple K-subalgebras of A is extended to an inner automorphism of A, that is, there exists an invertible element a ∈ A, such that φ (x) = axa−1 for all x ∈ A.

Proof. Let V be a simple left A-module. Then V becomes a left D⊗K B-module by the rule (d ⊗ b) v = dbv, for d ∈ D, b ∈ B and v ∈ V . Moreover, V becomes a left D ⊗K Be-module by the rule d ⊗ eb v = debv, for d ∈ D, eb ∈ Be and v ∈ V . Since φ : B −→ Be is an isomorphism of K-algebras, then the map

1 ⊗ φ : D ⊗K B −→ D ⊗K Be is also an isomorphism of K-algebras. Now we can deﬁne a new action of

D ⊗K B on V denoted by

(d ⊗ b) ∗ v = dφ (b) v, for d ∈ D, b ∈ B and v ∈ V . Let us denote by Ve = (V, ∗). We consider the map θ : V −→ Ve dbv 7→ (d ⊗ b) ∗ θ (v) = dφ (b) θ (v) .

Then θ is an isomorphism of D ⊗K B-modules, taking in account that V and

Ve are simple left D ⊗K B-modules and they have the same K-dimension.

Taking b = 1, then θ = EndD (V ) = A. This means that θ is the left multiplication by an element a ∈ A. This element a is invertible, since θ is an isomorphism. Hence

a (dbv) = dφ (b) av, for all v ∈ V . For d = 1 we get

abv = φ (b) av, 1.2. THE BRAUER GROUP 9 for all v ∈ V . Hence ab = φ (b) a, for all b ∈ B, since V is a faithful left A-module. Therefore φ (b) = aba−1, for all b ∈ B, and the theorem has been proved.

An immediate consequence from the Skolem-Noether Theorem is the next result.

Corollary 4 Let A be a central simple K-algebra. Then (i) Every K-automorphism of A is inner. (ii) If L and L0 are isomorphic subﬁelds of A, then L0 = aLa−1, for some invertible element a ∈ A, i.e. L and L0 are conjugate.

Theorem 7 (Wedderburn) Every ﬁnite division algebra is a ﬁeld.

Proof. Let D be a finite division algebra with center K, and let D* = D − {0}, K* = K − {0}. If L is a maximal subfield of D, then (D : K) = n2, and (L : K) = n. Any other maximal subfield L0 is a finite field, and (L0 : K) = n, so card L = card L0. Thus L0 is K-isomorphic to L, and by Corollary 4(ii), L0 is conjugate to L. Since every x ∈ D lies in a subfield K (x) of D, and hence in some maximal subfield, we take the result that [ D* = aL*a−1, where a ranges over some set of invertible elements of D. But for x ∈ L*,

(ax) L* (ax)−1 = aL*a−1, so in our ﬁrst relation we need to let a range over the left coset represen- − tatives of L* in D*. Hence there are D* : L* sets {aL*a 1} that occur in this relation. Each of them has cardinality card L*. These are not disjoint, since each contains 1. Thus D* cannot be their union, unless there is only one such set, that is, D* = L*. Thus D = L = K.

1.2 The Brauer Group

In this section, L always denotes a finite extension of the algebraic number field K, D is a finite dimensional division K-algebra and A and B are central simple K-algebras. 10CHAPTER 1. CENTRAL SIMPLE ALGEBRAS AND THE BRAUER GROUP

We now consider the set B (K) of all classes [A], where A is a central simple K-algebra. In B (K) we deﬁne a multiplication by the rule

[A][B] = [A ⊗K B] for two central simple K-algebras A and B. This multiplication is well deﬁned, because of Theorem 2. So B (K) is a semigroup, because of the associativity of the tensor product. Moreover, since A ⊗K K A, we get that

[A][K] = [A], so [K] is the identity element of B (K). Let now Ao be the opposite ring of A; then o o [A][A ] = [A ⊗K A ] = [K], by Corollary 3. Hence [Ao] is the inverse of the class [A], and so B (K) is a group, hereafter called the Brauer group of K. From the Wedderburn’s Structure Theorem, we know that for each central simple K-algebra A, A = Mn (D) for some n and for a unique (up to isomorphism) central division K-algebra D, that is, [A] = [D].

Definition 4 A subfield L of D is called a maximal subfield if it is not prop- erly contained in a larger subfield of D.

The next theorem gives the whole information on the maximal subﬁelds of a central simple K-algebra.

Theorem 8 Let D be a central division K-algebra, m2 = (D : K), and E a ﬁnite extension of K. Then (i) If E splits D, then m | (E : K). (ii) There exists a smallest positive integer r for which there is an embedding

E ⊂ Mr (D). Furthermore, the centralizer CMr (D) (E) of E in Mr (D) is a division algebra, and E is a maximal subﬁeld of Mr (D) if and only if E = CMr (D) (E).

Proof. (i) Let E split D. We may form the central simple E-algebra E ⊗K D, and let V be a simple left E ⊗K D-module. V is also a left D-vector space, say of D-dimension (V : D) = r. Of course, E and D commute elementwise in E ⊗K D, so there is an embedding

E,→ EndD (V ) Mr (D) . 1.2. THE BRAUER GROUP 11

We shall prove that r is minimal. Let

E,→ Mt (D) be another embedding. Then there exists another D-vector space W , with (W : D) = t, and

Mt (D) EndD (W ) , thus

E,→ EndD (W ) .

If we view W as a left E ⊗K D-module, then M W V (k), for some k. Because of the fact that W is a direct sum of copies of V , we have that t must be a multiple of r. This proves that r is indeed minimal. Now

E ⊗K D Mm (E) , and since we chose V to be simple, we get that

(m) E ⊗K D V as left E ⊗K D-modules. Passing at the dimensions, we have that

(m) (E ⊗K D : D) = V : D = mr and (E ⊗K D : D) = (E : K) , thus (E : K) = mr, which exactly means that m divides (E : K). (ii) We have already proved in (i) that there exists an integer r, which is minimal, and for which there is an embedding E,→ Mr (D). Now, let

CMr (D) (E) be the centralizer of E in the central simple K-algebra Mr (D). If f is an element of this centralizer, then f must also belong in EndE⊗K D (V ), and the converse also holds true, since EndE⊗K D (V ) ⊂ E. We have thus obtained that CMr (D) (E) = EndE⊗K D (V ). Moreover, by Schur’s Lemma, EndE⊗K D (V ) is a division algebra, so CMr (D) (E) is also a division algebra. We also know that each ring belongs in its centralizer. Thus

CMr (D) (E) = E if and only if E is a division algebra of Mr (D). The last thing that remains to be proved is that E is a maximal subﬁeld of Mr (D) if and only if E coincides 12CHAPTER 1. CENTRAL SIMPLE ALGEBRAS AND THE BRAUER GROUP

with its centralizer in Mr (D). First, we assume that E < Mr (D) and we choose an element

x ∈ CMr (D) (E) − E. Then x commutes with every element of E, since it belongs to the centralizer of E. This means that E (x) is a field, which is contained in the division algebra Mr (D). Also, obviously E < E (x), thus E is not a maximal subfield of Mr (D). On the other hand, if E is a maximal subfield of Mr (D), then

E = CMr (D) (E) . We have o Mr (D) ⊗K E Mm CMr (D) (E) . And since o E = CMr (D) (E) = E , then E splits Mr (D).

One of the important assertions that arise from the preceding theorem is that we cannot ﬁnd any extension of K with K-dimension less than m, which split the central division K-algebra D. Also, in the case where

E = CA (E), we shall call E a self-centralizing maximal subﬁeld of A, for a central simple K-algebra A.

Now, let B be a central simple K-algebra. Then B Ms (D) for a central division K-algebra D and a positive number s. By Theorem 5, there exists a maximal subﬁeld F of D such that F/K is a separable extension, and F splits D. Let (D : K) = m2, then (F : K) = m. Let L be the normal closure of K. Then L/K is a normal and separable extension, so L/K is a Galois extension of K containing F. Let (L : F) = t. We remark that

L ⊗K Mt (D) L ⊗F F ⊗K Mt (D)

L ⊗F Mt (F ⊗K D) L ⊗F Mt (Mm (F))

L ⊗F Mtm (F) Mtm (L ⊗F F) Mtm (L) . 2 2 So we see that L is embedded in Mtm (L). Moreover (Mt (D) : K) = t m = 2 2 (tm) = (L : K) so L is a maximal subﬁeld of Mt (D) and by 8, it is self centralizing, that is, L = CMt (D) (L). From the above, the central simple K-algebra Mt (D) has a maximal subﬁeld L, such that the extension L/K is a Galois extension and Mt (D) belongs in the same class with B. So we get the next theorem. 1.2. THE BRAUER GROUP 13

Theorem 9 Let [B] ∈ B (K). There exists a central simple K-algebra A such that [A] = [B] and A has a maximal subﬁeld L, such that the extension L/K is a Galois extension, and L is self-centralizing in A.

Let E be a ﬁeld extension of K and we consider the Brauer groups B (K) and B (E). We deﬁne the map

φ : B (K) −→ B (E) , [A] 7→ [E ⊗K A],

[A] ∈ B (K). By Corollary 1, E ⊗K A is a central simple E-algebra. Now, if B is a central simple K-algebra, and A ∼ B, we get that

0 A ⊗K Mr (D) B ⊗K Ms D , thus 0 E ⊗K A ⊗K Mr (D) E ⊗K B ⊗K Ms D , that is,

E ⊗K A ∼ E ⊗K B, and the map is well deﬁned. Also,

E ⊗K (A ⊗K B) (E ⊗K A) ⊗E (E ⊗K B) .

o o Finally, we obtain that E ⊗K A (E ⊗K A) . Taking all of these remarks into account, we have shown that φ is a homomorphism of groups, with kernel

B (E/K) := {[A] ∈ B (K) :[E ⊗K A] = [E]}. We observe that for a central simple K-algebra A, [A] ∈ B (E/K) if and only if [E ⊗K A] = [E], i.e. E ⊗K A Ms (E) for some s. That is, [A] is an element of the kernel if and only if E splits A. So there exists an exact sequence of groups 1 → B (E/K) → B (K) → B (E) . Now, by 9, if [B] ∈ B (K) there exists a finite Galois extension L/K such that [B] = [A] and L is a self-centralizing maximal subfield of A, splitting A. Hence [ B (K) = B (L/K) , L where L is a finite Galois extension of K. 14CHAPTER 1. CENTRAL SIMPLE ALGEBRAS AND THE BRAUER GROUP

Example1 (i) Let us remark at this point that the Brauer group B (C) over the field of complex numbers has one element [C], since there is no division algebra over C, except C itself. (ii)(Frobenius) The next example is a theorem of Frobenius. Let D be a central division K-algebra over the field of real numbers R. Then, D R or D C or D H, where H is the algebra of real quaternions. Indeed, there exists a maximal subfield L of D, with

R ⊆ K ⊆ L ⊆ D.

Further, the extension L/R is algebraic. Thus the only possibilities for this extension are the ﬁelds R and C. Therefore, we get that

dimK (L) ≤ dimR (L) ≤ 2.

But, we know that dimK (L) = dimL (D) and dimK (D) = dimK (L). Thus either dimK (D) = 1 or dimK (D) = 4. If dimK (D) = 1, then dimK (L) = 1, which exactly means that L = K = D. Therefore

D R or D C.

On the other hand, if dimK (D) = 4, then dimK (L) = 2 = dimL (D). Therefore

K = R and L = C.

Furthermore, because dimL (D) = 2 and since L = C, we get that D is noncommutative, otherwise D would be an algebraic extension of the ﬁeld C of degree 2, which is impossible. Thus D is a division algebra with dimC (D) = 2, and C is a maximal subﬁeld of D. Of course, C = R (i), where i2 = −1. The extension C/R is a Galois extension, with Galois group G = Gal (C/R) = {1, σ}, where σ (a + bi) = a − bi, for a, b ∈ R. By Skolem- Noether Theorem, we get that the R-automorphism σ of C is extended to an inner automorphism t of D, that is, there exists an element 0 , d ∈ D, such that t (x) = dxd−1, for all x ∈ D. But

σ (i) = σ (−i) = −i, and t (i) = did−1.

Therefore di = −id. Moreover, d2i = id2, that is, d2 commutes with every element in C. If d2 < C, then the ﬁeld C d2 is an algebraic extension of C, 1.2. THE BRAUER GROUP 15

and C is contained in C d2 , which is a contradiction. Thus d2 ∈ C. We also observe that t d2 = dd2d−1 = d2.

Thus d2 ∈ R, since it is ﬁxed under the inner automorphism t. If d2 > 0, then d ∈ R. But then, since

t (i) = did−1 = i, t is the identity map in C, which means that σ is the identity map in C, again a contradiction. Therefore d2 < 0, thus d2 = −r2, for some r ∈ R. This means that d 2 = −1. r We set d = j and k = ij. r Then, the set of symbols {1, i, j, k} forms an R-basis for D, thus D H. 16CHAPTER 1. CENTRAL SIMPLE ALGEBRAS AND THE BRAUER GROUP Chapter 2

Galois Cohomology

In this chapter, we discuss some basic theory about crossed products. These are the construction we make in order to produce a central simple K- algebra, which contains a certain Galois extension as a maximal subﬁeld. We continue to denote by K an algebraic number ﬁeld. The main references are [Re], [Pi], [J] and [S1].

2.1 Crossed-product algebras

Let B Mr (D) be a central simple K-algebra, for an algebraic number√ ﬁeld K, and where D is a central division K-algebra, with index m = (D : K). We know that D contains as a maximal subﬁeld a separable extension F/K and (F : K) = m, due to Theorem 5. We can consider the normal closure

L of K, and let (L : F) = n. Then, the K-algebra Mn (D) is a central simple

K-algebra, and L is a maximal subﬁeld of Mn (D). We remark that

F ⊗K Mm (K) Mm (F) ,→ Mn (D) , and L is contained in Mn (D) by means of its regular representation over F in Mn (K). Moreover,

2 2 2 (Mn (D) : K) := (D : K)(Mn (K) : K) = m n = (L : K) .

