# Hereditary crossed product orders over discrete valuation rings

Contents Chapter 1.Introduction and background information 1 1.1.Crossed product algebras over ﬁelds 1 1.2.Discrete valuation rings and Dedekind domains 3 1.3.The Jacobson radical of a ring 5 1.4.The theory of orders 6 1.5.Weak crossed product algebras over ﬁelds 7 1.6.Weak crossed products orders over DVRs 9 1.7.Outline of results 12 Chapter 2.The local case 14 2.1.The Jacobson radical and structure of A f 14 2.2.HLS and Dedekind inertial subgroups 26 2.3.Ideals and orders containing A f 32 Chapter 3.The general case 45 3.1.The Jacobson Radical 46 3.2.Factorization of elements of Rad(A f ) 53 3.3.A local-to-global theorem for hereditarity of A f 55 Bibliography 63 Appendix A.Cocycle Generator Routine 65 ix

CHAPTER 1 Introduction and background information In this thesis,we will present results concerning weak hereditary crossed products over discrete valuation rings.This chapter will provide the reader with some prereq- uisite background and terminology.We will state results without proof and provide standard references where appropriate.Then we will present a brief summary of pre- vious work in the theory of hereditary and maximal crossed product orders and give an outline of the results contained in this thesis. 1.1.Crossed product algebras over ﬁelds Let K/F be a Galois extension of ﬁelds with automorphism group G.The con- ventional notation in the context of crossed products is to use exponential notation for the action of G on K:for a ∈ K and σ ∈ G, a σ := σ(a). Let Σ be the (left) K-vector space on a basis indexed by the elements of G,so Σ:=

σ∈G Kx σ = {

σ∈G a σ x σ :a σ ∈ K}. We put a multiplication on Σ as follows: (1) For a ∈ K,we deﬁne x σ a:= a σ x σ . (2) For σ,τ ∈ G,deﬁne x σ x τ := f(σ,τ)x στ . (3) x 1 G x σ = x σ x 1 G = x σ for any σ ∈ G,so f(1 G ,σ) = f(σ,1 G ) = 1 K . (4) Extend this deﬁnition linearly. 1

1.INTRODUCTION AND BACKGROUND INFORMATION 2 Any unit-valued “coeﬃcient function” f:G×G →K −{0} may be used in (2),as long as the resulting multiplication turns out to be associative.This is equivalent to the requirement that f must be a 2-cocycle;that is,f must satisfy the identity f(σ 1 ,σ 2 )f(σ 1 σ 2 ,σ 3 ) = f(σ 2 ,σ 3 ) σ 1 f(σ 1 ,σ 2 σ 3 ). When (3) is imposed,the cocycle f is sometimes said to be normalized.This re- quirement does no harm as it does not aﬀect the isomorphism class of the resulting algebra.Henceforth,we decorate the notation Σ by adding a subscript f to indicate that the cocycle f deﬁnes the multiplication: Σ f :=

σ∈G Kx σ = {

σ∈G a σ x σ :a σ ∈ K}. The F-algebra Σ f is called a crossed product algebra. One can use any group that acts on K to index the basis elements for Σ f over K,but we will always use G = Gal(K/F) except where explicitly stated.When G = Gal(K/F),we have many classical results:in this case Σ f is an F-central simple algebra,and there is a sense (Brauer group equivalence) in which every F-central simple algebra is equivalent to a crossed product algebra.One can consider classes of cocycles that produce isomorphic crossed products for a given extension K/F.The collection of such classes is denoted H 2 (G,K × ).This collection has a group structure and is an invariant of the ﬁeld extension K/F.If one chooses other K-multiples of the x σ as basis elements for Σ f ,this is equivalent to the selection of a cocycle that is cohomologous to the original f. An accessible introduction to crossed product algebras is found in Farb and Dennis [3].

1.INTRODUCTION AND BACKGROUND INFORMATION 3 1.2.Discrete valuation rings and Dedekind domains This thesis will address crossed products with coeﬃcients and cocycle values in Dedekind domains.Therefore,we brieﬂy state some deﬁnitions and standard results about these types of rings. A local principal ideal domain (that is,a PID having exactly one maximal ideal m) is called a discrete valuation ring (DVR).Given a DVR R,let π R be any generator of the maximal ideal m.The generator π R is called a uniformizer of m and is deﬁned up to multiplication by a unit.One can deﬁne a function v m :R →Z by v m (r) = max{z ∈ Z:π R z divides r} = max{z ∈ Z:r ∈ m z }, called the valuation (or π R -adic valuation or m-adic valuation).For r ∈ R we say that v m (r) is the (m-adic) value of r.The invertible elements of R are precisely those elements r ∈ R for which v m (r) = 0.The valuation v m can be extended to the ﬁeld of fractions F of R,via v(a/b) = v m (a) −v m (b). This valuation deﬁnes a topology on R,wherein (informally speaking) two ele- ments r and r

