Equivalent generating pairs of an ideal of a commutative ring
aa r X i v : . [ m a t h . A C ] D ec EQUIVALENT GENERATING PAIRS OF AN IDEAL OF ACOMMUTATIVE RING
LUC GUYOT
Abstract.
Let R be a commutative ring with identity and let I be a two-generated ideal of R . We denote by SL ( R ) the group of × matricesover R with determinant . We study the action of SL ( R ) by matrix right-multiplication on V ( I ) , the set of generating pairs of I . Let Fitt ( I ) bethe second Fitting ideal of I . Our main result asserts that V ( I ) / SL ( R ) identifies with a group of units of R/ Fitt ( I ) via a natural generalization ofthe determinant if I can be generated by two regular elements. This resultis illustrated in several Bass rings for which we also show that SL n ( R ) actstransitively on V n ( I ) for every n > . As an application, we derive aformula for the number of cusps of a modular group over a quadratic order. Introduction
Rings are supposed unital and commutative. The unit group of a ring R is denoted by R × . Let M be a finitely generated R -module. We denote by µ ( M ) the minimal number of generators of M . For n ≥ µ ( M ) , we denote by V n ( M ) the set of generating vectors of M of length n , i.e., the set of n -tuplesin M n whose components generate M . We consider the action of GL n ( R ) on V n ( M ) by matrix right-multiplication. Let SL n ( R ) be the subgroup of GL n ( R ) of determinant matrices. Let E n ( R ) be the subgroup of SL n ( R ) generated bythe elementary matrices , i.e., the matrices which differ from the identity by asingle off-diagonal element. Let G be a subgroup of GL n ( R ) . Two generatingvectors m , m ′ ∈ V n ( M ) are said to be G -equivalent , which we also denoteby m ′ ∼ G m , if there exists g ∈ G such that m ′ = m g . Apart from Section3.1 where M is any faithful two-generated module, M will be a two-generatedideal of R . Our chief concern is the description of the quotient V ( M ) /G with G ∈ { SL ( R ) , GL ( R ) } . In some applications, we also address the case G = SL n ( R ) with n > , see Propositions 6.6 and 6.12. Date : December 11, 2020.2020
Mathematics Subject Classification.
Primary 13E15, Secondary 13F05.
Key words and phrases.
Finitely generated ideals, special linear group, Fitting invariants,Bass rings; Hilbert modular group.
Let us highlight earlier results pertaining to this topic. When M = R ,the elements of V n ( M ) are called unimodular rows of size n . The orbit set V n ( R ) / GL n ( R ) is in one-to-one correspondence with the isomorphism classesof the stably free R -modules of type [Lam06, Proposition I.4.8]. It is eas-ily seen that GL ( R ) acts transitively on V ( R ) . For n > , the questionas to whether GL n ( R ) acts transitively on V n ( R ) is central in the resolu-tion of Serre’s problem on projective modules [Lam06, Corollary I.4.5]. Das,Tikader and Zinna have described the van der Kallen Abelian group structureof V n +1 ( R ) / SL n +1 ( R ) [vdK83] for R a smooth affine real algebra of dimension n ≥ [DTZ18, Theorem 1.2]; see also [Fas11] for seminal results of this flavor.Although no result in this paper refers directly to E n ( R ) , let us mentionthat V n ( R ) / E n ( R ) has been extensively studied and can be endowed with thestructure of an Abelian group if R is of finite stable rank and n is large enough[Lam06, Section VIII.5] . For R = S [ x , . . . , x m ] , the ring of polynomials in m indeterminates over a ring S , Murthy has determined conditions under whichGL n ( R ) acts transitively on V n ( P rk =1 Rx i ) for r ≤ n ≤ m [Mur03].To introduce further results, let us define the determinant det( M ) of afinitely generated R -module M as the exterior product V µ M where µ = µ ( M ) .The module det( M ) is a cyclic R -module whose annihilator is Fitt µ − ( M ) , the µ -th Fitting ideal of M (see Section 2.1 for definitions). The determinant map det : M n → V n M is defined by det( m , . . . , m µ ) = m ∧ · · · ∧ m n . It is imme-diate to check that det(V µ ( M )) ⊆ V (det( M )) . For R a quasi-Euclidean ringand M an arbitrary finitely generated R -module, we proved that the determi-nant map induces a bijection from V µ ( M ) / E µ ( R ) onto V (det( M )) [Guy17,Theorem A]. In this paper, we aim for a similar description of V ( I ) / SL ( R ) for I a two-generated ideal of an arbitrary ring R .To state our main theorem, we need the following definitions. An elementof a ring R is said to be regular if it is not a zero-divisor. An ideal of R is regular if it contains a regular element. The total ring of quotients K ( R ) of R is the localization R [ S − ] where S is the set of regular elements of R .An R -submodule I of K ( R ) is said to be a fractional ideal of R if there is aregular element d of R such that dI ⊆ R . For I a fractional ideal of R , we set I − + { x ∈ K ( R ) | xI ⊆ R } . Theorem A (Theorem 3.5) . Let I be a two-generated ideal of a ring R suchthat Fitt ( I ) = II − , e.g., I is generated by two regular elements. Then thedeterminant map induces an injection from V ( I ) / SL ( R ) into V (det( I )) ≃ ( R/ Fitt ( R )) × . We investigate in Section 4 conditions under which the determinant mapinduces a bijection. The main outcome is