Hence, by Theorem 9, [Mn (D)] = [B] in the Brauer group B (K). Further- more, the algebra Mn (D) contains L as a maximal subﬁeld, and L/K is a Galois extension. Therefore in any case we may consider a class [A] in

17 18 CHAPTER 2. GALOIS COHOMOLOGY

B (K), such that A is a central simple K-algebra and has as a maximal self-centralizing subﬁeld a Galois extension L/K. Let G = Gal (L/K) and σ ∈ G. Then σ is an automorphism of the subalgebra L of A, and by Skolem-Noether Theorem, there exists an invertible element uσ ∈ A, such that the inner automorphism

−1 A −→ A, a 7→ uσ auσ , for all a ∈ A, extends σ. Hence

−1 uσ luσ = σ (l) , for all l ∈ L. From Galois theory we know that the elements σ ∈ G are linearly independent over L and hence the elements uσ , σ ∈ G, are linearly independent over L. Now, since (A : L) = n = |G|, we get that the elements uσ , σ ∈ G, consist an L-basis of the L-vector space A.

Let, now, σ, τ ∈ G; then, since uσ l = σ (l) uσ , we get that for l ∈ L,

−1 −1 −1 −1 uσ uτluτ uσ = uσ τ (l) uσ = στ (l) = uστluστ . Thus −1 −1 −1 uστ uσ uτluτ uσ uστ = l, −1 for all l ∈ L. Therefore uστ uσ uτ ∈ L, since it commutes with all the elements of L, and L is a self-centralizing maximal subﬁeld of A. This means that there exists an element λσ,τ ∈ L, depending on σ, τ, such that

−1 uστ uσ uτ = λσ,τ ⇒ uσ uτ = uστλσ,τuσ uτ = στ λσ,τ uστ. In other words, there exists an element, say f (σ, τ), of L such that

uσ uτ = f (σ, τ) uστ, σ, τ ∈ G. Next, the algebra A is an associative K-algebra, so we get that

uρ (uσ uτ) = (uρuσ )uτ, for all ρ, σ, τ ∈ G. Combining, now, the two relations uσ l = σ (l) uσ and uσ uτ = f (σ, τ) uστ, we obtain uρf (σ, τ) uστ = f (ρ, σ) uρσ uτ ⇒ ρ (f (σ, τ)) f (ρ, στ) uρ(στ) = f (ρ, σ) f (ρσ, τ) u(ρσ)τ. But in G, ρ (στ) = (ρσ) τ, so we conclude to the relation ρ (f (σ, τ)) f (ρ, στ) = f (ρ, σ) f (ρσ, τ) , for ρ, σ, τ ∈ G. 2.1. CROSSED-PRODUCT ALGEBRAS 19

Deﬁnition 5 A map f : G × G −→ L*, (σ, τ) 7→ f (σ, τ), which satisﬁes the condition ρ (f (σ, τ)) f (ρ, στ) = f (ρ, σ) f (ρσ, τ), ρ, σ, τ ∈ G is called a factor set from G to L*.

This condition is called the factor set condition. Much of the work on the factor sets was done by Noether, so in many references of the literature factor sets are called Noether factor sets. So we have seen that the central simple K-algebra A can be written L ∈ ∈ as A = σ∈G Luσ , for the invertible elements uσ L, σ G to consist an L-basis, and the elements uσ are multiplied by the rules

uσ l = σ (l) uσ ,

uσ uτ = f (σ, τ) uστ, and ρ (f (σ, τ)) f (ρ, στ) = f (ρσ) f (ρσ, τ) , l ∈ L, ρ, σ, τ ∈ G. We denote the algebra A by (L/K, f ). According to all the above, we can now give the following deﬁnition.

Deﬁnition 6 Let L/K be a Galois extension with Galois group G = Gal (L/K), L × −→ * and f : G G L be a factor set. The K-algebra A := (L/K, f ) := σ∈G Luσ , which is an L-vector space with L-basis the symbols {uσ }, and with multiplication given by the rules

uσ l = σ (l) uσ , uσ uτ = f (σ, τ) uστ, and ρ (f (σ, τ)) f (ρ, στ) = f (ρσ) f (ρσ, τ) , l ∈ L, and ρ, σ, τ ∈ G, is called a crossed product algebra.

In the sequel we examine some properties of the factor sets from G to L*. If f and g are factor sets, so is fg, where by deﬁnition

(fg)(σ, τ) = f (σ, τ) g (σ, τ) ,

σ, τ ∈ G. We call trivial factor set, and we denote it by 1, the factor set that takes the value 1 ∈ L* for all (σ, τ) ∈ G × G. L Now let A = Luσ as above, let {cσ : σ ∈ G} be any set of elements of * L L , and put vσ = cσ uσ , σ ∈ G. Then A = Lvσ , and

vσ x = σ (x) vσ , 20 CHAPTER 2. GALOIS COHOMOLOGY

vσ vτ = g (σ, τ) vστ, x ∈ L, σ, τ ∈ G. But, remembering that vσ = cσ uσ , we get

vσ vτ = (cσ uσ )(cτuτ) = cσ (uσ cτ) uτ = cσ (σ (cτ) uσ ) uτ = cσ σ (cτ) f (σ, τ) uστ.

But also, since vσ = cσ uσ , we obtain that vστ = cστuστ. Thus,

−1 g (σ, τ) vστ = vσ vτ = cσ σ (cτ) cστ f (σ, τ) vστ, which exactly means that

−1 g (σ, τ) = cσ σ (cτ) cστ f (σ, τ) ,

σ, τ ∈ G. The map δc : G × G −→ L* ,given by

−1 (δc)(σ, τ) = cσ σ (cτ) cστ ,

σ, τ ∈ G, is a factor set, and is called a principal factor set. We have thus shown that g = (δc) f , and that (L/K, f ) (L/K, (δc) f ), for each c : G −→ L*. We call a factor set f : G × G −→ L* normalized if f (σ, 1) = f (1, σ) = 1, for all σ ∈ G. This deﬁnition is needed to make simpler the various calcu- lations.

Example 2 We return to Frobenius Theorem, Example 1(ii). Let t denote the element j, then we may deﬁne the 2-cocycle f : G × G −→ C*, with f (σ, 1) = f (1, σ) = 1, and f (σ, σ) = −1. Moreover H = R ⊕ Ri ⊕ Rt ⊕ Rit, with multiplication t · t = f (σ, σ) and t · i = σ (i) t.

In other words, the R-algebra H is a crossed product (C, G, idG, f ).

2.2 Galois Cohomology

We consider the group Z 2 G, L* = {f : G × G −→ L* : f is a factor set } of all factor sets from G to L*, and we deﬁned the multiplication of two elements f, g ∈ Z 2 G, L* by the rule (fg)(σ, τ) = f (σ, τ) g (σ, τ), for σ, τ ∈ 2.2. GALOIS COHOMOLOGY 21

G. It is easy to see that Z 2 G, L* becomes an abelian group under this multiplication. Let B2 G, L* = {f : G × G −→ L* : f = δc} be the subgroup of Z 2 G, L* consisting of all principal factor sets from G to L*. The factor group H 2 G, L* := Z 2 G, L* /B2 G, L* is called the second cohomology group from G to L*. Two factor sets f, g ∈ Z 2 G, L* are called equivalent, if they have the same image in H 2 G, L* , that is, if [f ] = [g] in H 2 G, L* . The next theorem shows the usefulness of normalized factor sets.

Theorem 10 Every factor set f : G × G −→ L* is equivalent to a normalized factor set.

−1 Proof. Let c1 = f (1, 1) and cσ = 1, for σ ∈ G − {1}. We make use of the −1 formula (δc)(σ, τ) = cσ σ (cτ) cστ . Thus, we obtain

−1 −1 −1 (δc)(1, 1) = c1 · 1 (c1) · c1 = c1 · c1 · c1 = c1 = f (1, 1) ,

−1 −1 −1 (δc)(σ, 1) = cσ · σ (c1) · cσ = 1 · σ f (1, 1) · 1 = σ f (1, 1) , and

−1 =1 −1 −1 (δc)(1, σ) = c1 · 1 (cσ ) · cσ = f (1, 1) · cσ · cσ = f (1, 1) . We set g = (δc) f . Since g is the product of two factor sets from G to L*, then g is a factor set from G to L*. Moreover, because of the fact that g = (δc) f , we get that [g] = [f ] in H 2 G, L* , that is, f and g are equivalent. We shall use the deﬁnition of multiplication between factor sets (fg)(σ, τ) = f (σ, τ) g (σ, τ), σ, τ ∈ G to prove that g is a normalized factor set. We have

g (1, 1) = ((δc) f )(1, 1) = (δc)(1, 1) f (1, 1) = f (1, 1)−1 f (1, 1) = 1.

Further, g (σ, 1) = ((δc) f )(σ, 1) = (δc)(σ, 1) f (σ, 1) = σ f (1, 1)−1 f (σ, 1) , and

g (1, σ) = ((δc) f )(1, σ) = (δc)(1, σ) f (1, σ) = f (1, 1)−1 f (1, σ) . 22 CHAPTER 2. GALOIS COHOMOLOGY

Next, we remember the factor set condition ρ (f (σ, τ)) f (ρ, στ) = f (ρ, σ) f (ρσ, τ). Putting ρ = σ, σ = 1 and τ = 1, we get

σ (f (1, 1)) f (σ, 1) = f (σ, 1) f (σ, 1) thus σ (f (1, 1)) = f (σ, 1), which means that σ f (1, 1)−1 = f (σ, 1)−1. There- fore g (σ, 1) = f (σ, 1)−1 f (σ, 1) = 1. Finally, setting ρ = 1, σ = 1 and τ = σ, we obtain

f (1, σ) f (1, σ) = f (1, 1) f (1, σ) , thus f (1, σ) = f (1, 1). Hence

g (1, σ) = f (1, 1)−1 f (1, 1) = 1.

Consequently, we have that g is normalized, thus every factor set f is equivalent to a normalized factor set. L If (L/K, f ) = Luσ is a crossed-product algebra in which f is normalized, then u1 is the unity element of the ring (L/K, f ), and each uσ is a unit in this ring. We shall always identify L with the subring Lu of (L/K, f ). 1 Our aim is to ﬁnd a relation between the groups H 2 G, L* and B (L/K). The next theorem gives more information on the central simple algebras.

Theorem 11 Let A = (L/K, f ) and B = (L/K, g) be two crossed products, for f, g ∈ Z 2 G, L* . Then A is a central simple K-algebra. Moreover, A B as K-algebras if and only if g = (δc) f , for some principal factor set δc.

Proof. We first prove tht K is the center of A, and L = CA (L). Because of the fact that uσ l = σ (l) uσ , the center of A is consisted from elements that stay fixed under the action of σ, and since L/K is a Galois extension, the fixed field of G is K itself. In other words, the center of A is K. Now, by Theorem

10, we may consider both f and g normalized and identify L with Lu1. Let uσ ∈ A is an element which commutes with each x ∈ L, then uσ must lie in L. This means that L is its own centralizer in A, and by Theorem 8(ii), L is a maximal subﬁeld of A. Now, we prove that A is a simple K-algebra. For this, let I , 0 be a two-sided ideal of A. Then, there exists an element 0 , x ∈ I, of the form

x = lσ1 uσ1 + ... + lσr uσr , 2.2. GALOIS COHOMOLOGY 23 with minimal r. Let us suppose that r > 1. We can choose an element l ∈ L, such that

σ1 (l) , σ2 (l) ,

−1 since σ1 , σ2. Then the element x − σ1 (l) xl is a nonzero element of I, and is shorter than x. This is impossible, because of the choice of x and so r = 1; but this means that I contains the invertible element lσ1 uσ1 of A, and so we obtain that necessarily I = A, which shows that A is simple. L About the second part of the theorem, let B = Lvσ , where {vσ } multiply according to the factor set g. Let φ : A −→ B be a K-algebra isomorphism. Then φ has to preserve the identity element, thus

φ (u1) = v1.

Therefore 0 φ (Lu1) = L v1, where L0 is a ﬁeld which is K-isomorphic to L. If

0 uσ = φ (uσ ) , σ ∈ G, then 0 0 φ (Luσ ) = L v1uσ = 0 0 = L uσ . L 0 0 Thus B = σ∈G L uσ , and

0 0 uσ φ (x) = φ (σx) uσ ,

0 0 0 uσ uτ = φ (f (σ, τ)) uστ , 0 for x ∈ L, and σ, τ ∈ G. On the other hand, we have that L v1 and Lv1 are K-isomorphic simple subalgebras of B, so by Skolem-Noether Theorem, there exists an inner automorphism θ of B such that

θ (φ (x)) = x, x ∈ L.

We apply the automorphism θ to the preceding equations and we also put

0 wσ = θ uσ , σ ∈ G. 24 CHAPTER 2. GALOIS COHOMOLOGY

L Thus we obtain B = Lwσ ,

wσ x = σ (x) wσ ,

wσ wτ = fσ,τwστ.

−1 But wσ vσ commutes with each x ∈ L, for all σ ∈ G and we remember that

L is its own centralizer in B. Thus wσ = cσ vσ , for some cσ ∈ L. Also, the set of elements {wσ } forms an L-basis for B,since the set of elements {vσ } forms an L-basis for B, so necessarily every cσ , 0. Finally, we have to prove that f is equivalent to g. We remind the equalities

wσ wτ = f (σ, τ) wστ, vσ vτ = g (σ, τ) vστ and wσ = cσ vσ .

So,

wσ wτ = (cσ vσ )(cτvτ) = cσ (vσ cτ) vτ = cσ (σ (cτ) vσ ) vτ = cσ σ (cτ)(vσ vτ) =

= cσ σ (cτ) g (σ, τ) vστ.

But also, since wσ = cσ vσ , then wστ = cστvστ. Thus we obtain that

f (σ, τ) cστvστ = wσ wτ = cσ σ (cτ) g (σ, τ) vστ.

−1 Multiplying by vστ on both right sides, we get the desired result that f = (δc) g, which exactly means that f is equivalent to g, and so A B.

The preceding theorem is one of the most classical results in the study of crossed products. Its importance arises from the assertion that crossed products not only are central simple algebras, but also we can choose the maximal subﬁelds of a central simple K-algebra to be a Galois extension of K.

Corollary 5 Let n = (L : K). Then, the crossed product algebra (L/K, f ) is isomorphic to Mn (K) if and only if f is equivalent to the trivial factor set.