are “close” if v m (r −r

) is large.The completion ˆ R of R with respect to this topology is a ring that contains an embedded copy of R. Three standard examples of DVRs are (1) the localization of Z at a prime ideal, (2) the p-adic integers Z p ,and (3) the ring k[[t]] of formal power series in the inde- terminate t with coeﬃcients in some ﬁeld k. A Dedekind domain is an integral domain S for which the localization S P at any prime ideal P is a DVR.A localization argument shows that prime ideals in a Dedekind domain are always maximal.A Dedekind domain for which there are only ﬁnitely many maximal ideals is said to be semilocal.Semilocal Dedekind domains are always principal ideal domains.A local Dedekind domain has a single maximal ideal and is therefore a DVR.

1.INTRODUCTION AND BACKGROUND INFORMATION 4 Let (R,m) be a DVR with ﬁeld of quotients F = Frac(R) and maximal ideal m. Let K/F be a Galois extension and S:= {s ∈ K:s satisﬁes a monic polynomial in R[x]} be the integral closure of R in K.Then S is a semilocal Dedekind domain and K is the ﬁeld of fractions of S.One can show that the maximal ideals M 1 ,...,M g of S satisfy mS = (M 1 · · · M g ) e (product of ideals) for some integer e.Each M i is principally generated,with generator π i deﬁned up to multiplication by a unit of S; thus,we have π R S = (π 1 · · · π g ) e S.If e > 1,the extensions S/R and K/F are said to be ramiﬁed extensions;if e = 1,we have unramiﬁed extensions.Accordingly,e is called the ramiﬁcation index. Each maximal ideal M i of S gives an M i -adic valuation v M i on S:Given s ∈ S, deﬁne v M i (s):= max{z ∈ Z:π i z divides s}. This valuation is related to that of R by the equation v M i (r) = v R (r) e for any r ∈ R. As in the local case,each v M i extends to a valuation on the ﬁeld of quotients K of S via v M i (a/b) = v M i (a) −v M i (b). The elements of G:= Gal(K/F) act transitively on the prime ideals of S.More- over,given σ ∈ G for which σ(M i ) = M j ,we have v M j (σ(a)) = v M i (a) for every a ∈ S. The ideal structure of S gives rise to two important classes of subgroups of G. Consider a maximal ideal M i of S.Those automorphisms of K that map M i onto itself formthe decomposition group for M i .For M = M i ,we will use the notations D M ,D M i ,and D i for this group.(We will attempt to use the least fussy notation for a given context.) Every σ ∈ D i induces an automorphism ¯σ of the extension of the residue ﬁelds (the ﬁelds S/M i and R/m);in fact,the correspondence σ →¯σ is a surjection from D i onto Gal[(S/M i )/(R/m)] (Janusz [9] Theorem III.1.4).Those σ ∈ D i that become trivial on S/M i form what is called the inertia group for M i .

1.INTRODUCTION AND BACKGROUND INFORMATION 5 In this paper,we will use the notation U M i or U i for the inertia group of M i ,so U i := {σ ∈ D i :¯σ = Id S/M i } = ker(D i →Gal[(S/M i )/(R/m)]).If S is local,we omit the subscript from the U notation. The theory of Dedekind domains can be found in any standard graduate-level algebra text.A good introduction with concise proofs is given in sections I.3,I.6,I.7, II.1,III.1 of Janusz [9]. 1.3.The Jacobson radical of a ring Given a ring A,one deﬁnes the Jacobson radical Rad(A) to be the ideal of A consisting of elements of A that annihilate every simple left A-module.The ring A is said to be semisimple if Rad(A) = 0.The computation of the Jacobson radical for certain rings will be essential in this paper. Let us recall two deﬁnitions before we state facts about Rad(A).An ideal I of a ring A is said to be nilpotent if I n = 0 for some n ∈ Z + ,where I n denotes the ideal generated by {a 1 a 2 · · · a n :a i ∈ I}.We say that I is nil if every element of I is nilpotent.It is clear from the deﬁnitions that every nilpotent ideal is nil. We now list some standard results concerning the Jacobson radical,which are to be found in Lam [12] and Reiner [14]: (1) The ring A/Rad(A) is semisimple. (2) The intersection of the maximal left ideals of A is equal to Rad(A). (3) Every nil ideal of A is contained in Rad(A). (4) If k is a ﬁeld and Ais a ﬁnite-dimensional k-algebra,then Rad(A) is nilpotent. (5) In a left artinian ring,left nil ideals are also nilpotent.