QUIVALENT GENERATING PAIRS OF AN IDEAL OF A COMMUTATIVE RING 3
Corollary B (Corollary 4.4) . Let R be a ring whose quotients by regular idealshave stable rank . Let I be an ideal of R which can generated by two regularelements. Then the determinant map induces a bijection from V ( I ) / SL ( R ) onto V (det( I )) . In the remainder of this introduction, we shall illustrate consequences ofTheorem A in several Bass rings (see Section 6 for definition and background).In such rings, every ideal can be generated by two elements.Our first application generalizes a theorem of Maass [vdG88, PropositionI.1.1] asserting that the number of cusps of the Hilbert modular group PSL ( O ) ,with O the ring of integers of a totally real algebraic number field, is equalto the class number of O . To introduce this result, further definitions arerequired.The ring of mutlipliers ̺ ( I ) of a factional ideal I of R is the overring { x ∈ K ( R ) | xI ⊆ I } . The ideal class [ I ] of a fractional ideal I of R is the equivalenceclass of I for the following relation on fractional ideals: I ∼ J if there are tworegular elements r, s ∈ R such that rI = sJ . We denote by Cl( R ) the monoidof the ideal classes of R . We define the projective line P ( K ( R )) over K ( R ) as the quotient of K ( R ) × K ( R ) \ { (0 , } by the equivalence relation definedthrough: v ∼ w if there are two regular elements r, s ∈ R such that r v = s w .There is a natural action of SL ( R ) on P ( K ( R )) from the right. The set of cusps of R is the orbit set P ( K ( R )) / SL ( R ) . Corollary C (Propositions 5.1 and 5.4) . Let R be an order in a number field.Then the number of cusps of R is X [ I ] ∈ Cl( R ) , µ ( I ) ≤ [( R/ Fitt ( I )) × : ǫ ( I )] . where ǫ ( I ) is the natural image of ̺ ( I ) × in ( R/ Fitt ( I )) × , see Definition 5.3. The finiteness of
Cl( R ) for R an order in a number field is a direct conse-quence of [BS66, Theorem 2.6.3]. As a result, such an order has finitely manycusps, which is a special case of [Shi63, Theorem 5]. We use Corollary C toeffectively compute the number of cusps of a quadratic order. To state thisresult, we shall denote by Pic( R ) , the Picard group of R , that is, the group ofthe invertible elements of Cl( R ) . Theorem D.
Let O be the ring of integers of a quadratic number field. Let f > be a rational integer and let O f be the unique order of index f in O .Then the number of cusps of O f is X f ′ | f ϕ ( f /f ′ )2 ǫ ( f,f ′ ) | Pic( O f ′ ) | LUC GUYOT where ϕ is the Euler totient function and ǫ ( f, f ′ ) = 1 if O f ′ has a unit of norm − and f /f ′ > , else ǫ ( f, f ′ ) = 0 . By [Neu99, Theorem 12.12], we have | Pic( O f ′ ) | = | Pic( O ) || O × / O × f ′ | | ( O / O f ′ ) × || ( O f ′ / O f ′ ) × | and moreover | Pic( O ) | divides | Pic( O f ′ ) | . Thus | Pic( O ) | , which is the numberof cusps of O , divides the number of cusps of O f . Note that the computationof ǫ ( f, f ′ ) is a well-studied and solved problem, see e.g., [Fur59, Theorem 2],[Jen62] and the seminal work of Rédei [R´53].Theorem D relies on the computation of Fitt ( I ) for an arbitrary ideal I of R = O f . This computation allows us moreover to describe V ( I ) / SL ( R ) , Aut R ( I ) \ V ( I ) / SL ( R ) and V ( I ) / GL ( R ) and to show that SL n ( R ) acts transitively on V n ( I ) for every n > in Section6.1 below. The results of Section 6.1 enable us to solve in [Guy20b] a problem incombinatorial group theory, namely the description of the Nielsen equivalenceclasses of generating vectors in semi-direct products of the form Z ⋊ Z .We present now an analysis similar to the one of Section 6.1 for anotherBass domain: the coordinate ring R of the curve defined by the equation y = x n over an arbitrary field K where n ≥ is an odd integer. It iswell-known and easy to show that R is isomorphic to K [ x , x n ] . For thisexample, the computation of ( R/ Fitt ( I )) × yields also an explicit descriptionof V ( I ) / SL ( R ) . The following definition is required to state this last result.For S ⊆ K ( R ) , a ring containing R , the conductor ( R : S ) of R in S is the R -ideal { r ∈ R | rS ⊆ R } . Proposition E (Lemma 6.11 and Proposition 6.12) . Let K be a field and let R = K [ x , x n ] where n ≥ is an odd rational integer. Let I be an ideal of R and let f be the smallest positive odd rational integer such that x f I ⊆ I . Thenthe following hold. ( i ) ̺ ( I ) = K [ x ] + K [ x ] x f − . ( ii ) Fitt ( I ) = ( R : ̺ ( I )) = K [ x ] x n − f + K [ x ] x n − . ( iii ) The ring R/ Fitt ( I ) is isomorphic to K [ x ] /K [ x ] x n − f . ( iv ) The group SL k ( R ) acts transitively on V k ( I ) for every k > . Thanks to Corollary B and Proposition E. iii , we can identify V ( I ) / SL ( R ) with the group ( K [ x ] /K [ x ] x n − f ) × which in turn is isomorphic to K n − f − × K × where K + is the additive group of K . QUIVALENT GENERATING PAIRS OF AN IDEAL OF A COMMUTATIVE RING 5
Final remarks.