Proof. Let f : G × G −→ L* be equivalent to the trivial factor set. Then, L by Theorem 11, (L/K, f ) (L/K, 1), and let (L/K, 1) σ∈G Luσ , where uσ x = σ (x) uσ , and uσ uτ = 1uστ = uστ, for x ∈ L, and σ, τ ∈ G. If x ∈ L, let xL be the left multiplication by x, where this action is considered on the ﬁeld 2.2. GALOIS COHOMOLOGY 25

L. We deﬁne the map φ : (L/K, 1) −→ EndK (L), by the rule xuσ 7→ xL ◦ σ. Then,

φ (xuσ )(yuτ) = φ (xσ (y) uστ) = (xσ (y))L ◦ στ = xL σ (y)L ◦ σ ◦ τ = xL ◦ σ (yL ◦ τ) =

= φ (xuσ ) φ (yuτ) , hence φ is a homomorphism. Moreover, the kernel of φ is a two-sided ideal of (L/K, 1), and since (L/K, 1) is a simple algebra, we get that Ker φ = {0}, that is, φ is a monomorphism. Finally, ((L/K, 1) : K) = (L : K)2 = 2 2 n . Also, EndK (L) MN (K), therefore, dimK EndK (L) = dimK Mn (K) = n . Hence, (L/K, 1) and EndK (L) have the same K-dimension. Consequently φ is an isomorphism, and (L/K, 1) EndK (L) Mn (K). Next, we prove that if (L/K, f ) M (K), then f is equivalent to the trivial factor set. Let L n ∈ (L/K, f ) = σ∈G Luσ , where uσ x = σ (x) uσ , and uσ uτ = f (σ, τ) uστ, for x L, σ, τ ∈ G. Since (L/K, 1) Mn (K), the map φ deﬁned as above, must be a homorphism. We have

φ ((xuσ )(yuσ )) = φ (xσ (y) f (σ, τ) uστ) = f (σ, τ) φ (xσ (y) uστ) =

= f (σ, τ) φ (xuσ ) φ (yuτ) , therefore f (σ, τ) = 1, for each σ, τ ∈ G, that is, f is the trivial factor set.

Theorem 12 Let f, g be factor sets from G to L*. Then

[(L/K, f ) ⊗K (L/K, g)] = [(L/K, fg)], in B (L/K). L L Proof. Let A = (L/K, f ) = Luσ and B = (L/K, g) = Lvσ . We consider the K-algebra A ⊗K B. Since both A and B are central simple K-algebras, by Theorem 2, A ⊗K B is a central simple K-algebra, thus we can write

A ⊗K B Mr (D) , for some r and a central division K-algebra D. Let e be an idempotent element of A ⊗K B, and let V be a left D-vector space. Then

V = eV ⊕ (1 − e) V 26 CHAPTER 2. GALOIS COHOMOLOGY is a D-decomposition of V . This means that for a suitable D-basis of V , e is represented by a diagonal matrix of the form

diag (1,..., 1, 0 ..., 0) , where the number of 1’s that occur in this representation is s. So, we obtain that

e (A ⊗K B) e Ms (D) ∼ D.

And also obviously D ∼ A ⊗K B. Therefore

e (A ⊗K B) e ∼ A ⊗K B.

What we need to do now, is find an idempotent element e ∈ L ⊗K L, such that e (A ⊗K B) e (L/K, fg). At this point we remark that L ⊗K L ⊂ A ⊗K B, and also that the subfields of L⊗K L, L⊗K 1 and 1⊗K L commute elementwise. Further, L is a Galois extension of K, thus it is a separable extension of K, so there exists an element a ∈ L, such that L = K (a). Let n = (L : K), then the minimal polynomial of a in K has degree n, i.e. deg min. pol.K (a) = n. We define the element Y Y e = (a ⊗ 1 − 1 ⊗ σ (a)) / (a − σ (a)) ⊗ 1. σ∈G−{1} σ∈G−{1} Q Because σ , 1G, a , σ (a), thus a − σ (a) , 0, so (a − σ (a)) ⊗ 1 , 0, that is, the denominator is not zero. Also, the elements {ai ⊗ 1 : 1 ≤ i ≤ n − 1} are linearly independent over L ⊗K 1, thus the numerator is not zero in L ⊗K L. Therefore e , 0 in L ⊗K L. Also, it is well known that Y f (X) = min. pol.K (a) = (X − a) (X − σ (a)) . σ∈G−{1}

Putting a ⊗ 1 instead of X in f (X), and identifying a with 1 ⊗ a, we obtain Y f (a ⊗ 1) = (a ⊗ 1 − 1 ⊗ a) (a ⊗ 1 − 1 ⊗ σ (a)) , σ∈G−{1} therefore Y (a ⊗ 1 − 1 ⊗ a) (a ⊗ 1 − 1 ⊗ σ (a)) − f (a ⊗ 1) = 0. σ∈G−{1} 2.2. GALOIS COHOMOLOGY 27

This means that (a ⊗ a) e = (a ⊗ 1) e in L ⊗K L. We shall prove by induction on i that (ai ⊗ 1) e = (a ⊗ ai ) e. We assume that this holds true for i = k. So ak ⊗ 1 e = a ⊗ ak e. Then, ak+1 ⊗ 1 e = aak ⊗ 1 e = a ak ⊗ 1 e = a a ⊗ ak e = a ⊗ ak+1 e,

Ln−1 i ⊗ as desired. Now, L = i=0 Ka , and multiplication in L K L is commutative. Thus

(x ⊗ 1) e = e (x ⊗ 1) = e (1 ⊗ x) = (1 ⊗ x) e, x ∈ L.

Consequently Y Y e2 = e (a ⊗ 1 − a ⊗ σ (a)) / (a − σ (a)) ⊗ 1 = σ∈G−{1} σ∈G−{1} Y Y = e (a − σ (a)) ⊗ 1/ (a − σ (a)) ⊗ 1 = e. σ∈G−{1} σ∈G−{1}

Thus e is an idempotent in L ⊗K L. It remains for us to prove that e (A ⊗K B) e (L/K, fg). We have

e (A ⊗K B) e = M = e ((Luσ ) ⊗K (Lvτ)) e = σ,τ∈G M = e (L ⊗K L)(uσ ⊗ vτ) e = σ,τ∈G M = e (L ⊗K 1) e · e (1 ⊗K L) e · e (uσ ⊗ vτ) e. σ,τ∈G 0 0 But e (1 ⊗K L) e = e (L ⊗K 1) e, and e (L ⊗K 1) e = L , where L is a ﬁeld which is K-isomorphic to L. We make use of the formulas

uσ x = σ (x) uσ and vτx = τ (x) vτ, x ∈ L, σ, τ ∈ G, to compute e (uσ ⊗ vτ) e. So

e (uσ ⊗ vτ) e =    Y Y  =  (σ (a) ⊗ 1 − 1 ⊗ τρ (a)) / (σ (a) − σρ (a)) ⊗ 1 (u ⊗ v ) e =   σ τ ρ∈G−{1} ρ∈G−{1} 28 CHAPTER 2. GALOIS COHOMOLOGY    Y Y  = (u ⊗ v )  (σ (a) ⊗ 1 − 1 ⊗ τρ (a)) / (σ (a) − σρ (a)) ⊗ 1 e. σ τ   ρ∈G−{1} ρ∈G−{1} And since (x ⊗ 1) e = e (x ⊗ 1) = e (1 ⊗ x) = (1 ⊗ x) e, x ∈ L, we obtain that Y Y e (uσ ⊗ vτ) e = (uσ ⊗ vτ) e (σ (a) − τρ (a)) ⊗ 1/ (σ (a) − σρ (a)) ⊗ 1. ρ∈G−{1} ρ∈G−{1}

If σ , τ, we may put ρ = τ−1σ ∈ G − {1}. Then we have that

σ (a) − τρ (a) = σ (a) − ττ−1σ (a) = 0.

So e (uσ ⊗ vτ) e = 0. On the other hand, when σ = τ we obtain

e (uσ ⊗ vσ ) e = (uσ ⊗ vσ ) e.

In the same way, we prove that e (u ⊗ v ) e = e (u ⊗ v ). Thus we have L σ σ σ σ ⊗ 0 shown that e (A K B) e = σ∈G L wσ , where 0 L = e (L ⊗ 1) e, and wσ = e (uσ ⊗ vσ ) e,

σ ∈ G. We have also shown that wσ = (uσ ⊗ vσ ) e = e (uσ ⊗ vσ ), σ ∈ G. Clearly, due to the fact that L and L0 are K-isomorphic ﬁelds, we get the fact that Gal (L0/K) Gal (L/K) = G. Now, for x ∈ L,

wσ · e (x ⊗ 1) e = e (uσ ⊗ vσ )(x ⊗ 1) e = e (σx ⊗ 1) e · wσ .

0 This means that wσ acts as σ on L . Further, for σ, τ ∈ G,

wσ wτ = e (uσ uτ ⊗ vσ vτ) e = e (f (σ, τ) uστ ⊗ g (σ, τ) vστ) e =

= e ((f (σ, τ) g (σ, τ) ⊗ 1) uστvστ) e = e (f (σ, τ) g (σ, τ) ⊗ 1) ewστ. Therefore the w’s multiply according to the factor set fg. Consequently,

0 e (A ⊗K B) e = L /K, fg ,

0 and since e (A ⊗K B) ∼ A ⊗K B, and L L, we obtain that

A ⊗K B ∼ (L/K, fg) , that is, [(L/K, f ) ⊗K (L/K, g)] = [(L/K, fg)] in B (L/K). 2.2. GALOIS COHOMOLOGY 29

Example 3 Let K = Q,L = Q (i) , i2 = −1. Then G = Gal (L/K) is cyclic of order 2, with generator σ, where

σ (a + bi) = a − bi, a, b ∈ Q. Let f : G × G −→ L* be the factor set given by

f (1, 1) = f (σ, 1) = f (1, σ) = 1, f (σ, σ) = −1.

Then (L/K, f ) = Lu1 ⊕ Luσ , where u1 is the unity element of (L/K, f ), and where

uσ x = σ (x) uσ , x ∈ L 2 uσ = uσ uσ = f (σ, σ) uσ2 = −1. Thus (L/K, f ) is isomorphic to the quaternion algebra

(L/K, f ) Q ⊕ Qi ⊕ Qj ⊕ Qk over Q, by identifying Q ⊕ Qi with L, and j with uσ . On the other hand, if we had chosen f normalized, with f (σ, σ) = +1, then by Corollary 5,

(L/K, f ) M2 (K) , since in this case, f is the trivial factor set.

Theorem 13 (Auslander-Goldman) Let L be a ﬁnite Galois extension of K, with Galois group G. Then H 2(G, L*) B (L/K), and the isomorphism is given by [f ] ∈ H 2 G, L* 7→ [(L/K, f )] ∈ B (L/K) .

Proof. This map is well defined, by Theorem 11, and is a group monomorphism by Theorem 12. Thus we need to show that it is also an epimorphism. Let B be a central simple K-algebra, which is split by the field L. By Theorem 9, there exists a central simple K-algebra A, such that [B] = [A] in B (L/K), and L is a self-centralizing maximal subfield of A. Also, if (L : K) = n, then (A : K) = n2. Now, let σ ∈ G. Then, the map σ : L −→ L is obviously a K-isomorphism. And since L is a simple subalgebra of A, the conditions of Skolem-Noether Theorem are satisfied. Thus, there exists u ∈ u (A), such that u xu −1 = σ (x), x ∈ L. What we need to prove now, σ L σ σ is that A = σ∈G Luσ . At first, we need to show that the expression on the 30 CHAPTER 2. GALOIS COHOMOLOGY right side is indeed a direct sum. We assume the opposite, that this sum is not direct. Let

s = aσ1 uσ1 + ... + aσk uσk = 0,

* aσi ∈ L , be a relation with minimal k. Since uσ is a unit in A, for every * σ ∈ G, then k > 1. Let b ∈ L be such an element that σ1 (b) , σ2 (b). Then σ1 (b) s−sb = σ1 (b) aσ1 uσ1 + ... + σ1 (b) aσk uσk − aσ1 uσ1 b + ... + aσk uσk b = 0.

Thus σ (b)s − sb = 0 gives a shorter relation concerning u , . . . , u , which 1 σ1 L σk is a contradiction. Thus, indeed, the sum is direct and A = σ∈G Luσ . At this point, we make use of the formula uσ · x = σ (x) · uσ , where x ∈ L, and −1 σ ∈ G. Let σ, τ ∈ G, and form uσ uτuστ . If x ∈ L, then

−1 −1 −1 −1 −1 −1 −1 uσ uτuστ x = uσ uτ uστ (x) = uσ uτ (στ) (x) uστ = uσ ττ σ (x) uτuστ =

−1 −1 −1 = uσ σ (x) uτuστ = x uσ uτuστ .

−1 Thus uσ uτuστ commutes with each x ∈ L. But L is its own centralizer in A, so

uσ uτ = f (σ, τ) uστ, f (σ, τ) ∈ L*. Multiplication in A is associative, hence f : G × G −→ L* is a factor set. Consequently, A (L/K, f ), for some f .

The preceding theorem shows that every class [f ] of H 2(G, L*) is isomorphic to a class [(L/K, f )] of the Brauer group B(L/K). But since every crossed product algebra of this form is a central simple K-algebra, then by Theorem 1, (L/K, f ) Mr (D), for some r and some central division K- algebra D. Therefore, [(L/K, f )] = [Mr (D)] = [D]. In other words, for every central division K-algebra D, we can always ﬁnd a crossed product algebra, which is isomorphic to D. However, Amitsur proved in 1972 that if D is an arbitrary division algebra, not necessarily K-central simple, then this assertion is not true, because there exist division algebras which are not isomorphic a crossed product.

Let L/K be a Galois extension and E/K an arbitrary extension. Let EL be the composite of the fields E and L. Then EL is an extension field of K, 2.2. GALOIS COHOMOLOGY 31 which contains both L and E. Let F = E ∩ L. By Galois theory, we know that EL/E is a finite Galois extension, and we have a diagram

L E

H = Gal (EL/E) Gal (L/F) ⊂ Gal (L/K) = G. This means that the factor set f : G × G −→ L* restricts to a factor set 0 f : H × H −→ L*. The map res : H 2 G, L* −→ H 2 H, L* is called the restriction of f to f 0. The restriction map gives us the ability to construct the crossed products (L/F, f 0) and (EL/E, f 0). The next theorem describes the change of ﬁelds in crossed product algebras.

Theorem 14 (Hasse) Let L/K be a ﬁnite Galois extension, and E/K an ar- 0 0 bitrary extension. Then E ⊗K (L/K, f ) ∼ (EL/E, f ), where f is obtained from f by restriction. P Proof. Let C(L/K,f ) (F) be the centralizer of F in (L/K, f ). Let σ∈G aσ uσ ∈ C(L/K,f ) (F), aσ ∈ L. Then, for all x ∈ F, X X X x aσ uσ = aσ uσ x = aσ σ (x) uσ ⇒ xaσ = aσ σ(x), for all σ ∈ G. If aσ , 0, then σ (x) has to be equal to x for all x ∈ F. This means that σ ∈ H = Gal (L/F). Further,

M 0 C(L/K,f ) (F) = Luσ = L/F, f . σ∈H Now, by Corollary 3, we have that

F ⊗K (L/K, f ) ∼ C(L/K,f ) (F) .

