Sets of lengths in maximal orders in central simple algebras
Let be a holomorphy ring in a global field , and a classical maximal -order in a central simple algebra over . We study sets of lengths of factorizations of cancellative elements of into atoms (irreducibles). In a large majority of cases there exists a transfer homomorphism to a monoid of zero-sum sequences over a ray class group of , which implies that all the structural finiteness results for sets of lengths—valid for commutative Krull monoids with finite class group—hold also true for . If is the ring of algebraic integers of a number field , we prove that in the remaining cases no such transfer homomorphism can exist and that several invariants dealing with sets of lengths are infinite.
Key words and phrases:sets of lengths, maximal orders, global fields, Brandt groupoid, divisorial ideals, Krull monoids
2010 Mathematics Subject Classification:16H10, 16U30, 20M12, 20M13, 11R54
defipropdefi \aliascntresetthedefiprop \newaliascntdefilemmadefi \aliascntresetthedefilemma \newaliascntthmdefi \aliascntresetthethm \newaliascntlemmadefi \aliascntresetthelemma \newaliascntpropdefi \aliascntresettheprop \newaliascntcordefi \aliascntresetthecor \newaliascntremarkdefi \aliascntresettheremark \newaliascntexmdefi \aliascntresettheexm
Let be a (left- and right-) cancellative semigroup and its group of units. An element is called irreducible (or an atom) if with implies that or . If , then is a length of if there exist atoms with , and the set of lengths of , written as , consists of all such lengths. If there is a non-unit with , say , then for every , we have , which shows that sets of lengths become arbitrarily large. If is commutative and satisfies the ACC on divisorial ideals, then all sets of lengths are finite and non-empty.
Sets of lengths (and all invariants derived from them, such as the set of distances) are among the most investigated invariants in factorization theory. So far research has almost been entirely devoted to the commutative setting, and it has focused on commutative noetherian domains, commutative Krull monoids, numerical monoids, and others (cf. [1, 12, 27, 28, 26, 20, 7]). Recall that a commutative noetherian domain is a Krull domain if and only if the monoid of non-zero elements is a Krull monoid and this is the case if and only if the domain is integrally closed. Suppose that is a Krull monoid (so completely integrally closed and the ACC on divisorial two-sided ideals holds true). Then the monoid of divisorial two-sided ideals is a free abelian monoid. If is commutative (or at least normalizing), this gives rise to the construction of a transfer homomorphism , where is the monoid of zero-sum sequences over a subset of the class group of . Transfer homomorphisms preserve sets of lengths, and if is finite, then is a finitely generated commutative Krull monoid, whose sets of lengths can be studied with methods from combinatorial number theory. This approach has lead to a large variety of structural results for sets of lengths in commutative Krull monoids (see [27, 24] for an overview).
Only first hesitant steps were taken so far to study factorization properties in a non-commutative setting (for example, quaternion orders are investigated in [19, 18, 16]), semifirs in ([14, 15], semigroup algebras in ). The present paper provides an in-depth study of sets of lengths in classical maximal orders over holomorphy rings in global fields.
Let be a commutative Krull domain with quotient field , a central simple algebra over , a maximal order in , and the semigroup of cancellative elements (equivalently, is a PI Krull ring). Any approach to study sets of lengths, which runs as described above and involves divisorial two-sided ideals, is restricted to normalizing Krull monoids ([25, Theorem 4.13]). For this reason we develop the theory of divisorial one-sided ideals. In Section 3 we fix our terminology in the setting of cancellative small categories. Following ideas of Asano and Murata  and partly of Rehm [45, 46], we provide in Section 4 a factorization theory of integral elements in arithmetical groupoids, and introduce an abstract transfer homomorphism for a subcategory of such a groupoid (Theorem 3). In Section 5 the divisorial one-sided ideal theory of maximal orders in quotient semigroups is given, and Section 5 establishes the relationship with arithmetical groupoids. Theorem 4 is a main result in the abstract setting of arithmetical maximal orders (Remarks LABEL:*rem:ideal-struct.LABEL:*rem:ideal-struct:normalizing and LABEL:*rem:rtr.LABEL:*rtr:normalizing reveal how the well-known transfer homomorphisms for normalizing Krull monoids fit into our abstract theory). For maximal orders over commutative Krull domains, we see that all sets of lengths are finite and non-empty (Section 5.2). In Section 6 we demonstrate that classical maximal orders over holomorphy rings in global fields fulfill the abstract assumptions of Theorem 4, which implies the following structural finiteness results on sets of lengths.