1.INTRODUCTION AND BACKGROUND INFORMATION 6 1.4.The theory of orders Let R be a discrete valuation ring with ﬁeld of fractions F.Let Σ be an F-central simple algebra.By an R-order in Σ,we mean a subring A ⊆ Σ (with 1 A = 1 Σ ) for which (1) A is a ﬁnitely generated R-module,and (2) A contains an F-basis for Σ;equivalently,F · A = Σ. An R-module satisfying (1) and (2) is called an R-lattice (or full R-lattice in Reiner [14]),so R-orders are lattices that have an additional ring structure. Let Rad(A) denote the Jacobson radical of A.The Jacobson radical of R is equal to the maximal ideal m = π R R.By Theorem 6.15 of Reiner [14],π R A ⊆ Rad(A). Thus,F · Rad(A) contains F · π R A = F · A = Σ,so F · Rad(A) = Σ,and Rad(A) is an R-lattice in Σ.Given any R-lattice W ⊆ Σ,one has the so-called left order O l (W) deﬁned by O l (W):= {x ∈ Σ:xW ⊆ W}. This is an R-order in Σ,c.f.Section 8 of Reiner [14].Several of the results in this paper rely on certain facts about the left order of W = Rad(A). A hereditary ring A is one for which every two sided ideal is a projective A- module.By this we mean that given (1) an ideal I ⊆ A,(2) A-modules M 1 and M 2 , and (3) a diagram I ↓ M 2 → M 1 → 0 of A-module homomorphisms,there exists a lifting I →M 2 of the map I →M 1 .A hereditary order is one which is hereditary as a ring. There are other ways to characterize hereditarity of an R-order when R is a DVR.

1.INTRODUCTION AND BACKGROUND INFORMATION 7 (1) If A is a ﬁnitely generated torsion-free R-module,then:Rad(A) is projective if and only if A is hereditary (Williamson [15] p.106,Lemma). (2) Some power of Rad(A) is principally generated by an element of Σ (that is, [Rad(A)] t = xA for some x ∈ Σ and t > 0) if and only if A is hereditary (Harada [8] p.72,Lemma 3). (3) If A is noetherian (so that every A-module is ﬁnitely presented) then:A⊗ R ˆ R is hereditary if and only if A is hereditary,where ˆ R is the completion of R at its maximal ideal (Reiner [14] 2.21-2.22). Additionally,every hereditary R-order A satisﬁes O l (Rad(A)) = A (complete case: Reiner [14] 39.11-39.12;general case:Kauta [10] 1.4-1.5).We will restate these facts wherever they are used. Let K/F be a Galois extension of ﬁelds and let Σ f :=

σ∈G Kx σ be a crossed product algebra.Let S be the integral closure of R in K,and let A f be the subset of Σ f deﬁned by A f :=

σ∈G Sx σ . If f(σ,τ) ∈ S for every σ,τ ∈ G,then A f is an R-order in Σ f .One of the main results of this thesis is a characterization of the hereditarity of A f in terms of the properties of the cocycle f. The standard reference for the theory of orders is Reiner [14]. 1.5.Weak crossed product algebras over ﬁelds Let Σ f :=

σ∈G Kx σ be a crossed product algebra.Recall that (in the classical case) one deﬁnes multiplication of basis elements by x σ x τ := f(σ,τ)x στ ,where the cocycle f is a unit-valued function fromG×Ginto K.One may relax the requirement that f is unit-valued,allowing the value 0 to be taken by the cocycle f.The resulting algebra is known as a weak crossed product algebra,the adjective weak indicating