0 Since we have shown that C(L/K,f ) (L/F, f ), we get the result that 0 F ⊗K (L/K, f ) ∼ L/F, f . 32 CHAPTER 2. GALOIS COHOMOLOGY

Next, we consider the F-algebra (L/F, f 0), and we deﬁne the homomorphism of F-algebras

φ : E ⊗F L −→ EL, by the rule x ⊗ y 7→ xy. We see that Im φ = {φ (x ⊗ y) : x ∈ E, y ∈ L} = = {xy : x ∈ E, y ∈ L} = EL, thus this homomorphism is an epimorphism. Moreover

dimE E ⊗F L = (L : F) = dimE EL, thus φ is an isomorphism. Consequently,

0 M M E ⊗F L/F, f = (E ⊗F L)(1 ⊗ uσ ) ELvσ , σ∈H σ∈Gal(EL/E)

0 where the set of symbols {vσ } multiply according to the factor set f . So, we get that

0 0 E ⊗K (L/K, f ) E ⊗F (F ⊗K (L/K, f )) ∼ E ⊗F L/F, f ∼ EL/E, f , as desired.

If L/K is a finite Galois extension, and F is an intermediate field, let G = Gal (L/K) ,H = Gal (L/F). Of course, H is a subgroup of G, thus we may define the homomorphism res : H 2 G, L* −→ H 2 H, L* .

But also, there exists a homomorphism F ⊗K · : B (L/K) −→ B (L/F). 0 The relation F ⊗K (L/K, f ) ∼ (L/F, f ) is equivalent to the assertion that the following diagram commutes:

res H 2 G, L* / H 2 H, L*

F⊗ · B (L/K) K / B (L/F) .

Finally, we give a few details about another variant, the exponent of a class in Brauer group, which along with the index, plays an important role in the study of algebras. 2.2. GALOIS COHOMOLOGY 33

Deﬁnition 7 For [A] ∈ B (K), let exp[A] denote the exponent of [A] in B (K), that is, the least positive integer t such that

[A]t = 1 in B (K). On the other hand, deﬁne the index of [A] to be the index of the division algebra part of A. This means that for [A] ∈ B (K), p ind[A] = ind[D] = (D : K), where D is a central division K-algebra, such that A ∼ D.

Theorem 15 (Schur) For every [A] ∈ B (K), we have

[A]ind[A] = 1 in B (K). Therefore exp[A] divides ind[A].

Proof. Let A be a central simple√ K-algebra, and D be a central division K- algebra, with index ind D = (D : K) = m, such that A ∼ D. Then [A] = [D] in B (K), and by Theorem 9, there exists a ﬁnite Galois extension ﬁeld L of K, such that L splits A. And since A ∼ D, then L also splits D. Now, let V be a simple right L ⊗K D-module, with, say D-dimension r. Then, from the proof of Theorem 8, there exists an embedding

L ⊂ EndD (V ) Mr (D) , and (L : K) = mr. But we may also view V as a left Mr (D)-module, and we have (r) Mr (D) V as left Mr (D)-modules. Thus

r (V : L) = (Mr (D) : L) = (Mr (D) : K) / (L : K) = mr, which gives us (V : L) = m. Next, L is a self-centralizing maximal subﬁeld of the central simple K-algebra Mr (D), and by Theorem 13, there exists a * factor set f : G × G −→ L , such that Mr (D) (L/K, f ). Thus, passing at the Brauer group B (K), we obtain the result that

m m m m [A] = [Mr (D)] = [(L/K, f )] = [(L/K, f )]. 34 CHAPTER 2. GALOIS COHOMOLOGY

So, our aim is to prove that f m is a principal factor set. Let G = Gal (L/K).

Every element σ ∈ G, determines an element uσ ∈ (L/K, f ), as we already know. But since Mr (D) (L/K, f ), then every σ ∈ G, determines an element uσ ∈ Mr (D). Since V is a left MR (D)-module, then we can assume that uσ acts on V . If we also remember that (V : L) = m, then the action of uσ on (σ) an L-basis of V , is given by an element P ∈ Mm (L). In other words, we deﬁne a mapping

G −→ Mm (L) , by the rule σ 7→ P(σ). Lm ∈ Now, we may write V = i=1 Lvi . Let σ G, and let us put

X (σ) uσ vj = pij vi , i

(σ) (σ) P = pij ∈ Mm (L) . Then   X  X (u u ) v = u  p(τ)v  = σ p(τ) · p(σ)v . σ τ j σ  kj k kj ik i k i,k But also X (στ) (uσ uτ) vj = (f (σ, τ) uστ) vj = f (σ, τ) · pij vi . i Thus X (στ) X (τ) (σ) f (σ, τ) · pij vi = σ pkj · pik vi , i i,k which gives us X (στ) X (τ) (σ) f (σ, τ) · pij = σ pkj · pik . i i,k

(σ) (σ) But since P = pij , we get f (σ, τ) P(στ) = P(σ) · σ P(τ) , where σ P(τ) is obtained from P(τ) by applying σ to each of its entries. Next, (στ) (σ) (τ) (σ) * we take determinants in f (σ, τ) P = P · σ P . Let cσ = det P ∈ L . Then we obtain m (f (σ, τ)) cστ = cσ σ (cτ) , 2.2. GALOIS COHOMOLOGY 35

σ, τ ∈ G, that is,

m −1 (f (σ, τ)) = cσ σ (cτ) cστ ,

σ, τ ∈ G. Therefore f m = δc, that is, f m is a principal factor set, and so [A]m = [B]m = [(L/K, f m )] = 1 in B (K).

The technique of proving that f m is a principal factor set is due to Schur, who ﬁrst proved this theorem. The next theorem gives us useful information about the factorization in exp[A] and ind[A].

Theorem 16 (Brauer) For each [A] ∈ B (K), the integers exp[A] and ind[A] have the same prime factors, apart from multiplicities.

Proof. Since every central simple K-algebra A is equivalent to a central division K-algebra D, thus [A] = [D] in B (K), and hence they have the same exponent and index, we can make the proof for D. So, let D be a central division K-algebra, with index ind[D] = m. By Theorem 15, exp[D] divides m. So, we prove that every rational prime p which divides m, also divides exp[D]. Let p be a rational prime. We assume that p divides m, but p does not divide exp[D], and this will lead us to a contradiction. By Theorem 9, there exists a ﬁnite Galois extension L/K such that L splits D. Let G = Gal (L/K), and let H by a Sylow p-subgroup of G, then

|H| = pr , r ≥ 0, and [G : H] = q, with p - q.

Now, let E be a subﬁeld of L, which is ﬁxed under the action of the group H. Then

(E : K) = [G : H] = q.

Since m does not divide q, by Theorem 8(i), E does not split D. We form the E-algebra E ⊗K D. Then, E is a central simple E-algebra, and because of the fact that E does not split D, we get that

[E ⊗K D] , 1 36 CHAPTER 2. GALOIS COHOMOLOGY in B (E). We consider the diagram

r p E ⊗K D m2

q D m2

We deﬁne the homomorphism

E ⊗K · : B (K) −→ B (E) , [D] 7→ [E ⊗K D].

Let exp[D] = t; this means that [D]t = 1 in B(K). Then we have

t t t t [E ⊗K D] = [E] [D] = [E] [K] = [K], which means that exp[E ⊗K D] divides t = exp[D]. And because p does not divide exp[D], we get that p does not divide exp[E ⊗K D]. On the other hand,

L ⊗E (E ⊗K D) L ⊗K D Mm (L) , since L splits D. This means that L also splits E ⊗K D, so ind[E ⊗K D] divides

(L : E), by Theorem 8(i). Hence ind[E ⊗K D] is a power of p. But we know that exp[E ⊗K D] divides ind[E ⊗K D], by Theorem 15, so exp[E ⊗K D] is also a power of p, which is a contradiction, since we have proved that p does not divide exp[E ⊗K D]. Therefore

[E ⊗K D] = 1 in B (E) , which is a contradiction to our hypothesis. Consequently, if p is a rational prime which divides ind[D], then p also divides exp[D], and our theorem is proved. 2.3. CYCLIC ALGEBRAS AND DIVISION ALGEBRAS OF SMALL DEGREES37

2.3 Cyclic Algebras and Division Algebras of Small Degrees

Let A = (L/K, f ) be a crossed product algebra, and assume further that the Galois group G = Gal (L/K) is a cyclic group generated by the element σ of order n. As usual, K is an algebraic number ﬁeld. As we know, Ln i j i i j i+j ≤ ≤ A = i=1 Luσ , where uσ l = σ (l) uσ and uσ uσ = f σ , σ uσ , for 1 i, j n and l ∈ L. We remark that

2 uσ uσ = f (σ, σ) uσ2 ⇒ uσ = f (σ, σ) uσ2 ,

3 2 2 uσ = uσ uσ = f (σ, σ) uσ2 uσ = f (σ, σ) f σ .σ uσ3 , and so on n n−1 2 n−1 uσ = uσ uσ = f (σ, σ) f σ , σ · f σ , σ uσn ,

Ln i n where uσ = 1. Hence we see that we can write A = i=1 Luσ , with multiplication i j i+j uσ uσ = uσ , for i + j < n, and for i + j > n, let i + j = n + k, then

i j n+k n k 2 n−1 k uσ uσ = uσ = uσ uσ = f (σ, σ) f σ , σ ··· f σ , σ uσ .

− Let us denote a = f (σ, σ) f σ2, σ ··· f σn 1, σ , then the crossed product Ln i (L/K, f ) is isomorphic to the crossed product i=1 Luσ , with factor set g : hσi × hσi −→ L*, given by   1, for i + j ≤ n g σi , σj =  , 1 ≤ i, j, ≤ n.  a, for i + j > n

* n+1 We remark that a ∈ K , since uσn uσ = uσ = uσ u1.

Deﬁnition 8 The crossed product K-algebra A = (L/K, f ) for a normalized factor set f , for which the Galois extension L/K is cyclic is called cyclic algebra, and is denoted by (L/K, σ, a).

From the above discussion we get the next result. 38 CHAPTER 2. GALOIS COHOMOLOGY

Theorem 17 Let G = Gal (L/K) = hσi be a cyclic group of order n, and let A = (L/K, f ) be a crossed product algebra, where f : G × G −→ L* is a normalized factor set. Then

(L/K, f ) (L/K, σ, a) , where Yn a = f (σi , σ) ∈ K*. i=1

The next result describes the main properties of the cyclic algebras. For a proof we refer to [Re, Theorem (30.4) and Corollary (30.7)].

Theorem 18 Let G = Gal(L/K) = hσi be a cyclic group of order n, and a, b ∈ K* := K\{0}. Then (i) (L/K, σ, a) (L/K, σs, as) for s ∈ Z and (s, n) = 1. (ii) (L/K, σ, 1) Mn(K). * (iii) (L/K, σ, a) (L/K, σ, b) if and only if b = NL/K (c) a, for some c ∈ L . In * particular (L/K, σ, a) ∼ K if and only if a ∈ NL/K (L ).

(iv) (L/K, σ, a) ⊗K (L/K, σ, b) ∼ (L/K, σ, ab). t * (v) The exp[(L/K, σ, a)] is the least positive integer t such tht a ∈ NL/K (L ). If exp[(L/K, σ, a)] = (L : K), then (L/K, σ, a) is a division algebra.

A division algebra of degree two is necessarily cyclic because separable quadratic extensions are always cyclic. We now prove that every division algebra of degree three is cyclic. This result was ﬁrst established by Wed- derburn in 1921 [W].

Theorem 19 If D is a division algebra of degree three, then D is cyclic.

Proof. Let D be a division algebra of degree 3 over the field K. D is cyclic if and only if there exists an element a ∈ K, such that K(a1/3) (for a proof of this we refer to [Pi, Corollary 15.5]). In other words, we have to show that D has a splitting field of the form K(1/3), where a ∈ K − K3. Let F be a maximal separable subfield of D. We can always find a separable extension of K, thus indeed, there exists a field F, which is the maximal element of all separable extension of K, that is, F is a maximal separable subfield of D. Then, the field F is a maximal subfield of D (for a proof we refer to [Pi, 2.3. CYCLIC ALGEBRAS AND DIVISION ALGEBRAS OF SMALL DEGREES39

Proposition 13.5]), and (F : K) = 3. Moreover, because F/K is a separable extension, there exists an element v ∈ D, such that

F = K (v) .

Furthermore, it is know by Galois theory that we can choose the element v so that its minimal polynomial over K is of the form

3 f (X) = min. pol.K (v) = x + b1x + b.

Let E be a splitting field of f over K, with F ⊆ E. We define the Galois group G = Gal (E/K). If G is abelian, then F/K is cyclic, and from the definition of a cyclic algebra, our proof is finished. Thus, we may assume that G is not abelian. Then (E : K) = 6, and G is the group of all permutations of the three roots of f . Now, let

Gal (E/F) = hτi .

If σ is the generator of the normal Sylow 3-subgroup of G, then

G = hσ, τi , where τ has order 2, σ has order 3, and

τστ = σ2.

Let L be the ﬁxed ﬁeld of σ. Then L/K is a Galois extension of degree 2, and we get that

Gal (L/K) = hτ |L i and Gal (E/L) = hσi .

E = L (v)

F = K (v)

K 40 CHAPTER 2. GALOIS COHOMOLOGY

Therefore, v ∈ E − L, so E = L (v). Also, the three roots of f are v, σ (v), and σ2 (v). In particular,

v + σ (v) + σ2 (v) = 0 and

v · σ (v) · σ2 (v) = b ∈ K.

Moreover, the algebra D ⊗K L is a cyclic L-algebra, because it contains the maximal subﬁeld E = L (v), and E/L is cyclic, since its Galois group is generated by the element σ, with order 2. Thus,

2 D ⊗K L = E ⊕ Euσ ⊕ Euσ , where

uσ x = σ (x) uσ , for all x ∈ E.