Within the course of the proofs of Theorem D and PropositionE, we established a common fact that we couldn’t generalize to arbitrary Bassdomains:
Fitt ( I ) = ( R : ̺ ( I )) for I a faithful ideal of R . In a Bass ring, thisimplies that Fitt ( I ) lies in a finite set of conductors as R has only finitelymany overrings contained in its normalization [LW85, Proposition 2.2. ii ]. Wealso noticed for each Bass ring example in this article the following properties: • Fitt (Fitt ( I )) = Fitt ( I ) , • ̺ (Fitt ( I )) = ρ ( I ) . Question 1.1.
To which extent can these observations be generalized?
Layout.
This paper is organized as follows. Section 2 introduces notationand background results. It encloses some of the fundamental properties ofthe determinant map and of the Bass stable rank. Section 3 presents a local-global criterion to study the action of SL ( R ) and then the proof of TheoremA. Section 4 presents several conditions under which the determinant mapinduces a bijection in Theorem A. It includes in particular Corollary B and itsproof. Section 5 is dedicated to the proof of Corollary C. Sections 6.1 and 6.2address in details the orders of a quadratic number field and the affine domain K [ x , x n ] . These two sections contain respectively the proof of Theorem D andof Proposition E. Acknowledgments.
We are grateful to Justin Chen, François Couchot andJean Fasel for their encouragements. We are specially thankful to Henri Lom-bardi and Wilberd van der Kallen for helpful comments and references.2.
Notation and background
We assume throughout that R is a commutative unital ring and M is afinitely generated R -module. If p is a prime ideal of R , we denote by R p thelocalization of R at p . Similarly, we denote by M p its localization at p , whichwe identify with M ⊗ R R p . Given an element m in M , we abuse notation indenoting also by m its image m ⊗ R ∈ M p . If M and N are two submodules ofa given R -module, we denote by ( M : N ) the ideal consisting of the elements r ∈ R such that rN ⊆ M . If Rm and Rm ′ are two cyclic submodules of M ,we simply write ( m : m ′ ) for ( Rm : Rm ′ ) . Matrices over R and M . Given two × n row vectors m = ( m , . . . , m n ) ∈ M n and r = ( r , . . . , r n ) ∈ R n , we denote by m ⊤ and r ⊤ the n × columnvectors obtained by transposition and we define the dot products mr ⊤ = rm ⊤ + P i r i m i . Based on these identities, the product of any two matricesover M and R , with compatible numbers of rows and columns, is uniquelydefined. We denote respectively by m × n and by n the m × n zero matrix andthe n × n identity matrix over R . LUC GUYOT
Fitting ideals and the determinant module.
The determinant mapis the invariant on which our study of V ( M ) / SL ( R ) hinges. This map canbe effectively computed by means of the second Fitting ideal. Fitting ideals.
Let F ϕ −→ G → M → be an exact sequence such that F and G are free modules over R and G isfinitely generated. Let I i ( ϕ ) the ideal of R generated by the i × i minors ofthe matrix of ϕ : F → G , agreeing that I ( ϕ ) = R . Then the ( i + 1) -thFitting ideal of M is defined as Fitt i ( M ) + I µ ( G ) − i ( ϕ ) . The Fitting ideals areindependent of ϕ by Fitting’s lemma [Eis95, Corollary 20.4]. They commutewith base change , i.e, Fitt i ( M ⊗ R S ) = Fitt i ( M ) ⊗ R S for any R -algebra S [Eis95, Corollary 20.5], and hence with localization. The determinant of a finitely generated module.
Let µ = µ ( M ) . Re-call that the determinant det( M ) of M is the exterior product V µ M . Thisis a cyclic R -module whose annihilator is Fitt µ − ( M ) [Eis95, Exercise 20.9.i].Given m = ( m , . . . , m µ ) ∈ V µ ( M ) , we denote by φ m : det( M ) → R/ Fitt µ − ( M ) the R -isomorphism induced by the map m ∧· · ·∧ m µ → µ − ( M ) . Givenanother generating pair m ′ = ( m ′ , . . . , m ′ µ ) ∈ M µ , we definedet m ( m ′ ) + φ m ( m ′ ∧ · · · ∧ m ′ µ ) . It is easily checked that the image of V µ ( M ) under det m is a subgroup of ( R/ Fitt µ − ( M )) × which does not depend on the choice of m .The following lemma is straightforward. Lemma 2.1.
Let m , m ′ , m ′′ ∈ V µ ( M ) . Then the following assertions hold. ( i ) det m ( m ′ A ) = det( A ) det m ( m ′ ) , for every µ × µ matrix A over R . ( ii ) det m ( m ′′ ) = det m ( m ′ ) det m ′ ( m ′′ ) . ( iii ) det m ( m ′ ) = 1+Fitt µ − ( M ) , if and only if, det m ( m ′ ) = 1+Fitt µ − ( M p ) for every maximal ideal p of R , where m and m ′ denote (abusively)their natural images in V µ − ( M p ) in the left-hand side of the identity. (cid:3) When there is no risk of ambiguity, we simply denote by the identityelement of R/ Fitt µ − ( M ) . The ideal Fitt µ − ( M ) has a convenient descriptionwhich makes the computation of det m effective when a workable presentationof M is given. Given m ∈ V µ ( M ) , we say that an element r is involved in arelation of m if there is ( r , . . . , r µ ) ∈ R µ such that P µi =1 r i m i = 0 and r = r i for some i . Lemma 2.2.