1.INTRODUCTION AND BACKGROUND INFORMATION 8 that noninvertible values may be taken by f.In this case,one often calls f a weak cocycle.(In [6],the term cosickle is used.) The adjective weak may be omitted if the context is clear.The adjective classical may be used to indicate that a cocycle takes invertible values only. The notion of a weak crossed product algebra over a ﬁeld was introduced by Haile,Larson,and Sweedler (HLS) in [6].They identiﬁed a subgroup H of G called the inertial subgroup,deﬁned by H:= {σ ∈ G:f(σ,σ −1 ) = 0}.Clearly,H contains precisely those elements σ ∈ G for which the corresponding basis element x σ is invertible in Σ f .This subgroup gives a decomposition of Σ f (as F-modules) known as the Wedderburn splitting: Σ f = B f ⊕Rad(Σ f ), where B f =

h∈H Kx h is a classical crossed product algebra and Rad(Σ f ) =

σ∈H Kx σ is the Jacobson radical of Σ f , It is also shown in HLS [6] that there is a partial ordering on G/H given by g 1 H ≤ g 2 H if and only if f(g 1 ,g −1 1 g 2 ) is a unit of K. This partial ordering ≤ enjoys a remarkable property called lower subtractivity: Given g 1 ,g 2 ,g 3 ∈ G with g 1 H ≤ g 3 H,we have g 1 H ≤ g 2 H ≤ g 3 H if and only if g 1 −1 g 2 H ≤ g 1 −1 g 3 H. There is a unique minimum element with respect to ≤,namely H itself.One may thus study a weak crossed product algebra Σ f by looking at the directed graph given by the partial ordering on G/H.The ideal structure of Σ f and several examples are given in Haile [4]. Weak crossed products arise naturally in the context of classical crossed products over certain ﬁelds.Let K/F be a Galois extension,and suppose that F is the ﬁeld

1.INTRODUCTION AND BACKGROUND INFORMATION 9 of fractions of a discrete valuation ring R and that S is the integral closure of R in K.It is well known that a cocycle f:G×G →K × for a classical crossed product algebra Σ f =

σ∈G Kx σ is cohomologous to one that takes values in S.Hence,let us assume that f does takes values in S.We can then obtain a crossed product algebra Σ ¯ f =

σ∈G (S/M)x σ for each maximal ideal M of S,where ¯ f denotes f followed by passage to the residue ﬁeld S/M.This construction produces a weak crossed product algebra over the ﬁeld S/M if the original classical cocycle f takes any values in M. 1.6.Weak crossed products orders over DVRs Let R,S,F,K,G,and Σ f be as in the previous paragraph.Assume again that f takes values in S.We may then consider the R-order A f :=

σ∈G Sx σ .Because the values of f are units of K,we know that f does not take the value 0.The values of f need not be units of S,however.We again use the adjective weak to describe both the crossed product order A f and the cocycle f,indicating that f may take values in S −{0} that are not invertible in S. In Haile [5],Haile gives conditions for A f to be maximal with respect to inclusion among R-orders in Σ f when the extension S/R is unramiﬁed.One can again deﬁne the inertial subgroup H = {σ ∈ G:f(σ,σ −1 ) is a unit of S}.In the case where S is a DVR,the subalgebra B f :=

σ∈H Sx σ is Azumaya with center S H ,and Rad(A f ) = MB f +

σ∈H Sx σ ,where M is the maximal ideal of S.Thus,we have structure results for A f that are analogous to the Wedderburn splitting theorem for weak crossed products over ﬁelds.We also have the same lower subtractive partial ordering on G/H,namely g 1 H ≤ g 2 H if and only if f(g 1 ,g 1 −1 g 2 ) is a unit of S. These facts are used to prove the following theorem concerning the maximality of A f :

1.INTRODUCTION AND BACKGROUND INFORMATION 10 Theorem 1.6.1.(Haile [5] 2.3) Assume that S/R is unramiﬁed and that S is a DVR.Then A f is a maximal order (with respect to inclusion among the R-orders in Σ f ) if and only if H = G or all of the following conditions are satisﬁed: (1) The inertial subgroup H is normal in G and G/H is cyclic; (2) There exists σ ∈ G for which σH = G/H and v S (f(σ,σ −1 )) = 1;and (3) The partial ordering ≤ is a total ordering,with H ≤ σH ≤ σ 2 H ≤ · · · ≤ σ |G/H|−1 H. Moreover,when H = G and (1)-(3) are satisﬁed,one has Rad(A f ) = x σ A f = A f x σ . Haile also gives a local-to-global criterion for maximality of A f in the case where S is semilocal: Theorem 1.6.2.(Haile [5] 3.11,3.2) Let S/R be unramiﬁed,and assume that,for every maximal ideal M of S,there is a set of right coset representatives σ 1 ,...,σ r of the decomposition group D M in G with f(σ i ,σ i −1 ) ∈ M for all i. Under these conditions,the crossed product order A f is a maximal order in Σ f if and only if A f M =