Now, we deﬁne ρ = idD ⊗ (τ |L ) ∈ AutK (D ⊗K L). Since τ (v) = v, we get that

ρ (x) = (idD ⊗ (τ |L )) (1 ⊗ x) = 1 ⊗ τ (x) = τ (x) , for all x ∈ E. Hence

ρ (uσ ) v = ρ (uσ v) = ρ (σ (v) uσ ) = (id ⊗τ)(1 ⊗ σ (v) uσ ) =

= (id ⊗τ)(1 ⊗ σ (v)) (id ⊗τ)(1 ⊗ uσ ) = τσ (v) ρ (uσ ) , and since τστ = σ2 ⇒ τσ = σ2τ, we get

2 ρ (uσ ) v = σ (τ (v)) ρ (uσ ) , but also τ (v) = v, thus

2 ρ (uσ ) v = σ (v) ρ (uσ ) .

Moreover, for c ∈ L, we get

ρ (uσ ) c = ρ (uσ c) = ρ (σ (c) uσ ) = ρ (cuσ) = cρ (uσ ) ,

−1 since by Corollary 1, L = Z (D ⊗K L). We deﬁne the element wσ = uσ ρ (uσ ). We observe that, for x ∈ E,

−1 −1 −1 −1 −1 −1 wσ x = uσ ρ (uσ ) ρρ (x) = uσ ρ uσ ρ (x) = uσ ρ σ ρ (x) uσ = 2.3. CYCLIC ALGEBRAS AND DIVISION ALGEBRAS OF SMALL DEGREES41

−1 −1 −1 −1 −1 2 = uσ ρσρ (x) ρ (uσ ) = uσ τστ (x) ρ (uσ ) = uσ σ (x) ρ (uσ ) =

−1 2 −1 −1 = σ σ (x) uσ ρ (uσ ) = σ (x) uσ ρ (uσ ) = σ (x) wσ . Also, −1 −1 2 ρ (wσ ) = ρ uσ ρ (uσ ) = ρ uσ ρ (uσ ) ,

2 2 but since τ = idE, we also get that ρ = id, therefore

−1 −1 −1 −1 ρ (wσ ) = ρ (uσ ) uσ = uσ ρ (uσ ) = wσ .

Now, according to all the previous observations, we obtain that

−1 −1 2 −1 wσ v = uσ ρ (uσ ) v = uσ σ (v) ρ (uσ ) = σ (v) uσ ρ (uσ ) = σ (v) wσ , also

−1 −1 2 −1 3 −1 wσ σ (v) = uσ ρ (uσ ) σ (v) = uσ σ (σ (v)) ρ (uσ ) = σ σ (v) uσ ρ (uσ ) =

2 = σ (v) wσ , and in the same way, we get

2 −1 2 −1 2 2 wσ σ (v) = uσ ρ (uσ ) σ (v) = uσ σ σ (v) ρ (uσ ) , and since σ has order 3, we have

2 −1 −1 wσ σ (v) = uσ σ (v) ρ (uσ ) = vuσ ρ (uσ ) = vwσ .

Further, we know that 3 wσ = d ∈ L, where 3 3 −1 3 3 −1 −1 τ(d) = ρ(d) = ρ(wσ ) = ρ (wσ ) = (wσ ) = (wσ ) = d . We may further remark that

−1 wσ v = ρ (wσ ) v = ρ (wσ v) = ρ (σ (v) wσ ) = τσ (v) ρ (wσ ) =

2 −1 2 −1 = σ τ (v) wσ = σ (v) wσ . −1 At this point, we deﬁne the element z = v 1 + wσ + wσ . Then, we have

2 −1 −1 −1 −1 z = v 1 + wσ + wσ v 1 + wσ + wσ = v + vwσ + vwσ v + vwσ + vwσ = 42 CHAPTER 2. GALOIS COHOMOLOGY

2 2 2 −1 −1 −1 −1 −1 −1 = v +v wσ +v wσ +vwσ v+vwσ vwσ +vwσ vwσ +vwσ v+vwσ vwσ +vwσ vwσ = 2 2 2 −1 2 2 −1 2 2 −2 = v +v wσ +v wσ +vσ (v) wσ +vσ (v) wσ +vσ (v)+vσ (v) wσ +vσ (v)+vσ (v) wσ = 2 2 2 2 2 2 −1 2 −2 = v + vσ (v) + vσ 2 (v) + v + vσ (v) wσ +vσ (v) wσ + v + vσ (v) wσ +vσ (v) wσ =

2 2 2 −1 2 −2 = v v + σ (v) + σ (v) +v (v + σ (v)) wσ +vσ (v) wσ +v v + σ (v) wσ +vσ (v) wσ , and since v + σ (v) + σ2 (v) = 0, we get that

2 2 2 −1 2 −2 z = v (v + σ (v)) wσ + vσ (v) wσ + v v + σ (v) wσ + vσ (v) wσ .

Now,

3 2 2 −1 2 −2 −1 z = v (v + σ (v)) wσ + vσ (v) wσ + v v + σ (v) wσ + vσ (v) wσ v 1 + wσ + wσ =

2 2 −1 2 −2 −1 = v (v + σ (v)) wσ + vσ (v) wσ + v v + σ (v) wσ + vσ (v) wσ v + vwσ + vwσ =

−1 2 = v (v + σ (v)) wσ v + v (v + σ (v)) wσ vwσ + v (v + σ (v)) wσ vwσ + vσ (v) wσ v+ 2 2 −1 2 −1 2 −1 +vσ (v) wσ vwσ + vσ (v) wσ vwσ + v v + σ (v) wσ v + v v + σ (v) wσ vwσ +

2 −1 −1 2 −2 2 −2 2 −2 −1 +v v + σ (v) wσ vwσ + vσ (v) wσ v + vσ (v) wσ vwσ + vσ (v) wσ vwσ . We compute this relation in parts. Then

−1 v (v + σ (v)) wσ v + v (v + σ (v)) wσ vwσ + v (v + σ (v)) wσ vwσ =

2 = v (v + σ (v)) σ (v) wσ + v (v + σ (v)) σ (v) wσ + v (v + σ (v)) σ (v) = 2 = vσ (v)(v + σ (v)) wσ + vσ (v)(σ + σ (v)) wσ + vσ (v)(v + σ (v)) . But since v + σ (v) + σ2 (v) = 0, we have that v + σ (v) = −σ2 (v). Also, we remind that vσ (v) σ2 (v) = b. Thus

−1 v (v + σ (v)) wσ v + v (v + σ (v)) wσ vwσ + v (v + σ (v)) wσ vwσ =

2 2 2 2 = −vσ (v) σ (v) wσ − vσ (v) wσ − vσ (v) σ (v) = −bwσ − bwσ − b. In the same way

2 2 2 −1 vσ (v) wσ v + vσ (v) wσ vwσ + vσ (v) wσ vwσ =

−1 = vσ (v) wσ wσ v + vσ (v) wσ wσ vwσ + vσ (v) wσ wσ vwσ = 2 = vσ (v) wσ σ (v) wσ + vσ (v) wσ σ (v) wσ + vσ (v) wσ σ (v) = 2.3. CYCLIC ALGEBRAS AND DIVISION ALGEBRAS OF SMALL DEGREES43

2 2 2 3 2 = vσ (v) σ (v) wσ + vσ (v) σ (v) wσ + vσ (v) σ (v) wσ =

2 = bwσ + bd + bwσ ,

3 since wσ = d,

2 −1 2 −1 2 −1 −1 v v + σ (v) wσ v + v v + σ (v) wσ vwσ + v v + σ (v) wσ vwσ =

2 2 −1 2 2 2 2 −2 = v v + σ (v) σ (v) wσ + v v + σ (v) σ (v) + v v + σ (v) σ (v) wσ , and since v + σ (v) + σ2 (v) = 0, we take that v + σ2 (v) = −σ (v), thus

2 −1 2 −1 2 −1 −1 v v + σ (v) wσ v + v v + σ (v) wσ vwσ + v v + σ (v) wσ vwσ =

2 −1 2 2 −2 = −vσ (v) σ (v) wσ − vσ (v) σ (v) − vσ (v) σ (v) wσ =

−1 −2 = −bwσ − b − bwσ . Finally, 2 −2 2 −2 2 −2 −1 vσ (v) wσ v + vσ (v) wσ vwσ + vσ (v) wσ vwσ =

2 −1 −1 2 −1 −1 2 −1 −1 −1 = vσ (v) wσ wσ v + vσ (v) wσ wσ vwσ + vσ (v) wσ wσ vwσ =

2 −1 2 −1 2 −1 2 2 −1 2 −2 = vσ (v) wσ σ (v) wσ + vσ (v) wσ σ (v) + vσ (v) wσ σ (v) wσ =

2 −2 2 −1 2 −3 = vσ (v) σ (v) wσ + vσ (v) σ (v) wσ + vσ (v) σ (v) wσ =

2 4 −2 2 4 −1 2 4 −3 = vσ (v) σ (v) wσ + vσ (v) σ (v) wσ + vσ (v) σ (v) wσ =

−2 −1 −1 = bwσ + vwσ + bd , since σ3 = 1. Therefore

3 2 2 −1 −2 −2 −1 −1 z = −bwσ −bwσ −b+bwσ +bd +bwσ −bwσ −b−bwσ +bwσ +bwσ +bd = = bd + bd−1 − 2b = b d + d−1 − 2 ∈ K.

− If we set a = b d + d 1 − 2 , then the cubic polynomial X 3 −a is irreducible over L. If it is not irreducible, due to the fact that E/L is a Galois extension, we obtain that z ∈ E, and so wσ ∈ E. However, this is a contradiction 2 2 because of the relations wσ v = σ (v) wσ , wσ σ (v) = σ (v) wσ , wσ σ (v) = vwσ , and because the roots of f are distinct. 44 CHAPTER 2. GALOIS COHOMOLOGY

E = L (v)

L (z)

Therefore, the extension L (z) is a maximal subﬁeld of D ⊗K L, such that L (z) splits D ⊗ L. Since (L/K) = 2 and L splits D, then K (z) splits D. K Therefore, we found an element a ∈ K, such that K a1/3 = K (z) splits D. Consequently, D is cyclic.

Corollary 6 If D is a divison algebra of degree six, then D is cyclic.

Proof. Let D be a division algebra of degree six. Then, by Primary De- composition Theorem, D is of the form D = D1 ⊗K D2, where D1 and D2 are division algebras, with degrees deg D1 = 2 and deg D2 = 3. Hence, D1 s and D2 are both cyclic algebras. So, let D1 = (L/K, σ, a ) = Lu1 ⊕ Luσ and r 2 D2 = (F/K, τ, a ) = Fv1 ⊕ Fvτ ⊕ Fvτ , where r, s ∈ Z, so that 2r + 3s = 1. Then, M M i j k D = D1 ⊗K D2 = (L ⊗K F) uσ ⊗ uτ = (L ⊗K F)(uσ ⊗ vτ) , i<2,j<3 k<6 where

(uσ ⊗ vτ)(l ⊗ f ) = uσ l ⊗ vτf = σ (l) uσ ⊗ τ (f ) vτ = (σ (l) ⊗ τ (f )) (uσ ⊗ vτ) , and also 6 3s 2r 3s+2r (uσ ⊗ vτ) = a ⊗ a = (1 ⊗ 1) a = (1 ⊗ 1) a.

Thus D = D1 ⊗K D2 (L ⊗ F, (σ, τ) , a), that is, D is cyclic. Chapter 3

The Albert-Brauer-Hasse-Noether Theorem

The aim of this chapter is to prove the Albert-Brauer-Hasse-Noether The- orem for algebraic number ﬁelds.

3.1 Valuations and Completions

We refer to [Re] and [Pi] for the context of this section. Let us recall the main deﬁnitions and results. As usual, K is an algebraic number ﬁeld, and R is the ring of algebraic integers of K.

Deﬁnition 9 A valuation of K is a map φ : K −→ R+, the set of nonnegative real numbers, such that: (i) φ(a) = 0 if and only if a = 0. (ii) φ(ab) = φ(a)φ(b). (iii) φ(a + b) ≤ φ(a) + φ(b). If the valuation φ satisﬁes the condition (iv) φ(a + b) ≤ max{φ(a), φ(b)}, then it is called non-archimedean. A valuation φ is called archimedean if it is not non-archimedean.

If φ(0) = 0 and φ(a) = 1 for all a ∈ K\{0}, then φ is called trivial. The value group of the valuation φ is the multiplicative group {φ(a): a ∈ K−{0}}. If this group is inﬁnite cyclic, then φ is called discrete rank one valuation, and non-archimedean.

45 46 CHAPTER 3. THE ALBERT-BRAUER-HASSE-NOETHER THEOREM

In the set of valuations φ : K −→ R+ it is deﬁned an equivalence relation by the rule φ ∼ ψ if and only if φ(a) ≤ 1 ⇔ ψ(a) ≤ 1.

The ring Rφ = {a ∈ K : φ(a) ≤ 1} is called the valuation ring of φ, with a unique maximal ideal Pφ = {a ∈ K : φ(a) < 1}. Let us examine some certain valuation of K. For an element a ∈ K\{0}, let Y aR = P−vP (a) P be the expression of the principal ideal aR in K as a product of prime ideals of R. Let κ ∈ R+ be some ﬁxed element; then the map

+ −vP (a) φP : K −→ R , a 7→ κ , for a , 0 and φP (0) = 0, is a discrete rank one valuation, hence non- archimedean, called P-adic valuation. Moreover, another valuation is de-

ﬁned, the exponential valuation vP associated with P as follows.

vP : K −→ Z ∪ {∞}, a 7→ vP (a).

Then, we get

(i) vP (a) = ∞ if and only if a = 0.

(ii) vP (ab) = vP (a) + vP (b). (iii) vP (a + b) ≥ min{vP (a), vP (b)}, and if vP (a) , vP (b), then the equality holds in this relation.

We remark that the valuation ring of vP is the ring

RP := {r/s : r ∈ R, s ∈ R\P}, which is the localization of R at P, and RP has a unique maximal ideal

PRP . It is proved that RP is a discrete rank one valuation ring, and it is a principal ideal domain with a unique maximaL ideal PRP = πRP , for an element π ∈ RP called prime element of RP . Moreoever,

R/P RP /πRP is a finite field. All the non-archimedean valuations of K are arise from discrete P-adic valuations and their classes are called finite primes of K. 3.2. THE ALBERT-BRAUER-HASSE-NOETHER THEOREM 47

The archimedean valuations of an algebraic number ﬁeld K arise from embedding of K in C. That is, if µ : K −→ C is an embedding, then the map

φ : K −→ R+, a 7→ φ(a) = |µ(a)| is an archimedean valuation. Certainly, let K = Q(a) and f (X) = irrQ,a (X) = Qs i=1(X − ai ). Then, for g(X) ∈ Q[X], we deﬁne

µi : K −→ C, g(a) 7→ g(ai ).