Let m ∈ V µ ( M ) . Then Fitt µ − ( M ) is the set of the elements of R involved in a relation of m . QUIVALENT GENERATING PAIRS OF AN IDEAL OF A COMMUTATIVE RING 7 (cid:3)
Given m ∈ V µ ( M ) , let m = π ( m ) be the image of m ∈ V µ ( M ) by thenatural map π : M → M/ Fitt µ − ( M ) M and let e be the canonical basis of ( R/ Fitt µ − ( M )) µ . Then the map m e induces an isomorphism ϕ m from M/ Fitt µ − ( M ) M onto ( R/ Fitt µ − ( M )) µ . The following lemma shows howthe map π can be used to compute det m . Lemma 2.3.
Let m , m ′ ∈ V µ ( M ) . Then det m ( m ′ ) is the determinant of theunique µ × µ matrix A over R/ Fitt µ − ( M ) satisfying π ( m ′ ) = π ( m ) A , that isthe matrix of ϕ m ◦ π ( m ′ ) with respect to e . (cid:3) Ranks.
Following [MR87, Section 6.7.2], we define the Bass stable rankof a finitely generated R -module M . An integer n > lies in the stablerange of M if for every m = ( m , . . . , m n +1 ) ∈ V n +1 ( M ) , there is r ∈ R such that ( m + rm n +1 , . . . , m n + rm n +1 ) belongs to V n ( M ) . If n lies in thestable range of M , then so does k for every k > n [MR87, Lemma 11.3.3].The stable rank sr ( M ) of M is the least integer in the stable range of M .Extending naturally the linear rank introduced in [MR87, Section 11.3.3] tofinitely generated modules, we define the linear rank glr ( M ) of M as the leastinteger n ≥ µ ( M ) such that GL k ( R ) , acts transitively on V k ( M ) for every k > n . Note that the analogous definition based on SL k ( R ) yields also therank glr ( M ) . It is easily checked that SL ( R ) acts transitively on V ( R ) .Hence glr ( R ) = 2 is equivalent to glr ( R ) = 1 . A ring R is said to be a Hermitering if glr ( R ) = 1 . A ring R is semi-local if it has only finitely many maximalideals. Proposition 2.4.
The following hold: ( i ) sr ( R ) = 1 if R is semi-local [Bas64, Corollary 10.5] . ( ii ) sr ( R ) ≤ dim Krull ( R ) + 1 [Hei84, Corollary 2.3] or [CLQ04, Theorem2.4] . A ring R is K-Hermite if for every n ≥ and every r ∈ R n , there is γ ∈ GL n ( R ) such that r γ = (0 , . . . , , d ) for some d ∈ R . The following lemmawill be used to show that SL n ( R ) acts transitively on V n ( I ) for R and I as inPropositions 6.6 and 6.12. Lemma 2.5.
Let R be a ring which is a free module of rank k over a subring S . Assume that S is a K-Hermite ring. Let I be an ideal of R such that µ ( I ) ≤ k and let n > k . Then for every m ∈ V n ( I ) , there is σ ∈ SL n ( R ) suchthat m σ = ( m ′ , , . . . , for some m ′ ∈ V k ( I ) . The proof is a straightforward induction on k consisting mainly in unrollingdefinitions. It is therefore omitted. LUC GUYOT Generating pairs and the determinant map
A local-global principle.
In this section M denotes a two-generated R -module. We shall prove that the SL ( R ) -equivalence of two generating pairsof M is a local property when M is faithful. This property is key to general-ize Theorem A to every finitely generated R -modules of a ring R with someDedekind-like properties [Guy20a, Theorem A]. Proposition 3.1.
Let M be a two-generated faithful R -module. Let m , m ′ ∈ V ( M ) . Then the following are equivalent: ( i ) m ∼ SL ( R ) m ′ , ( ii ) m ∼ SL ( R p ) m ′ for every maximal ideal p of R . As an immediate consequence of Proposition 3.1, we observe that SL ( R ) acts transitively on V ( M ) if M is a two-generated faithful projective mod-ule of constant rank . This well-known fact is instrumental in the proof ofMaass theorem on the number of cusps of the Hilbert modular group [vdG88,Proposition I.1.1]. More generally, we have Corollary 3.2.
Let M be a two-generated locally cyclic R -module. If thenatural homomorphism SL ( R ) → SL ( R/ ann( M )) is surjective, then SL ( R ) acts transitively on V ( M ) . (cid:3) Remark 3.3.
The natural map SL n ( R ) → SL n ( R ) is surjective for every n and every quotient R of R if sr ( R ) ≤ [EO67, Corollary 8.3] . See also thepartial stability result [EO67, Theorem 8.3] when R is a ring of univariatepolynomials over a principal ideal domain. The proof of Proposition 3.1 relies essentially on
Lemma 3.4.
Let M be an R -module and let A = ( a ij ) be an m × n matrixwith coefficients in M . Let b = ( b i ) an n × column vector with coefficients in R . Then the following are equivalent: ( i ) The linear system Ax = b has a solution in R m , ( ii ) For every maximal ideal p of R , the linear system Ax = b has a solutionin R m p . Lemma 3.4 is a simple variation of [HSG86, Proposition 1]. A proof isprovided for the convenience of the reader.
Proof of Lemma 3.4.