Let be a holomorphy ring in a global field , a central simple algebra over , and a classical maximal -order of . Suppose that every stably free left -ideal is free. Then there exists a transfer homomorphism , where
is a ray class group of , and is the monoid of zero-sum sequences over . In particular,
The set of distances is a finite interval, and if it is non-empty, then .
For every , the union of sets of lengths containing , denoted by , is a finite interval.
There is an such that for every the set of lengths is an AAMP with difference and bound .
Thus, under the additional hypothesis that every stably free left -ideal is free, we obtain a transfer homomorphism to a monoid of zero-sum sequences over a finite abelian group. Therefore, sets of lengths in are the same as sets of lengths in a commutative Krull monoid with finite class group.
If satisfies the Eichler condition relative to , then every stably free left -ideal is free by Eichler’s Theorem. In particular, if is a number field and is its ring of algebraic integers, then satisfies the Eichler condition relative to unless is a totally definite quaternion algebra. Thus in this setting Theorem 1 covers the large majority of cases, and the following complementary theorem shows that the condition that every stably free left -ideal is free is indeed necessary.
Let be the ring of algebraic integers in a number field , a central simple algebra over , and a classical maximal -order of . If there exists a stably free left -ideal that is not free, then there exists no transfer homomorphism , where is any subset of an abelian group. Moreover,
For every , we have .
The proof of Theorem 2 is based on recent work of Kirschmer and Voight ([39, 40]), and will be given in Section 7. If is a commutative Krull monoid with an infinite class group such that every class contains a prime divisor, then Kainrath showed that every finite subset of can be realized as a set of lengths (, or [27, Section 7.4]), whence and for all . However, we explicitly show that in the above situation no transfer homomorphism is possible, implying that the factorization of cannot be modeled by a monoid of zero-sum sequences. A similar statement about sets of lengths in the integer-valued polynomials, as well as the impossibility of a transfer homomorphism to a monoid of zero-sum sequences, was recently shown by Frisch .
Let denote the set of positive integers and put . For integers , let denote the discrete interval. All semigroups and rings are assumed to have an identity element, and all homomorphisms respect the identity. By a factorization we always mean a factorization of a cancellative element into irreducible elements (a formal definition follows in Section 3). In order to study factorizations in semigroups we will have to investigate their divisorial one-sided ideal theory, in which the multiplication of ideals only gains sufficiently nice properties if one considers it as a partial operation that is only defined for certain pairs of ideals. This is the reason why we introduce our concepts in the setting of groupoids and consider subcategories of these groupoids.
Throughout the paper there will be many statements that can be either formulated “from the left” or “from the right”, and most of the time it is obvious how the symmetric statement should look like. Therefore often just one variant is formulated and it is left to the reader to fill in the symmetric definition or statement if required.
2.1. Small categories as generalizations of semigroups
Let be a small category. In the sequel the objects of play no role, and therefore we shall identify with the set of morphisms of . We denote by the set of identity morphisms (representing the objects of the category). There are two maps such that two elements are composable to a (uniquely determined) element if and only if .
A semigroup may be viewed as a category with a single object (corresponding to its identity element), and elements of the semigroup as morphisms with source and target this unique object. In this way the notion of a small category generalizes the usual notion of a semigroup ( is a semigroup if and only if ). We will consider a semigroup to be a small category in this sense whenever this is convenient, without explicitly stating this anymore. For we write for the set of all possible products, and if , then and .
An element is called left-cancellative if it is an epimorphism ( implies for all ), and it is called right-cancellative if it is a monomorphism ( implies for all ), and cancellative if it is both. The set of all cancellative elements is denoted by , and is called cancellative if . The set of isomorphisms of will also be called the set of units, and we denote it by . A subcategory is wide if .
In line with the multiplicative notation, if and are two small categories, we call a functor a homomorphism (of small categories). Explicitly, a map is a homomorphism if and whenever with then also is defined (i.e., ) and .