If a1, a2, . . . , ar1 are the real roots of f (X), and ar1+1, ar1+1, . . . , ar1+r2 , ar1+r2 are the complex roots of f (X), for r1 + 2r2 = s, then there are r1 + r2 non- equivalent valuations φi , deﬁned by

φi (a) = ai , 1 ≤ i ≤ r1 + r2.

It is proved that all the archimedean valuations of K arise in this manner, and the equivalence classes of archimedean valuations of K are called inﬁnite primes of K. Any valuation φ : K −→ R+ deﬁnes a topology. The completion of K in accordance to this topology is denoted by Kˆφ. In particular, the completion of K in accordance to the P-adic topology is denoted by KP .

Let KP denote the completion of K relative to the P-adic valuation φP , and RˆP the valuation ring of KP . Then, the ﬁeld K is embedded in KP and the valuation φP is extended to a valuation φˆP on KP , which is also a non-archimedean valuation.

For any R-module M we can deﬁne its localization MP at P and its completion at P by the relation

MP := RP ⊗R M and MˆP := RˆP ⊗R M.

Similarly, if A is a K-algebra, then

AˆP := KP ⊗K A is deﬁned to be the completion of A at P.

3.2 The Albert-Brauer-Hasse-Noether Theorem

Let, now, A = Mr (D) be a central simple√ K-algebra with associated division algebra D, and index ind D = m = (D : K). Then, by Corollary 1, AˆP is a 48 CHAPTER 3. THE ALBERT-BRAUER-HASSE-NOETHER THEOREM

central simple KP -algebra. Also

KP ⊗K A KP ⊗K Mr (D) Mr (KP ⊗K D) , and KP ⊗K D is also by Corollary 1 a central simple KP -algebra. Thus, by 0 0 Theorem 1, KP ⊗K D Mt (D ), for some central division KP -algebra D . Then

0 AˆP Mrt D .

We denote by κP the number rt and we call it the local capacity of A at P, so ˆ 0 AP MκP D . 0 Moreover, we denote by mP the index of the division KP -algebra D , called the local index of A at P.

We remark that [AˆP ] = [KP ] in the Brauer group B(KP ) if and only if mP = 1. We say that the prime ideal P is ramiﬁed in A, if mP > 1. We, now, consider the map

B(K) −→ B(KP ), [A] 7→ [AˆP ].

From the deﬁnition of the Brauer group, we get that the above map is a group homomorphism with kernel

{[A] ∈ B(K):[AˆP ] = [KP ]}.

Thus it is easy for someone to see that

[A] = [K] ⇒ [AˆP ] = [KP ]}, for all the prime ideals P, since if A Mr (K), then KP ⊗K A Mr (KP ⊗K K) Mr (KP ). The Albert-Brauer-Hasse-Noether Theorem proves the inverse of the above relation. The cyclic algebras play a very important role in Theorem’s proof. Therefore we refer to some more of their properties. From the theory of maximal orders in central simple K-algebras it is know that there is only a ﬁnite number of prime ideals P for which mP = 1. For more about this, we refer to [Re, §22 §25 and (32.1)]. Let us consider a central division KP -algebra D, where KP is√ the completion of K in accordance to the P-adic valuation, and let m = (D : KP ) be the index of D. From the theory of KP -division algebras( see [Re, §14]), 3.2. THE ALBERT-BRAUER-HASSE-NOETHER THEOREM 49 we are going to express D as a cyclic algebra. For this, let |R/P| = q, and W be the unique unramiﬁed extension of KP of degree m. Then, W = KP (ω), m where ω is a primitive (q − 1)th root of unity over KP . The Galois group Gal(W/K) is cyclic of order m, generated by the map σ(ω) = ωq. We consider now a prime element π of RˆP and let πRˆP be the unique maximal ideal of RˆP . Then, W can be embedded in D and there exists an element z ∈ D, such that Mm−1 D = Wzj, j=1 where zaz−1 = σr (a), for a ∈ W, zm = π, and (r, m) = 1. Moreover, D determines r mod m uniquely. Furthermore, for each pair r, m with (r, m) = 1 we get a central division KP -algebra D. From the construction of cyclic algebras we choose an element s, such that

rs ≡ 1 mod m, so (s, m) = 1, and then

D (W/K, σr , π) (W/K, σrs, πs) (W/K, σ, πs).

Hence, there are φ(m) such numbers s, where φ is the Euler function and we conclude the following.

Theorem 20 The cyclic algebras

s {(W/KP , σ, π ): 1 ≤ s ≤ m, (s, m) = 1} is a full set of non-isomorphic central simple KP -algebras with index m.

The next theorem is very important for the proof of the Albert-Brauer- Hasse-Noether Theorem in the case where the algebra A is a cyclic algebra.

Theorem 21 (Hasse Norm Theorem) Let L be a ﬁnite cyclic extension of an algebraic number ﬁeld K, and a ∈ K. Then

a ∈ NL/K (L) ⇔ a ∈ NLQ/KP (LQ), for each prime P of K, where Q is a prime of L that divides P. 50 CHAPTER 3. THE ALBERT-BRAUER-HASSE-NOETHER THEOREM

Theorem 22 Let A = (L/K, σ, a) be a cyclic algebra, where Gal(L/K) = hσi and a ∈ K*. Then

A ∼ K if and only if AˆP ∼ KP , for all primes P of K.

Proof. Let P be a prime of K, KP the P-adic completion of K and LQ the Q-adic completion of L for a prime of L dividing P. Then

KP ⊗K L ⊕Q|P LQ, and the extension LQ/KP is a Galois extension with Galois group a subgroup of Gal(L/K). Hence Gal(LQ/KP ) is also cyclic and generated by a power of σ, say σk. Moreover

k AˆP KP ∼ (LQ/KP , σ , a)

(see [Re, Theorem (30.8)]). Now, by Theorem 18(iii) we get that

ˆ * AP ∼ KP ⇔ a ∈ NLQ/KP (LQ) and the result follows from the Hasse Norm Theorem, since

* k ˆ A ∼ K ⇔ a ∈ NL/K (a) ⇔ a ∈ NLQ/KP (LQ) ⇔ (LQ/KP , σ , a) ∼ KP ⇔ AP ∼ KP , for each prime P of K.

Now the Albert-Brauer-Hasse-Noether Theorem can be reduced to the case of cyclic algebras.

Theorem 23 (Albert-Brauer-Hasse-Noether) Let K be an algebraic number ﬁeld and A be a central simple K-algebra. Then

A ∼ K ⇔ AˆP ∼ KP , for each prime P of K.

Proof. We only need to prove that if AˆP ∼ KP , then A ∼ K. We assume that

AˆP ∼ KP for each P, but m = ind[A] > 1. If ind[A] > 1, then the central simple K-algebra A is not in the same equivalence class as K, therefore A and K are not equivalent. Therefore, we will obtain a contradiction. By 3.2. THE ALBERT-BRAUER-HASSE-NOETHER THEOREM 51

Theorem 9, there exists a ﬁnite Galois extension L of K, such that L splits

A, that is, L ⊗K A ∼ L. Let G = Gal (L/K), and let p be a rational prime dividing m. We may form the Sylow p-subgroup H of G. Then, p does not divide [G : H], thus the order of H, |H| is a power of p. Furthermore, we know that there exists a descending chain of subgroups of the group H,

H = H0 ⊃ H1 ⊃ ... ⊃ Hn = 1, where [Hi : Hi+1] = p, and Hi+1 /Hi , for 0 ≤ i ≤ n − 1. Now, we consider the subfields Ei of L, which are fixed by the action of the elements of the group Hi , for 1 ≤ i ≤ n. Since Hn = 1, we obtain that the fixed subfield of L which is fixed by Hn is L itself. Moreover, each Ei+1/Ei is a cyclic extension of degree p.

L = En

F = En−1

Furthermore, E0 is a subﬁeld of L, which is ﬁxed by H0 = H, thus

(E0 : K) = [G : H] . 0 (modp) .

If we set

F = En−1, then L/F is a cyclic extension of degree p. We consider the F-algebra

F ⊗K A. By Corollary 1, since A is a central simple K-algebra, and F ⊃ K, then F ⊗K A is a central simple F-algebra. Now, we have

L ⊗F (F ⊗K A)

(L⊗F ) ⊗K A

L ⊗K A. 52 CHAPTER 3. THE ALBERT-BRAUER-HASSE-NOETHER THEOREM

Since L splits A, then L ⊗K A ∼ L. Thus

L ⊗F (F ⊗K A) ∼ L, which means that L splits also F ⊗K A. Hence, from the proof of Theorem

13, we get that F ⊗K A is equivalent to some central simple K-algebra B split by L, such that (B : K) = (L : K)2. We, now consider the Galois group Gal (L/F), and let Gal (L/F) = hσi .

By Theorem 9, we may embed L in B as a self-centralizing maximal sub- ﬁeld of B. From the proof of Theorem 13, we obtain that B is isomorphic to a crossed product (L/F, f ), for some [f ] ∈ H 2 Gal (L/F) ,L* . Finally, since every crossed product is isomorphic to a cyclic algebra, we obtain by Theorem 17 that B (L/F, f ) (L/F, σ, a) , where σ is the generator of Gal (L/F), and a ∈ F * is the element that is deﬁned by |Gal(YL/F)|−1 a = f (σi , σ) . i=0

Since F ⊗K A B = (L/F, σ, a), instead of proving that F ⊗K A ∼ F, we shall prove that B ∼ F. By Theorem 22 we know that

B ∼ F if and only if BˆP ∼ FP , for every prime P of F. But, if P is a prime of F, then its restriction to K is a prime of K. Thus, we have that

ˆ [ ˆ BP ∼ (F ⊗K A)P = FP ⊗F (F ⊗K A) FP ⊗K A FP ⊗KP AP .

ˆ ˆ And since FP ⊗KP AP is a central simple KP -algebra, then FP ⊗KP AP ∼ FP . Thus BˆP ∼ FP . In other words, B splits locally everywhere, which means that B ∼ F. Thus, we have proved that F ⊗K A ∼ F, that is, F splits A.

If we repeat this argument, using the cyclic extension F/En−2 in place of

L/F, we ﬁnd that En−2 also splits A. If we continue in this manner until we reach E0, we will obtain that E0 splits A. Therefore, by Theorem 8(i), m divides (E0 : K). But, in our hypothesis, p divides m, and p does not divide 3.2. THE ALBERT-BRAUER-HASSE-NOETHER THEOREM 53

(E0 : K), thus the assertion that m divides (E0 : K) is a contradiction. Con- sequently, ind[A] = 1, and A ∼ K.

For each prime P of K, there is a homomorphism

B (K) −→ B (KP ) , by the rule KP ⊗K ·.

Let [A] ∈ B (K), and let mP be the local index of A at P. Let [A] ∈ B (K), and let mP be the local index of A at P. Then mP = 1 almost everywhere, which means that [AˆP ] = 1 almost everywhere. Hence there is a well deﬁned homomorphism M B (K) −→ B (KP ) . P The Albert-Brauer-Hasse-Noether theorem asserts that the last deﬁned homomorphism is a monomorphism. 54 CHAPTER 3. THE ALBERT-BRAUER-HASSE-NOETHER THEOREM Chapter 4

The General Case

The aim of this chapter is to present concepts of homological algebra in crossed product algebras. In particular, we examine and compare the global dimensions of crossed products. Further, we study in more detail the case in which the globl dimension is zero. Let A be an R-algebra, and let M be an (A, A)-bimodule. We consider n n the set CR (A, M) = {Φ : A −→ M : Φ is multilinear } of all multilinear n n mappings from A to M. In CR (A, M), we deﬁne addition and scalar multiplication, by the rules

(Φ + Ψ)(x1, . . . , xn) = Φ (x1, . . . , xn) + Ψ (x1, . . . , xn) and

(Φc)(x1, . . . , xn) = Φ (x1, . . . , xn) c n Φ, Ψ ∈ CR (A, M). It is easy for someone to see that under these operations, n the set CR (A, M) becomes an R-module. We deﬁne a mapping

(n) n n+1 δ : CR (A, M) −→ CR (A, M) , by the rules δ(0)u (x) = xu − ux, and Xn (n) i δ Φ (x1, x2, . . . , xn, xn+1) = x1Φ (x2, . . . , xn+1)++ (−1) Φ (x1, . . . , xi xi+1, . . . , xn+1) + i=1 n+1 + (−1) Φ (x1, . . . , xn) xn+1, n ≥ 1. We denote

n (n) n (n−1) ZR (A, M) = Ker δ and BR (A, M) = Im δ ,

55 56 CHAPTER 4. THE GENERAL CASE

0 for n ≥ 1, with BR (A, M) = 0. The n’th cohomology module of A with coeﬃcients in M is the factor module

n n n HR (A, M) = ZR (A, M) /BR (A, M) .

We give the explicit forms of H 0,H 1, and H 2. First we compute H 0.

Z 0 = Ker δ(0) = {f ∈ M : σf = f, for all σ ∈ A} = {f ∈ M : σf = f, for allσ ∈ A} = M A, where by M A we denote the set which is contained by all elements of M which are ﬁxed by A. Moreover, by deﬁnition B0 = 0. Therefore, H 0(A, M) = M A. Now, (1) δ f (σ1, σ2) = σ1f (σ2) − f (σ1σ2) + f (σ1). Thus

1 (1) Z = Ker δ = {f : A −→ M : σ1f (σ2)−f (σ1σ2)+f (σ1) = 0, for all σ1, σ2 ∈ A} =

{f : A −→ M : f (σ1σ2) = σ1f (σ2) + f (σ1)}. B1 = Im δ(0) = {f : A −→ M : ∃m ∈ M, δ(0)m = f (σ), for all σ ∈ A} =

{f : A −→ M : f (σ) = σm − m, for a ﬁxed m ∈ A, for all σ ∈ A}. Finally,

2 (2) Z = Ker δ = {f : A×A −→ M : σ1f (σ2, σ3)−f (σ1σ2, σ3)+f (σ1, σ2σ3)−f (σ1, σ2) = 0} is actually the group of all factor sets, and

2 (1) B = Im δ = {f (σ1σ2) + f (σ1) − σ1f (σ2): f : A −→ M} is actually the group of all principal factor sets.