The implication ( i ) ⇒ ( ii ) is evident. Assuming that ( ii ) holds true, we shall prove that so does ( i ) . For every prime ideal p , we can findby hypothesis m elements x j ( p ) in R with j ∈ { , . . . , m } and two elements QUIVALENT GENERATING PAIRS OF AN IDEAL OF A COMMUTATIVE RING 9 s ( p ) , t ( p ) ∈ R \ p , such that t ( p ) (cid:16)P mj =1 x j ( p ) a ij (cid:17) = s ( p ) t ( p ) b i holds for every i ∈ { , . . . , n } . Since the ideal of R generated by all the elements s ( p ) t ( p ) is notcontained in any maximal ideal of R , these elements generate R . Therefore wecan find l elements λ , . . . , λ l ∈ R for some l ≥ and l prime ideals p , . . . , p l such that P lk =1 λ k s ( p k ) t ( p k ) = 1 . Setting x j + P lk =1 λ k t ( p k ) x j ( p k ) yields asolution x = ( x j ) of the linear system Ax = b in R m . (cid:3) If A is a square matrix over R , we denote by adj( A ) its adjugate matrix ,i.e., the transpose of the cofactor matrix of A . The adjugate matrix satisfiesthe identities A adj( A ) = adj( A ) A = det( A )1 n . If A = (cid:18) a a a a (cid:19) , then adj( A ) = (cid:18) a − a − a a (cid:19) . Proof of Proposition 3.1.
In order to prove the equivalence ( i ) ⇔ ( ii ) , it suf-fices to show that m ∼ G m ′ is equivalent to the existence of a solution of alinear system as in Lemma 3.4. Let σ be a × matrix over R such that m σ = m ′ . Since M is faithful and m generates M , the identity det( σ ) = 1 holds true if and only if m σ adj( σ ) = m , which is equivalent to m ′ adj( σ ) = m .Therefore m ∼ G m ′ holds true if and only if the following linear system withcoefficients in M (cid:26) m σ = m ′ , m ′ adj( σ ) = m has a solution σ ∈ R . (cid:3) Proof of Theorem A.
In this section, we shall prove Theorem A, thatis
Theorem 3.5.
Let I be a two-generated ideal of R such that Fitt ( I ) = II − ,e.g., I is generated by two regular elements. Let m , m ′ ∈ V ( I ) . Then thefollowing are equivalent. ( i ) m ∼ SL ( R ) m ′ , ( ii ) det m ( m ′ ) = 1 .In other words, the determinant map induces an injection from V ( I ) / SL ( R ) into V (det( I )) ≃ ( R/ Fitt ( R )) × . The proof of Theorem 3.5 relies on the following two lemmas.
Lemma 3.6.
Let I be a two-generated ideal of R . Let m , m ′ ∈ V ( I ) and let A be a × matrix over R such that m A = m ′ . Let N m be the set of × matrices N over R such that m N = (0 , . Let Φ A : N m → R be the mapdefined by Φ A ( N ) = det( A + N ) − det( A ) . Then we have II − ⊆ Φ A ( N m ) ⊆ Fitt ( I ) . If I is moreover faithful, then Φ A ( N m ) is an ideal of R . Lemma 3.7.
Let I be an ideal generated by two regular elements of R . Then Fitt ( I ) = II − .Proof. Let m = ( a, b ) ∈ V ( I ) . By Lemma 2.2, the ideal Fitt ( I ) is generatedby the components of the row vectors c such that mc ⊤ = 0 . Therefore Fitt ( I ) = ( a : b ) + ( b : a ) = (cid:16) R ∩ R ab (cid:17) + (cid:18) R ∩ R ba (cid:19) = (cid:18) Ra ∩ Rbab (cid:19) ( Ra + Rb ) = J I. where J + Ra ∩ Rbab . Since by definition I − = (cid:0) a R (cid:1) ∩ (cid:0) b R (cid:1) = J , the resultfollows. (cid:3) Proof of Lemma 3.6.
It is straightforward to check that Φ A ( N ) = det( N ) +trace(adj( A ) N ) . Let us fix N ∈ N m and an exact sequence F ϕ −→ R π m −−→ I → where F is a free R -module and π m is the R -module epimorphism mappingthe canonical basis of R to m . Since m N = 0 , the columns of N are R -linear combinations of the columns of the matrix of ϕ . Therefore, we have det( N ) ∈ I ( ϕ ) and trace(adj( A ) N ) ∈ I ( ϕ ) , which yields Φ A ( N ) ∈ Fitt ( I ) .Let us write m = ( a, b ) and m ′ = ( a ′ , b ′ ) and set N ( r, s ) + (cid:18) rb sb − ra − sa (cid:19) for r, s ∈ I − . Using the identity m ′ = m A , we easily infer that Φ A ( N ( r, s )) = rb ′ − sa ′ , which shows that II − ⊆ Φ A ( N m ) . If in addition I is faithful, thenwe have Fitt ( I ) = I ( ϕ ) = 0 [Eis95, Proposition 20.7.a]. As a result we get det( N ) = 0 and hence Φ A ( N ) = trace(adj( A ) N ) , so that Φ A ( N m ) is an idealof R . (cid:3) We are now in position to prove Theorem 3.5.
Proof of Theorem 3.5.
The implication ( i ) ⇒ ( ii ) is evident. Let us prove ( ii ) ⇒ ( i ) . Since det m ( m ′ ) = 1 , there is a × matrix A over R such that m A = m ′ and det( A ) ∈ ( I ) . As II − = Fitt ( I ) , we infer fromLemma 3.6 that Φ A ( N m ) = Fitt ( I ) . Hence we can find a matrix B such that m B = m ′ and det( B ) = 1 , which completes the proof. (cid:3) The following lemma provides a condition under which Theorem 3.5 applieswithout assuming the generators of I to be regular. Lemma 3.8.
Let I be a two-generated maximal ideal of R which is faithful butnot projective of constant rank . Then we have Fitt ( I ) = II − = I . QUIVALENT GENERATING PAIRS OF AN IDEAL OF A COMMUTATIVE RING 11
Proof.