If is a commutative semigroup, and is a subsemigroup, then a localization with an embedding exists whenever all elements of are cancellative, and in particular has a group of fractions if and only if is cancellative. If is a non-commutative semigroup and , then a semigroup of right fractions with respect to , , in which every element can be represented as a fraction with , , together with an embedding , exists if and only if is cancellative and satisfies the right Ore condition, meaning for all and . For a semigroup of left fractions, , one gets the analogous left Ore condition, and if satisfies both, the left and the right Ore condition, then every semigroup of right fractions is a semigroup of left fractions and conversely. In this case we write . If satisfies the left and right Ore condition, we also write for the corresponding semigroup of fractions.
The notion of semigroups of fractions generalizes to categories of fractions with analogous conditions (). Let be a small category, and a subset of the cancellative elements. Then admits a calculus of right fractions if is a wide subcategory of and it satisfies the right Ore condition, i.e., for all and with . In that case there exists a small category with and an embedding (i.e., is a faithful functor) with and such that every element of can be represented in the form with , and , and it is universal with respect to that property, i.e., if is any homomorphism with , then there exists a unique such that . We can assume and take to be the inclusion map, and we call the category of right fractions of with respect to . If also admits a left calculus of fractions, then is also a category of left fractions, and we write .
A monoid is a cancellative semigroup satisfying the left and right Ore condition (following the convention of ). Every monoid has a (left and right) group of fractions which is unique up to unique isomorphism. A semigroup is called normalizing if for all . It is easily checked that a normalizing cancellative semigroup is already a normalizing monoid.
Let be a directed multigraph (i.e., a quiver). For every edge of we write for the vertex that is its source and for the vertex that is its target. The path category on , denoted by , is defined as follows: It consists of all tuples with , vertices of and edges of with either and or , , for all and . The set of identities is the set of all tuples with , and given any tuple as above, and . Composition is defined in the obvious manner by concatenating tuples and removing the two vertices in the middle. We identify the set of vertices of with so that . Every subset of a small category will be viewed as a quiver, with vertices and for each a directed edge (again called ) from to .
A groupoid is a small category in which every element is a unit (i.e., every morphism is an isomorphism). If and there exist and , then
is a bijection.
For all the set is a group, called the vertex group or isotropy group of at . If and , then, taking , the map in (1) is a group isomorphism from to . If is abelian, it can be easily checked that this isomorphism does not depend on the choice of : If , then
In particular, if is connected (meaning for all ) and one vertex group is abelian, then all vertex groups are abelian, and they are canonically isomorphic.
In this case we define for and the set , and the universal vertex group as
indeed has a natural abelian group structure: For every there is a bijection inducing the structure of an abelian group on , and because the diagrams
commute for every choice of and , this group structure is independent of the choice of , yielding a canonical group isomorphism for every . We will use calligraphic letters to denote elements of . If , then the unique representative of in , , will be denoted by .
If is a groupoid, and is a subcategory, then denotes the set of all right fractions of elements of . Furthermore, is a subgroupoid if and only if satisfies the right Ore condition.
2.3. Krull monoids and Krull rings
A monoid is called a Krull monoid if it is completely integrally closed (in other words, a maximal order) and satisfies the ACC on divisorial two-sided ideals. A prime Goldie ring is a Krull ring if it is completely integrally closed and satisfies the ACC on divisorial two-sided ideals (equivalently, its monoid of cancellative elements is a Krull monoid; see ). The theory of commutative Krull monoids is presented in [32, 27]. The simplest examples of non-commutative Krull rings are classical maximal orders in central simple algebras over Dedekind domains (see Section 5.2). We discuss monoids of zero-sum sequences.
Let be an additively written abelian group, a subset and let be the (multiplicatively written) free abelian monoid with basis . Elements are called sequences over , and are written in the form where and . We denote by the length of . Such a sequence is said to be a zero-sum sequence if . The submonoid
is called the monoid of zero-sum sequences over . It is a reduced commutative Krull monoid, which is finitely generated whenever is finite ([27, Theorem 3.4.2]). Moreover, every commutative Krull monoid possesses a transfer homomorphism onto a monoid of zero-sum sequences, and thus provides a model for the factorization behavior of commutative Krull monoids ([27, Section 3.4]).