4.1 The General Crossed Products

In this section, we proceed to deﬁne the general case of the construction of crossed products. Let Γ be a group and K be a ﬁeld. We may consider an action of Γ on K via a given homomorphism

t : Γ −→ Aut (K) , g 7→ t (g) = σ (g) . 4.1. THE GENERAL CROSSED PRODUCTS 57

Thus, the multiplicative group K* becomes a Γ-module under the action of t. Let a ∈ H 2 Γ,K* and let f : Γ × Γ −→ K* be a factor set representing a. Then, we may form the crossed product (K, Γ, a, t) as follows. If we view (K, Γ, a, t) as a left K-vector space, then it is of the form M (K, Γ, a, t) = Kuσ , and the multiplication between the elements of (K, Γ, a, t) is deﬁned by the rule

(xuσ )(yuτ) = x (uσ y) uτ = xσ (y) uσ uτ = xσ (y) f (σ, τ) uστ, and where σ (y) is the action of t (σ) on y. We have already remarked in Chapter 2 that this multiplication is associative, since by deﬁnition, a crossed product is an associative ring. We may further remark that (K, Γ, a, t) depends only on the choice of a an not on the choice of f , since f is a representative of a. Moreover, we know by Theorem 10 that every factor set f is equivalent to a normalized one. Due to this fact, we can always assume that f is normalized. In this case, we remind that the unity element of (K, Γ, a, t) is u1.

Example 4 (i) Let Γ be finite. We also assume that Γ acts faithfully. Then, σ (y) = y, for all y ∈ K, and (K, Γ, a, t) is the classical crossed product. Furthermore, (K, Γ, a, t) is a central simple algebra, with center the fixed field KΓ. (ii) If t and a are both trivial, since there is no action, we get that t(x) = σ (x) = 1, for all x ∈ K, and since a is also trivial, we obtain that f (σ, τ) = 1, for all σ, τ ∈ Γ. Thus, we get an algebra of the form M A = Kuσ , σ∈Γ where

(xuσ )(yu1) = xyuσ and

uσ uτ = uσ τ, σ, τ ∈ Γ. In other words, we get the group ring KΓ. (iii) Now, let A be torsionfree and abelian, and let a be the exact sequence

a : 1 → A → Γ → G → 1. 58 CHAPTER 4. THE GENERAL CASE

As we know, a is also an element of H 2 (G, A). Let L be a ﬁeld, and let us form the sequence 1 → LA → LΓ → LG → 1. Since a is exact, we get that the non-zero elements of LA do not divide zero in LΓ. Thus, we may form the multiplicative set S = LA\{0}, and we can localize LΓ with respect to S and form the ring of fractions S−1LΓ. For a more analytic proof of this, we refer to [Ros]. In this case, S−1LA is equal to the ﬁeld of fractions of LA, which is K = L (A). Furthermore, the group G acts on K via its action on A. Considering the multiplicative group K*, we may view this action as the set of monomials. The inclusion map * * 2 * i : A,→ K then induces an element in K , i* (a) ∈ H G, K . We drop the i* from the notation, and so we may form the ring (K, G, a, t). It is proved that S−1LΓ = (K, G, a, t). In the case where a = 0 we denote the crossed product as (K, Γ, t) and in the case where Γ acts trivially, as (K, Γ, a).

Now, let Γ0 be a normal subgroup of Γ, let G = Γ/Γ0, and L be the ﬁxed ﬁeld L = KΓ0 . We know by Galois theory, that G acts on L by the rule

t : G −→ Aut (L) , t (σ)(x) = σ (x) , x ∈ L. Let R = (K, Γ, a, t). We denote by a0 the restriction of a to Γ0, and by t0 the restriction of t to Γ0. Then, we form the ring R0 = (K, Γ0, a0, t0). R0 is obviously a subring of R. Let σ ∈ Γ, and M, N be left R-modules.

Lemma 1 Let h ∈ HomR0 (M, N), σ ∈ Γ, and let x ∈ L. Then, there is an action of xuσ on h, by the rule

−1 xuσ h 7→ xuσ h uσ m and HomR0 (M, N) becomes an L, G, t -module under this action.

Proof. At ﬁrst we notice that multiplication of elements of HomR0 (M, N) by elements of L is permissible. We prove that this action is well deﬁned. Let

ρ = σµ, where µ ∈ Γ0. The element h is R0-linear, thus we get

−1 −1 −1 uσ uµhuµ uσ = uσ huσ .

But also, −1 −1 uρhuρ = uσµhuσµ . 4.1. THE GENERAL CROSSED PRODUCTS 59

Now, we observe that

uσ uµ = f (σ, µ) uσµ, thus −1 uσµ = f (σ, µ) uσ uµ, and

−1 −1 −1 uσµ = uµ uσ f (σ, µ) . Therefore

−1 −1 −1 −1 −1 −1 −1 uρhuρ = f (σ, µ) uσ uµhuµ uσ f (σ, µ) = f (σ, µ) f (σ, µ) uσ uµhuµ uσ .

−1 −1 −1 But uµhuµ = h, so uρhuρ = uσ huσ . Consequently, uσ is K-linear, and so the action is well deﬁned. Now, we assume that µ ∈ Γ0, and we prove that uσ h ∈ HomR0 (M, N). We have to show that uσ h uµ (m) = uµ (uσ h (m)) , for m ∈ M. We compute

−1 −1 uσ h uµ (m) = uσ uµhuσ (m) = uµuσ huσ (m) = uµuσ h (m) , that is,

uσ huµ = uµ (uσ h) . Finally, we have to prove that it is an action, that is,

xuσ (yuτh) = xσ (y)(uστh) , for x, y ∈ L and σ, τ ∈ Γ. We have

−1 −1 −1 −1 xuσ yuτh uτ uσ m = x (uσ y) uτhuτ uσ (m) =

−1 −1 −1 −1 = xσ (y)(uσ uτ) h uτ uσ (m) = xσ (y) f (σ, τ) uστhuστ f (σ, τ) (m) =

−1 −1 = f (σ, τ) f (σ, τ) xσ (y) uστhuστ (m) = xσ (y) uστ (m) , as wanted.

Let M be an R-module. A projective resolution of M is an exact sequence

φ2 φ1 φ0 ... → P2 → P1 → P0 → M → 0, 60 CHAPTER 4. THE GENERAL CASE

in which each Pi is R-projective. Given such a projective resolution, let L be an R-module and form

* * φ1 φ2 0 → HomR (P0,L) → HomR (P1,L) → HomR (P2,L) → ....

This sequence is a sequence of additive groups. Moreover, the following relation holds true. * * * φi+1 · φi = (φi · φi+1) = 0,

* * i ≥ 1. Therefore Im φi ⊂ Ker φi+1. We deﬁne the derived functors Ext as:

n * * ExtR (M, L) = Ker φ n+1/ Im φ n, n ≥ 1. Now, back to our concern, it is clear that Hom deﬁnes two functors R0 from the category of left R-modules to the category of left L, G, t -modules. One contravariant

M 7→ HomR0 (M, N) and one covariant

N 7→ HomR0 (M, N) . These functors are additive left exact functors. We deﬁne the derived functors of M 7→ HomR0 (M, N), which are the functors

Extn . R0 These functors have a natural L, H, t -module structure, in a way that connecting homomorphisms are L, G, t -linear maps. Nowm let k = KΓ be the fixed field of Γ. Then, the group ring kG is a subring of L, G, t . We define U = L, G, t ⊗kG k.

The next result is provides some useful information about the change of functors between algebraic structures.

Lemma 2 There are natural isomorphisms of functors Hom (M, N) Hom U, Hom (M, N) Hom k, Hom (M, N) . R (L,G,t) R0 kG R0 4.2. GLOBAL DIMENSIONS OF CROSSED PRODUCTS 61

Proof. We only prove the ﬁrst isomosphism. The proof of the second one is exactly the same. If h ∈ Hom U, Hom (M, N) we want to map it to (L,G,t) R0 h (1U ), where 1U = 1⊗1 ∈ U. So we have to show that h (1U ) ∈ HomR (M, N). Let σ ∈ Γ, m ∈ M. Then

−1 −1 −1 −1 h (1U ) uσ m = uσ uσ h (1U ) uσ m = uσ ((uσ h (1U )) (m)) =

−1 −1 = uσ (h (σ · 1U )(m)) = uσ (h (1U )(m)) . f Thus h (1U ) is in HomR (M, N). Moreover, the map h 7→ h (1U ) is natural. To deﬁne the inverse transformation let g ∈ Hom (M, N). An element of R Hom U, Hom (M, N) is characterised by its value at 1 , which must (L,G,t) R0 U be G-invariant; conversely such a G invariant element gives rise to an element of Hom U, Hom (M, N) . Since g is R-linear, it is G-invariant (L,G,t) R0 when considered in HomR0 (M, N), so we can deﬁne g˜ by

g˜ (1U )(m) = g (m) , for m ∈ M. Then, g˜ deﬁnes a natural transformation, inverse to the above. Indeed,

f g˜ (1U (m)) = f (g (m)) = g (1U )(m) = 1U (m) , that is f = g˜ −1, as desired.

As an application of the second isomorphism of the previous lemma, we get the following useful result.

· − · − Theorem 24 There is an isomorphism Ext kG (k, ) Ext (L,G,t) (U, ). − − Proof. We know by Lemma 2 that Hom(L,G,t) (U, ) HomkG (k, ). Therefore these two rings have the same projective resolution, thus Extn (U, −) (L,G,t) n ExtkG (kG, −).

4.2 Global Dimensions of Crossed Products

We deﬁne the homological dimension hdR M of a left R-module M as the least positive integer n for which there exists an R-exact sequence

0 → Xn → Xn−1 → · · · → X0 → M → 0,Xi projective. 62 CHAPTER 4. THE GENERAL CASE

We set hdR M = ∞ if no such n exists. Also we note that hdR M = 0 if and only if M is projective. The (left) global dimension of R is deﬁned as

gl. dim R = sup{hdR M : M is a left R-module}. Clearly, gl. dim R = 0 if and only if every left R-module is projective, that is, if and only if R is a semisimple Artinian ring.

Ln ∗ | | ∞ ⊗ Let Λ = S G = i=1 Suσ , G = n < , and consider K R S = L. The group G acts on L, with R = {s ∈ S : σ(s) = s}.

Theorem 25 (Yi) Let Λ = S ∗ G, as above, and let H be a subgroup of G. Then L (i) Λ := Su is an R-subalgebra of Λ, and for an element g ∈ G, H L σ∈H σ ΛH ug = σ∈H Suσg is a ﬁnitely generated projective left ΛH -module. In particular, Λ is a projective left ΛH -module.

(ii) gl. dim S ≤ gl. dim ΛH ≤ gl. dim Λ.

Proof. (i) We shall prove that ΛH is an R-subalgebra of Λ. Of course ΛH is an R-algebra, since it is an S-algebra, and R ⊂ S. Now, let |H| = m ≤ n < ∞, Pm Pm and consider the elements x = i=1 xi uσi , y = i=1 yi uσi ∈ ΛH . Then Xm Xm Xm

x + y = xi uσi + yi uσi = (xi + yi )uσi ∈ ΛH , and i=1 i=1 i=1     Xm  Xm  xy =  x u   y u  = (x u + ... + x u )(y u + ... + y u ) =  i σi   i σi  1 σ1 m σm 1 σ1 m σm i=1 i=1

= x1σ1(y1)u 2 + ... + xnσn(y1)uσ σ + ... + x1σ1(ym )uσ σ + ... + xm σm (ym )u 2 = σ1 m 1 1 m σm

= x1σ1(y1)u 2 + (x1σ1(y2) + x2σ2(y1)) uσ σ + ... + (x1σ1(ym ) + xm σm (y1)) uσ σ + σ1 1 2 1 m

... x σ y u 2 . + + m m ( m ) σm

But since σ1, . . . , σm are linearly independent and {σ1, . . . , σm } generates H, then each σjσk is one of σ1, . . . , σm . Therefore, xy = a1σ1 +...+amσm , where ai ∈ S, therefore xy ∈ ΛH , that is, ΛH is an R-subalgebra of Λ. Now, let Pm Pm r = i=1 ri uσi g, s = i=1 si uσi g ∈ ΛH ug. Then,

xr = (x1uσ1 + ... + xm uσm )(r1uσ1g + ... + xm uσm g) = 4.2. GLOBAL DIMENSIONS OF CROSSED PRODUCTS 63

= x1uσ1 r1uσ1g + ... + xm uσm r1uσ1g + ... + xm uσm rm uσm g =

= x1σ1(r1)u 2 + ... + xm σm (r1)uσ σ g + ldots + xm σm (rm )u 2 ∈ ΛH ug. σ1 g m 1 σm g Now,     Xm Xm  Xm  x(r + s) = x  r u + s u  = x  (r + s )u  =  i σi g i σi g  i i σi g i=1 i=1 i=1

= x(r1 + s1)uσ1g + ... + x(rm + sm )uσm g =

= xr1uσ1g + xs1uσ1g + ... + xrm uσm g + xsm uσm g = = x r1uσ1g + ... + rm uσm g + x s1uσ1g + ... + sm uσm g = xr + xs. Moreover,     Xm Xm  Xm  (x + y)r =  x u + y u  r =  (x + y )u  r =  i σi i σi   i i σi  i=1 i=1 i=1

= (x1 + y1)uσ1 r + ... +(xm + ym )uσm r = x1uσ1 r + y1uσ1 r + ... + xm uσm r + ym yσm r =

= (x1uσ1 + ... + xm uσm )r + (y1uσ1 + ... + ym uσm )r = xr + yr.

Further, the unity element of ΛH is uσ1 , and we have

Xm uσ1 r = uσ1 ri uσi g = uσ1 r1uσ1g + ... + rm uσm g = i=1

= σ1(r1)uσ1σ1g + ... + σ1(rm )uσ1σm g = r1uσ1g + ... + rm uσm g = r.

Therefore, ΛH ug is a left ΛH -module. Furthermore, since the elements

σ1, . . . , σm are linearly independent, then the elements σ1g, . . . , σm g are linearly independent, and hence the elements uσ1g, . . . , uσm g are linearly independent and they form a free basis for ΛH ug, so ΛH ug is a ﬁnitely generated left ΛH -module. Also, each direct summand of ΛH ug, Suσi g is generated by the element uσi , for 1 ≤ i ≤ m, therefore each Suσi g is a direct summand of the free ΛH -module ΛH ug, and so every Suσi g is projective, 1 ≤ i ≤ m. But this means that ΛH ug is projective. Finally, we have to show that Λ is a projective left ΛH -module. We remark that since ΛH ug is projective, then ΛH is projective too. In the same manner as before, we have Mn Mm Mn Mn

Λ = Suσi = Suσi ⊕ Suσi = ΛH ⊕ Suσi . i=1 i=1 i=m+1 i=m+1 64 CHAPTER 4. THE GENERAL CASE

Each Suσi is projective, and so Λ is a projective left ΛH -module. (ii) First we shall prove a general assertion. Let A, B be rings with A ⊆ B, and consider the left B-module M. Then,

hdA M ≤ hdB M + hdA B.