Since I ⊆ II − ⊆ Fitt ( I ) holds for any two-generated ideal I of R , itis sufficient to show that Fitt ( I ) ( R when I is not projective of constantrank . The latter follows from [Eis95, Propositions 20.7.a and 20.8]. (cid:3) We conclude this section with an example, namely the specialization of[Mur03, Corollary 1.2] to the case of two variables.
Example 3.9.
Let S = R [ x, y ] be the ring of bivariate polynomials over a ring R and let I = Sx + Sy . As x and y are regular, Theorem 3.5 applies. It is easilychecked that Fitt ( I ) = I so that the determinant map induces an injective map V ( I ) / SL ( S ) → R × which is clearly surjective. Thus V ( I ) / SL ( S ) ≃ R × . Conditions under which V ( I ) / SL ( R ) identifies with ( R/ Fitt ( I )) × In this section, we investigate conditions under which the induced map ofTheorem A is a bijection. This is certainly the case if the natural group ho-momorphism R × → ( R/ Fitt ( I )) × is surjective. The following lemma derivesa less restrictive condition under which the identification V ( I ) / SL ( R ) ≃ ( R/ Fitt ( I )) × can also be made. Lemma 4.1.
Let I be a two-generated ideal of R . Assume that we can find ( a, b ) ∈ V ( I ) such that the natural group homomorphism ( R/ ( a : b )) × → ( R/ Fitt ( I )) × is surjective Then det ( a,b ) : V ( I ) → ( R/ Fitt ( I )) × is surjective.Proof. Let u ∈ ( R/ Fitt ( I )) × . By hypothesis, we can find r ∈ R such that r + Fitt ( I ) = u and r + ( a : b ) is a unit of R + R/ ( a : b ) . Let ϕ : I/Ra → R be the R -isomorphism which maps b + Ra to a : b ) . Since ϕ ( rb + Ra ) = r + ( a : b ) , we deduce that I is generated by ( a, rb ) . We conclude the proof byobserving that det ( a,b ) ( a, rb ) = r det ( a,b ) ( a, b ) = u . (cid:3) We say that the ring R has property ( ∗ ) if every surjective ring homomor-phism R ։ S induces a surjective group homomorphism R × ։ S × . If thereis ( a, b ) ∈ V ( I ) such that R/ ( a : b ) has ( ∗ ) , then we infer from Lemma 4.1that det ( a,b ) is surjective. A ring R is called a semi-field if the Krull dimensionof R/ J ( R ) is zero where J ( R ) denotes the Jacobson radical of R . Semi-fieldsenjoy property ( ∗ ) [LQ15, Exercise IX.6.16] [Che20, Theorem 5.1]. semi-localrings and von Neuman regular rings belong to the class of semi-fields, whichis a subclass of the local-global rings, see e.g., [LQ15, Fact IX.6.2] and [FS01,Examples 4.1 and 4.2 of Section V.4]. Local-global rings have stable rank and it turns out that this property fully charaterizes rings with ( ∗ ) : Proposition 4.2. [EO67, Lemma 6.1]
The following are equivalent: ( i ) The ring R has property ( ∗ ) . ( ii ) sr ( R ) = 1 .Proof. Assume that the natural homomorphism R × → ( R/ a ) × is surjective forevery ideal a of R . Let ( a, b ) ∈ V ( R ) . Then a + a is unit of R/I with a = Rb .Therefore we can find u ∈ R × such that a + a = u + a , i.e., there is r ∈ R verifying a + rb = u . As a result sr( R ) = 1 . Let us assume now that sr( R ) = 1 .Let a be an ideal of R and let a + a ∈ ( R/ a ) × . Then we can find b ∈ a suchthat ( a, b ) ∈ V ( R ) . Because R is of stable rank , there is r ∈ R such that u + a + rb is a unit of R . (cid:3) We present now several criteria which guarantee that the induced map ofTheorem A is a bijection.
Proposition 4.3.
Let I be a two-generated ideal of R such that the two fol-lowing hold: • Fitt ( I ) = II − . • There is ( a, b ) ∈ V ( I ) such that the natural group homomorphism ( R/ ( a : b )) × → ( R/ Fitt ( I )) × is surjective.Then det ( a,b ) induces a bijection from V ( I ) / SL ( R ) onto ( R/ Fitt ( I )) × .Proof. Combine Theorem 3.5 and Lemma 4.1. (cid:3)
We say that a ring R is almost of stable rank if R/Rr has stable rank forevery regular element r ∈ R . This definition adapts Mac Govern’s definition[McG08, Section 4] to allow the existence of zero divisors in R . A ring ofKrull dimension at most is almost of stable rank . More generally, almostlocal-global rings in the sense of Couchot [Cou07] are almost of stable rank . Corollary 4.4.
Let R be a ring which is almost of stable rank . Let I be anideal generated by two regular elements. Then the determinant map induces abijection from V ( I ) / SL ( R ) onto V (det( I )) ≃ ( R/ Fitt ( I )) × .Proof. It suffices to check that the two conditions of Proposition 4.3 hold true.By Lemma 3.7, we have
Fitt ( I ) = II − . By assumption, the ring R/ ( a : b ) has stable rank and thus enjoys property ( ∗ ) . As a result, the natural map ( R/ ( a : b )) × → ( R/ Fitt ( I )) × is surjective, which completes the proof. (cid:3) Corollary 4.5.
Let R be an integral domain which is almost of stable rank .Assume moreover that R is a free module of rank over a K-Hermit subring S .Let I be a two-generated ideal of R . Then SL n ( R ) acts transitively on V n ( I ) for every n > . QUIVALENT GENERATING PAIRS OF AN IDEAL OF A COMMUTATIVE RING 13
Proof.