d∈D M S M x d is maximal order for some maximal ideal M,where f M :D M ×D M →S M −{0} denotes the restriction of f to D M ×D M followed by the inclusion of S in the localization S M . Haile’s hypothesis that S/R is unramiﬁed makes particular sense when one views these theorems to be a generalization of a result of Auslander and Rim.Namely,if f ≡ 1 is the trivial cocycle,then A f (which is sometimes called the twisted group algebra or trivial crossed product) is a maximal order in Σ f if and only if S/R is unramiﬁed (c.f.Auslander and Rim [1],discussion after Corollary 3.6).Additionally, when S/R is unramiﬁed and S is a DVR that has a perfect 1 residue ﬁeld S/M,Haile 1 The ﬁeld k is said to be perfect if (1) char(k) = 0 or if (2) char(k) = p and every element of k is a pth power.

1.INTRODUCTION AND BACKGROUND INFORMATION 11 has shown that every maximal R-order in a central simple algebra is equivalent (in the sense of Brauer groups) to some crossed product order A f (Haile [5] Proposition 1.5). A theorem of Williamson characterizes hereditarity of A f in the classical case in which the values taken by f are units of S: Theorem 1.6.3.(Williamson [15]) The following are equivalent: (1) A f is a hereditary order for every unit-valued cocycle f:G×G →S. (2) The trivial crossed product A f is hereditary,where f ≡ 1. (3) The extension S/R is tamely ramiﬁed;that is,the characteristic of R/π R R does not divide the ramiﬁcation index e. The implication (2)⇔(3) was proven by Auslander and Rim in [1] (paragraph after Corollary 3.6);in Reiner [14],Rosen is credited with (2)⇔(3).The implication (1)⇔(3) is misstated in Reiner [14] (the phrase “for every f” is missing);Example 4.1 of Braun,Ginosar,and Levy [2] shows that the hereditarity for a single A f does not imply tame ramiﬁcation for the extension S/R. Because of Williamson’s result,one might hope to get theorems about the hered- itarity of a weak crossed product A f arising from a tamely ramifed extension S/R. Recent results of Kauta [11] accomplish this goal when G is a cyclic group and A f is a cyclic algebra (which means that there is a generator σ ∈ G for G for which f(σ,σ i ) is a unit of S for i ∈ {0,1,2,...,|G| − 1},with the possibility that f(σ,σ |G|−1 ) is not invertible in S).Maximal orders are hereditary (Reiner [14] Theorem 17.3),so results concerning the hereditarity of a weak crossed product order A f will generalize Haile’s theorems about the maximality of A f in the unramiﬁed case.

1.INTRODUCTION AND BACKGROUND INFORMATION 12 1.7.Outline of results We now state the primary results of this thesis.Let R,F,K,S,G,and Σ f =

σ∈G Kx σ be as in the previous section.As usual,we assume that f takes values in S −{0} so that we may consider the R-order A f =

σ∈G Kx σ .We will not require the values of f to be invertible in S,so A f is a weak crossed product. In view of Theorem 1.6.3 above,we assume throughout this thesis that S/R is tamely ramiﬁed.We will investigate what conditions on the cocycle f are equivalent to the hereditarity of A f . We will see that many of Haile’s theorems concerning the maximality of A f in the unramiﬁed setting have generalizations concerning hereditarity of A f in the tamely ramiﬁed setting.The Jacobson radical of A f has a very similar description in the tamely ramiﬁed setting as in the unramiﬁed setting;in fact,the description is the same when S is local: Theorem1.If the extenstion S/R is tamely ramiﬁed and S is a DVR,then Rad(A f ) =

h∈H π S Sx h +

σ∈H Sx σ . Here H = {σ ∈ G:f(σ,σ −1 ) is a unit of S} is the inertial group.We also give a criterion to determine if A f is a hereditary order: Theorem 2.Let S/R be tamely ramiﬁed and let S be a DVR.The order A f is hereditary if and only if any of the following equivalent conditions is satisﬁed: (1) v S (f(σ,σ −1 )) ≤ 1 for all σ ∈ G,or equivalently,v S (f(σ 1 ,σ 2 )) ≤ 1 for all σ 1 ,σ 2 ∈ G. (2) Either H = G or the partial ordering on G/H is given by H ≤ τH ≤ τ 2 H ≤ ...≤ τ |G/H|−1 H for some τ ∈ G satisfying v S (f(τ,τ −1 )) = 1. (3) A f satisﬁes O l (Rad(A f )) = A f .