We shall make this proof by induction on hdB M = n. If n = 0, then BM is a projective left B-module, therefore it is a direct summand of a free B-module F. Hence

hdA M ≤ hdA F = hdA B = hdB M + hdA B.

If n > 0, then there is a short exact sequence of B-modules

0 → K → F → M → 0, where BF is free as left B-module, and hdK B = hdB(M)−1 = n−1. Therefore, by the induction hypothesis, we obtain that

hdA K ≤ hdK B + hdA B = n − 1 + hdK B.

But also, we have that hdA M = hdA K + 1 ⇒ hdA K = hdA M − 1. So

hdA M − 1 ≤ hdB M − 1 + hdA B ⇒ hdA M ≤ hdB M + hdA B.

Lm Now, since ΛH = i=1 Suσi , then the ring S Suσ1 is a direct summand of Lm ⊕ the (S, S)-bimodule ΛH , and we can write ΛH S I, where I = i=2 Suσi . Let SM be any left S-module. Then,

hdS M ≤ hdS M ⊗S I.

But also, we have

M ⊗S ΛH M ⊗S (S ⊕ I) (M ⊗S S) ⊕ (M ⊗S I) M ⊕ (M ⊗S I).

Therefore,

hdS(M ⊗S ΛH ) = hdS(M ⊕ (M ⊗S I)) = sup{hdS M, hdS M ⊗S I} = hdS M ⊗S I.

Combining these two relations we get that hdS M ≤ hdS M ⊗S I = hdS M ⊗S ΛH .

Now, since S ⊆ ΛH , we obtain the inequality

hdS M ≤ hdΛH M + hdS ΛH ≤ gl. dim ΛH + hdS ΛH , 4.2. GLOBAL DIMENSIONS OF CROSSED PRODUCTS 65 that is,

gl. dim S ≤ gl. dim ΛH + hdS ΛH .

But since ΛH is a projective S-module, then hdS ΛH = 0, and we obtain the desired result gl. dim S ≤ gl. dim Λ . In the same manner, we consider L H L ⊆ the rings ΛH and Λ. Since ΛH = σ∈H Suσ σ∈G Suσ = Λ, for any left Λ-module M, we obtain that

hdΛH M ≤ hdΛ M + hdΛH Λ ⇒ hdΛH M ≤ gl. dim Λ + hdΛH Λ, and it follows that

gl. dim ΛH ≤ gl. dim Λ + hdΛH Λ.

But since Λ is a projective ΛH -module, then hdΛH Λ = 0. Thus we have gl. dim ΛH ≤ gl. dim Λ. Consequently gl. dim S ≤ gl. dim ΛH ≤ gl. dim Λ.

Moreover, we deﬁne the cohomological dimension of a group Γ over K to be the global dimension gl. dim KΓ of the ordinary group ring KΓ, and we note it by cdK Γ.

We denote the functor HomkG(k, −) by Ψ. If k = Z, the derived func- n n tors ExtkG(k, −) become the cohomology groups H (G, −). We also denote

HomR0 (M, N) by ΦN (M). Under these changes, we make use of Lemma 2, and we remark that

HomR(M, N) HomkG(k, HomR0 (M, N)) = Ψ · ΦN (M).

We may also present some results in the theory of spectral sequences. By a theorem of Grothendieck for spectral sequences, we obtain a spectral sequence Ep,q = Extp k, Extq (M, N) → Extp+q(M, N). 2 kG R0 R Therefore, after this change of notation, we get that there exists a spectral sequence E2, with p,q p q E2 = R Ψ · R ΦN (M), which converges to n R (Ψ · ΦN )(M). 66 CHAPTER 4. THE GENERAL CASE

p p p If we replace R Ψ = ExtkG(k, −) by H (G, −), we obtain the convergent spectral sequence H p(G, Extq (M, N)) → Ext n(M, N). R0 R After these remarks and observations, we get to the next theorem, which is of signiﬁcant importance.

Theorem 26 (Aljadeﬀ-Rosset) gl. dim (K, Γ, a, t) ≤ gl. dim (K, Γ0, a0, t0) + gl. dim L, G, t . Proof. We assume that gl. dim (K, Γ0, a0, t0) = r, and gl. dim L, G, t = s. If p,q p + q = n > r + s, then either p > r or q > s. In both cases E2 = 0, so p,q n E∞ = 0 and therefore ExtR (M, N) = 0, which means that HomR0 (M, N) is cohomologically trivial. Therefore, gl. dim R = gl. dim(K, Γ, a, t) ≤ n = r+s = gl. dim(K, Γ0, a0, t0)+gl. dim(L, G, t).

Corollary 7 gl. dim (K, Γ, a, t) ≤ gl. dim (K, Γ, t).

Proof. If we take Γ0 = 1 in Theorem 26, then we get

gl. dim(K, Γ, a, t) ≤ gl. dim(K, 1, a0, t0) + gl. dim(L, G, t).

But, a0 is the restriction of a in Γ0 = 1 and t0 is the restriction of t in Γ0.

Therefore, the crossed product (K, 1, a0, t0) has global dimension equal to 0. Γ0 Moreover, G = Γ/Γ0, and L = K , so L = K. Furthermore, the associated map t of L is the map t itself. Hence the crossed product (L, G, t) is in fact (K, Γ, t). Consequently, the inequivalence of Theorem 26 becomes

gl. dim(K, Γ, a, t) ≤ gl. dim(K, Γ, t).

Theorem 27 (Aljadeﬀ-Rosset) gl. dim (K, Γ, t) ≤ gl. dim KΓ = cdK Γ.

Proof. Again, we take Γ0 = 1 in Theorem 26. Since Γ0 = 1, then R0 is a ﬁeld contained in R. Therefore, in the spectral sequence we have that

Extq (M, N) = 0 R0 4.3. GLOBAL DIMENSION 0 67 for q > 0. Thus the spectral sequence collapses to isomorphisms

p p H (G, HomK (M, N)) ExtR (M, N) .

Taking these into account, we obtain that

gl. dim(K, Γ, t) ≤ gl. dim KΓ, and by deﬁnition, gl. dim KΓ = cdK Γ, thus we obtain the desired result.

Combining Theorem 26, Corollary 7 and Theorem 27, we obtain the following important relation between the global dimensions of several algebraic structures.

gl. dim (K, Γ, a, t) ≤ gl. dim (K, Γ, t) ≤ gl. dim KΓ = cdK Γ.

Theorem 28 (Aljadeﬀ-Rosset) Let k = KΓ. If gl. dim(K, a, Γ, t) < ∞, then

gl. dim(K, Γ, t) = gl. dim KΓ = cdk Γ.

Proof. By Theorem 27, we know that gl. dim(K, Γ, t) ≤ gl. dim KΓ. More- over, we remark that if gl. dim(K, Γ, a, t) < ∞, then gl. dim(K, Γ, a, t) ≤ gl. dim(F, Γ, a, t), where F is a ﬁeld such that K ⊆ F, by Theorem 25(ii). By this result, we get that gl. dim(K, Γ, t) ≤ gl. dim(F, Γ, t). If we put k = KΓ instead of K, and replace F with K also, we obtain that gl. dim(KΓΓ) = cdk Γ ≤ gl. dim(K, Γ, t). Therefore, gl. dim(K, Γ, t) = gl. dim KΓ.

4.3 Global Dimension 0

Let Γ be a group, Γ0 a normal subgroup of Γ, and G = Γ/Γ0. Let t : Γ −→ Aut(K) be a homomorphism. Then, we can consider the homomorphism t : G −→ Aut(KΓ0 ). Mashke’s Theorem asserts that if G is ﬁnite and char K does not divide |G|, then KG is semisimple, hence gl. dim KG = 0. But, by Corollary 7 and Theorem 27, we have that gl. dim(K, G, a, t) ≤ gl. dim(KG), therefore, gl. dim(K, G, a, t) = 0. In the following result we show that in this case G cannot be a torsionfree group.

Theorem 29 If (K, G, a, t) is semisimple, then G is a torsion group. 68 CHAPTER 4. THE GENERAL CASE

Proof. Let x , 0 be a divisor of (K, G, a, t) Since this crossed product is semisimple, by the Structure Theorem we get that x is invertible. And because of the fact that a nonzero divisor of (K, G, a, t) is invertible, G must be a torsion group.

Corollary 8 If G is a free group, then gl. dim (K, G, a, t) = 1.

Proof. Let σ ∈ G. Since G is free, then the order of σ is inﬁnite. We shall prove that in this case 1 − uσ ∈ (K, G, a, t) is not invertible, and also it is not a zero-divisor. Let v , 0 be such that (1 − uσ )v = 0 ⇒ v = uσ v, and we can say that u1 appears in v. Then

v = uσ v = uσ (uσ v) = uσ2 v = uσ2 uσ v = uσ3 v = ....

Hence, there are infinitely many powers of σ, and so there are infinitely many powers of uσ that appear in v, which is a contradiction. Therefore, v = 0 and 1 − uσ is not a zero-divisor. Next, we assume that 1 − uσ is invertible. Let hσi be the infinite cyclic group generated by σ, and let {τi } be coset representatives in G for the cosets hσi x. Then,

 ∞  M M  (K, G, a, t) =  Kun u .  σ  τi i 0

Therefore, due to this fact we obtain that the inverse element of 1 − uσ , L∞ − −1 n (1 uσ ) must lie in n=0 Kuσ , that is, X∞ −1 n (1 − uσ ) = Kuσ . n=0 Then, X∞ X∞ X∞ X∞ −1 ∞ n n n+1 n n (1−uσ ) (1−uσ ) = σn=0Kuσ (1−uσ ) ⇒ 1 = Kuσ − Kuσ = Kuσ − Kuσ = n=1 n=0 n=1 n=1

0 = Kuσ , which is impossible, since σ is of inﬁnite order. Therefore, (1 − uσ ) is not invertible in (K, G, a, t). Since 1 − uσ is not invertible and is not a zero divisor, and G is a free group, we obtain the inverse of Theorem 29, that is,

gl. dim (K, G, a, t) > 0. 4.3. GLOBAL DIMENSION 0 69

On the other hand, it has been proved by Serre in [S2, Proposition 7] that cdK G = 1. Therefore, by Corollary 7 we get that gl. dim(K, G, a, t) ≤ 1. Hence, we have shown that 0 < gl. dim(K, G, a, t) ≤ 1, and since gl. dim is a positive integer, this leads to the fact that gl. dim(K, G, a, t) = 1.

Example 5 We know by Mashke’s Theorem that if the group G is of infinite order, then the group ring KG is not semisimple. We show that this is not the case for the crossed product (K, G, a, t). In other words, (K, G, a, t) can be semisimple, even if G is infinite. More than this, we this ring is a division ring. Let G be an infinite locally finite group. These groups exist, for example Q/Z is an infinite locally finite group. We can construct an exact sequence

π a : 1 → A → Γ → G → 1, such that Γ is torsionfree, and the group ring kΓ over a field k is an integral domain. Then, by Example 4(iii), under this conditions, we have that S−1kΓ (K, G, a, t), where S = kA\{0}, K is the field of fractions of kA, G acts on K via its action on A, and a comes from the inclusion A,→ K*. Given a finite set in G, this set is contained in a finite subgroup H. If −1 −1 π H = ΓH , then S kΓH (K, H, b, t), where b is the restriction from G to H of a. Then, the center of (K, H, b, t) is KH , which is a field. Moreover, this crossed product is domain and is finite dimensional over its center, therefore (K, H, b, t) is a division ring. Also, G is locally finite, thus we S obtain H (K, g, b, t) = (K, G, a, t), that is, (K, G, a, t) is a division ring. For the proof that (K, H, b, t) is a domain, we refer to [Pas, 5]. 70 CHAPTER 4. THE GENERAL CASE Bibliography

[ABGV] A. Auel, E. Brussel, S. Garibaldi and U. Vishne, "Open problems on central simple algebras", Transformation Groups 16(1) (2011), 219- 264.

[Am] S. A. Amitsur, "On central division algebras", Israel J. Math 12 (1972), 408-422.

[AR] E. Aljadeﬀ and S. Rosset, "Global Dimensions of Crossed Products", J.of Pure and Applied Algebra 40 (1986), 103-113.

[B] E. Brussel, " Noncrossed products and nonabelian crossed products over Q(t) and Q((t))", Amer. Jour. Math 117 (1995), 377-384.

[BS] V. I. Borevich and I. R. Shafarevich, Number Theory, Academic Press, New York, 1966.

[J] N. Jacobson, Finite-dimensional division algebras over ﬁelds, Springer-Verlag, New York, 1996.

[Pal] M. Palaisti, Ramiﬁcation Theory on Quadratic Number Fields, 2011.

[Pas] D.A. Passman, The Algebraic Structure of Group Rings, Wiley Inter- science, New York 1977.

[Pi] R. S. Pierce, Associative Algebras, Graduate Texts in Mathematics, vol. 88, Springer, 1982.

[Re] I. Reiner, Maximal Orders, Clarendon Press, Oxford, 2005.

[Rib] P. Ribenboim, Classical Theory of Algebraic Numbers, Universitext. Springer-Verlag, New York, 2001.

71 72 BIBLIOGRAPHY

[Ris] L. Risman, "Cyclic algebras, complete ﬁelds, and crossed products", Israel J. Math. 28 (1977), 113-128.

[Roq] P. Roquette The Brauer-Hasse-Noether Theorem in Historical Perspec- tive, Springer, 2006.

[Ros] S. Rosset, "A vanishing theorem for Euler characteristics", Math. Z. 185 (1984), 211-215.

[S1] J.-P. Serre, Local ﬁelds, Gradute Texts in Mathematics, vol.67, Springer-Verlag, New York-Heildeberg-Berlin, 1979.

[S2] J.-P. Serre, "Cohomologie de groupes discret", Prospects in Mathe- matics, Ann. of Math. Studies 70 (1971), 77-169.

[T] Th. Theohari-Apostolidi, Notes on crossed products, Unpublished Notes, 1997.

[Yi] Z. Yi, "Homological dimension of skew group rings and crossed products", J. Algebra 164 (1994), 101-123.

[W] J. H. M. Wedderburn, "On division algebras", Trans. Amer. Math. Soc. 22 (1921), 129-135.