We fix ( a, b ) ∈ V ( I ) and consider an arbitrary generating vector m ∈ V n ( I ) for n > . As R is a free S -module of rank , Lemma 2.5 applies. Hencewe can find σ ∈ SL n ( R ) such that m σ = ( m ′ , , . . . , for some m ′ ∈ V ( I ) .By Lemma 4.1 and its proof, we can find σ ′ ∈ SL ( R ) and r ∈ R such that m ′ σ ′ = ( a, rb ) . Thus we have ( m ′ , , . . . , ∼ SL n ( R ) ( a, rb, , . . . , ∼ E n ( R ) ( a, rb, b, , . . . , ∼ E n ( R ) ( a, b, , . . . , . Therefore m ∼ SL n ( R ) ( a, b, , . . . , . (cid:3) We denote by
Min( R ) the minimal prime spectrum of R , that is, the set ofminimal prime ideals of R . It is well-known that a Noetherian ring has a finiteminimal prime spectrum [Eis95, Theorem 3.1] and this holds more generallyfor rings with the ascending condition on radical ideals. Let R be a reducedring with minimal prime spectrum. Let I be a faithful ideal of R generated by µ elements. We shall establish that I can be generated by µ regular elements.Combining this fact with Corollary 4.4 above, we obtain: Corollary 4.6.
Assume that
Min( R ) is finite and that R is reduced and al-most of stable rank . Let I be a two-generated faithful ideal of R . Then thedeterminant map induces a bijection from V ( I ) / SL ( R ) onto V (det( I )) ≃ ( R/ Fitt ( I )) × . The following lemma resolves our debt with the proof of Corollary 4.6.
Lemma 4.7.
Let R be a reduced ring. Then the following hold. ( i ) The set of zero divisors of R is the union of the minimal prime idealsof R . ( ii ) The annihilator of an ideal of R is the intersection of the minimal primeideals that do not contain it. ( iii ) If R is Noetherian then the set Ass( R ) of associated primes of R andthe set Min( R ) of minimal prime ideals of R are finite and coincide. ( iv ) If Min( R ) is finite and I is a faithful ideal generated by m ∈ V n ( I ) ,then there is m ′ ∈ V n ( I ) , every component of which is regular, andsuch that m ∼ E n ( R ) m ′ .Proof. ( i ) . See for instance [Mat83, Proposition 1.1]. ( ii ) . This is routine. ( iii ) . As the set of zero divisors of R is the union of the associated primesof R [Eis95, Theorem 3.1.b], the result follows from ( i ) together with primeavoidance [Eis95, Lemma 3.3]. ( iv ) . Let E be a subset of Min( R ) such that T p ∈ E p = { } and T p ∈ F p = 0 for every proper subset F ( E . For each p in E , we pick r p ∈ (cid:16)T q ∈ E, q = p q (cid:17) \ { } . If r is a regular element, then rr p = 0 forevery p ∈ E . Hence r does not belong to any p in E . Thus S p ∈ E p is the set ofzero-divisors of R and it follows from ( i ) and prime avoidance [Eis95, Lemma E = Min( R ) . Let I be a faithful ideal of R that is generated by n elements, say r , . . . , r n . Let p ∈ Min( R ) . Then at least one generator, say r ,does not belong to p , since otherwise r p I = 0 would hold with r p as above.Replacing each r i that lies in p by r i + r p r yields a new generating vectorwith no component in p . By iterating this process over Min( R ) , we eventuallyobtain a vector of n regular generators of I . (cid:3) We conclude this section with two examples of Bass rings for which thedeterminant map provides a particularly simple description of V ( I ) / SL ( R ) for a faithful ideal I of R . Proposition 4.8. ( i ) Let R be the integral group ring of a cyclic group h c i with p elements, p > a prime number. Let I be a faithful idealof R which is not invertible. Then Fitt ( I ) = R ( c −
1) + Rp and V ( I ) / SL ( R ) ≃ ( Z /p Z ) × . ( ii ) Let K be field of characteristic distinct from . Let R = K [ x, y ] /K [ x, y ]( y − x ( x + 1)) , i.e., the coordinate ring of the nodal plane cubic curve y = x ( x + 1) over K . Let I be a non-zero ideal of R which is not invertible. Then Fitt ( I ) = Rx + Ry and V ( I ) / SL ( R ) ≃ K × where x and y are theimages of x and y in R . The proof of Proposition 4.8 relies on the following two lemmas.
Lemma 4.9.
Let R be a reduced ring with finite minimal spectrum. Let S = Π p ∈ Min( R ) R/ p and assume that each factor R/ p is a Dedekind ring. Let I be a two-generatedideal of R . Then the two following hold: ( i ) Fitt ( S/R ) ⊆ Fitt ( I ) . ( ii ) If S/R is a cyclic R -module, then Fitt ( S/R ) = ( R : S ) Proof. ( i ) . For p ∈ Min( R ) , denote by p ′ the product of the minimal primesof R distinct from p . It is easy to see that Fitt ( S/R ) = P p ∈ Min( R ) p ′ byconsidering a presentation of the R -module S/R whose generators are theobvious idempotents splitting S . Since Fitt i commutes with localization, itsuffices to show that Fitt ( S q /R q ) ⊆ Fitt ( I q ) for every maximal ideal q of R .Let q be such a maximal idea Then either ( R/ p ) q = { } or p ⊆ q in whichcase ( R/ p ) q is a local Dedekind ring, hence a principal ideal domain. Thus wecan assume, without loss of generality that S is a product of principal idealdomains. Let ( a, b ) ∈ V ( R ) and p ∈ Min( R ) . As R/ p is a principal idealdomain, there is E ∈ E ( R ) such that ( a p , b p ) E = ( c, where a p and b p are QUIVALENT GENERATING PAIRS OF AN IDEAL OF A COMMUTATIVE RING 15 the components of a and b in R/ p and c ∈ R/ p . Let ( a ′ , b ′ ) = ( a, b ) E . Thenwe have p ′ ⊆ ( a ′ : b ′ ) ⊆ Fitt ( I ) , so that Fitt ( S/R ) ⊆ Fitt ( I ) . ( ii ) . This follows from [Eis95, Proposition 20.7]. (cid:3) The next lemma comes in handy for the ring of Proposition 4.8. ii but alsoin Section 6.2. Lemma 4.10.