3. Arithmetical Invariants
In this section we introduce our main arithmetical invariants (rigid factorizations, sets of lengths, sets of distances) and transfer homomorphisms in the setting of cancellative small categories.
Throughout this section, let be a cancellative small category.
is reduced if . An element is an atom (or irreducible) if with implies or . By we denote the set of all atoms of , and call atomic if every can be written as a (finite) product of atoms. A left ideal of is a subset with , and a right ideal of is defined similarly. A principal left (right) ideal of is a set of the form () for some . If is a commutative monoid, then is a prime element if implies or for all .
If satisfies the ACC on principal left and right ideals, then is atomic.
We first note that if then if and only if with , and similarly if and only if with . [We only show the statement for the right ideals. The non-trivial direction is showing that implies . Since implies and with , we get and . Since is cancellative, this implies and , hence and therefore .]
If , then there exist and such that .
Proof of Claim A.
Assume the contrary. Then the set
is non-empty, and hence, using the ascending chain condition on the principal right ideals, possesses a maximal element with . Then , and therefore with . But since , and thus maximality of in implies with and . But then , a contradiction. ∎
We proceed to show that every is a product of atoms. Again, assume that this is not the case. Then
is non-empty, and hence possesses a maximal element with (this time using the ascending chain condition on principal left ideals). Again as otherwise it would be a product of atoms. By Claim A, with and . Since , . Moreover, since and therefore with and . Thus is a product of atoms, a contradiction. ∎
Let denote the path category on atoms of . We define
and define an associative partial operation on as follows: If with ,
then the operation is defined if , and
while if . In this way is again a cancellative small category (with identities that we identify with again, and ). We define a congruence relation on it as follows: If with as before, then if , and either or there exist and such that
The category of rigid factorizations of is defined as
For with we write and the operation on is also denoted by . The length of is . There is a surjective homomorphism , induced by multiplying out the elements of the factorization in , explicitly . For , we define to be the set of rigid factorization of .
To simplify the notation, we make the following conventions:
If, for a rigid factorization , we have (i.e., ), then the unit can be absorbed into the first factor (replacing it by ), and we can essentially just work in , with defined to match the equivalence relation on .
If is reduced but , we often still write instead of the shorter , as is allowed and in the path category there is a different empty path for every .
If is reduced, then .
If is not reduced, the factor allows us to represent trivial factorizations of units, and the equivalence relation allows us to deal with trivial insertion of units. In the commutative setting these technicalities can easily be avoided by identifying associated elements and passing to the reduced monoid . Unfortunately, associativity (left, right or two-sided) is in general no congruence relation in the non-commutative case.
If is a commutative monoid, then , where is the free monoid on , while a factorization in this setting is usually defined as an element of the free abelian monoid , implying in particular that factorizations are unordered while rigid factorizations are ordered. The homomorphism obviously factors through the multiplication homomorphism , and the fibers consist of the different permutations of a factorization.
In the following we will only be concerned with invariants related to the lengths of factorizations, which may as well be defined using rigid factorizations.
the set of lengths of .
The system of sets of lengths of is defined as .
A positive integer is a distance of if there exists an such that and . The set of distances of is the set consisting of all such distances and is denoted by . The set of distances of is defined as
We define for .
is half-factorial if for all (equivalently, is atomic and ).
We write if and similarly if .
Definition & Lemma \thedefilemma.
Let be subcategories of a groupoid. The following are equivalent:
For all , implies ,
is called right-saturated if these equivalent conditions are fulfilled.
(a) (b): Let with , and . Then , i.e., and hence also . Since the left factor is uniquely determined as , it follows that .
(b) (a): Let . There exists with , and thus . Therefore , hence . ∎
Let be a reduced cancellative small category. A homomorphism is called a transfer homomorphism if it has the following properties:
If , and , then there exist such that , and .
The notion of a transfer homomorphism plays a central role in studying sets of lengths. It is easily checked that the following still holds in our generalized setting (cf. [27, §3.2] for the commutative case, [25, Proposition 6.4] for the non-commutative monoid case).
If is a transfer homomorphism, then for all and hence all invariants defined in terms of lengths coincide for and . In particular,
for all ,
for all , and .
Let be a cancellative small category, a finite abelian group and a transfer homomorphism. Then is half-factorial if and only if . If , then we have
is a finite interval, and if it is non-empty, then ,
for every , the set is a finite interval,
there exists an such that for every the set of lengths is an almost arithmetical multiprogression (AAMP) with difference and bound .
By the previous lemma it is sufficient to show these statements for the monoid of zero-sum sequences over a finite abelian group . is half-factorial if and only if by [27, Proposition 2.5.6]. The first statement is proven in , the second can be found in [24, Theorem 3.1.3]. For the definition of AAMPs and a proof of 3 see [27, Chapter 4]. ∎
4. Factorization of integral elements in arithmetical groupoids
In this section we introduce arithmetical groupoids and study the factorization behavior of integral elements. In Section 5 we will see that the divisorial fractional one-sided ideals of suitable semigroups form such groupoids. Thus in non-commutative semigroups arithmetical groupoids generalize the free abelian group of divisorial fractional two-sided ideals familiar from the commutative setting (see Section 4 and Section 4). This abstract approach to factorizations was first used by Asano and Murata in . We follow their ideas and also those of Rehm in [45, 46], who studies factorizations of ideals in rings in a different abstract framework. The notation and terminology for lattices follows , a reference for l-groups is . Section 4 is the main result on factorizations of integral elements in a lattice-ordered groupoid (due to Asano and Murata). We introduce an abstract norm homomorphism , and as the main result in this section, we present a transfer homomorphism to a monoid of zero-sum sequences in Theorem 3.
A lattice-ordered groupoid is a groupoid together with a relation on such that for all
is a lattice (we write and for the meet and join),
is a lattice (we write and for the meet and join),
is a sublattice of both and . Explicitly: For all it holds that and .
If and we write and . If we write and . By 3 this is unambiguous if and both hold. The restriction of to any of , or will in the following simply be denoted by again. (Keep in mind however that need not be a partial order on the entire set , and and do not represent meet and join operations on the entire set in the order-theoretic sense.)
An element of a lattice-ordered groupoid is called integral if and , and we write for the subset of all integral elements of .
A lattice-ordered groupoid is called an arithmetical groupoid if it has the following properties for all :
For , if and only if .
and are modular lattices.
If for and , then . Analogously, if and , then .
For every non-empty subset , exists, and similarly for . If moreover then .
contains an integral element.
and satisfy the ACC on integral elements.
For the remainder of this section, let be an arithmetical groupoid.
P5 implies in particular for all , i.e., is connected. If and , then is an order isomorphism by P3, and similarly every induces an order isomorphism from to . P2 could therefore equivalently be required for a single and a single . Moreover, since the map is also an order isomorphism (Lemma LABEL:*lemma:gb.LABEL:*gb:inv) and the property of being modular is self-dual, it is in fact sufficient that one of and is modular for one .
Using P5 we also observe that it is sufficient to have the ACC on integral elements on one and one : If, say, is an ascending chain of integral elements in and is integral, then is an ascending chain of integral elements in (Lemma LABEL:*lemma:gb.LABEL:*gb:integral), hence becomes stationary, and multiplying by from the left again shows that the original chain also becomes stationary.
We summarize some basic properties that follow immediately from the definitions.
holds if either or . In particular, for the following are equivalent: (a) ; (b) ; (c) ; (d) .
Let . If and are integral, then and .
If , and , then
and if ,
and if .
Let and . If exists, then also exists, and . Moreover, then also exists and . Analogous statements hold for and .
, , and in particular are conditionally complete as lattices.
The set of all integral elements forms a reduced wide subcategory of , and is the groupoid of (left and right) fractions of this subcategory.
For every , there exist and with and .
(a) (b) and (c) (d) by P1. For (a) (c) set .
Since , we have by P3. Similarly, .
We show (i), (ii) is similar. Since and , P3 implies and , thus . Therefore
and multiplying by from the left gives . Dually, .
Let . Then for all , , hence is an upper bound for . If is another upper bound for , then for all , hence and thus . Therefore .
For we have for all if and only if (in ), and follows.
We show the claim for , for the proof is similar. Let be bounded, say for some and all . Then is integral, hence exists by P4, and