Let
R, S be rings such that R ⊆ S ⊆ K ( R ) and assume that S is a principal ideal domain. Let I be a two-generated ideal of R . Then Fitt ( I ) contains ( R : S ) .Proof. We can assume that I = { } since the result is trivial otherwise. Byhypothesis, there are d, d ′ ∈ S such that ( R : S ) = Sd and SI = Sd ′ . Let a and b be two generators the fractional R -ideal I/d ′ . As Sa + Sb = S , it followsthat ( a : b ) + ( b : a ) is generated by the elements of the form sa and sb where s ∈ S is such that sa and sb belong to R . Since ( R : S ) = Sda + Sdb , we inferthat ( R : S ) ⊆ Fitt ( I/d ′ ) = Fitt ( I ) . (cid:3) The normalization ˜ R or R is the normal closure of R in K ( R ) . Proof of Proposition 4.8. ( i ) . The ring R is a Dedekind-like ring [Lev85, The-orem 1.2], hence a Bass ring (alternatively use the characterization [LW85,Theorem 2.1. iii ]). Thus I can be generated by two elements. The normaliza-tion ˜ R of R identifies with Z × Z [ e iπ/p ] and R embeds into ˜ R through the map r ( r + R ( c − , r + R Φ p ( c )) where Φ p ( x ) = 1 + x + · · · + x p − is the p thcyclotomic polynomial. From now on, we identifies R with its image in ˜ R . ByLemma 4.7, the ideal I can be generated by two regular elements. By Lemma3.7, we have Fitt ( I ) = II − . By Lemma 4.9, the ideal Fitt ( I ) contains ( R : ˜ R ) , which is the maximal ideal R ( c −
1) + R Φ p ( c ) = R ( c −
1) + Rp . Since I is not invertible, we have Fitt ( I ) = R ( c −
1) + Rp . Finally, the bijection V ( I ) / SL ( R ) ≃ ( Z /p Z ) × results from Corollary 4.6. ( ii ) . The ring R is Bass ring [Gre82, Theorem 2.1], hence I can be generatedby two elements. It is easily checked that the map x z − , y z ( z − induces a ring isomorphism from R onto K [ z − , z ( z − ( K [ z ] + ˜ R andwith this identification ˜ R is the normalization of R . Since ˜ R is a principalideal ring and I = { } , we have ( R : ˜ R ) = K [ z ]( z − ⊆ Fitt ( I ) by Lemma4.10. As K [ z ]( z − is maximal in R and I is not invertible, we infer that Fitt ( I ) = K [ z ]( z −
1) = R ( z −
1) + Rz ( z − . Finally, the bijection V ( I ) / SL ( R ) ≃ K × results from Corollary 4.4. (cid:3) Remark 4.11. • With the notation of Proposition 4.8. i , a non-faithfulideal I of R is annihilated by one of the two minimal prime ideals of R , say p . Hence such an ideal I identifies with an ideal of R/ p ≃ Z or Z [ e iπ/p ] . The natural map SL ( R ) → SL ( R/ p ) is surjective by Propo-sition 2.4. ii and Remark 3.3. It follows that SL ( R ) acts transitivelyon V ( I ) . • For both assertions of Proposition 4.8, we have actually shown that
Fitt ( I ) = ( R : ˜ R ) provided that I is faithful. The group
Aut R ( I ) and the line P ( K ( R )) This section is dedicated to the proof of Corollary C. We proceed by provingfirst a result valid in any ring, namely Proposition 5.1 below. To do so, weconsider the diagonal action of the R -automorphism group Aut R ( I ) of I fromthe left: φ ( a, b ) + ( φ ( a ) , φ ( b )) , ( a, b ) ∈ V ( I ) . This action clearly commutes with the action of GL ( R ) from the right. Themultiplication by a unit u ∈ ̺ ( I ) × gives rise to an R -automorphism of I sothat ̺ ( I ) × naturally identifies with a subgroup of Aut R ( I ) .We define an action of SL ( R ) on P ( K ( R )) from the right in the followingway: [ r : s ] σ = [ ar + cs : br + ds ] with [ r : s ] ∈ P ( K ( R )) and σ = (cid:18) a bc d (cid:19) ∈ SL ( R ) . Proposition 5.1.
Let R be a ring. Then P ( K ( R )) / SL ( R ) is equipotent withthe disjoint union G [ I ] ∈ Cl( R ) , µ ( I ) ≤ ̺ ( I ) × \ V ( I ) / SL ( R ) Proof.
Let ψ : P ( K ( R )) / SL ( R ) → { [ I ] ∈ Cl( R ) | µ ( I ) ≤ } be defined by [ a : b ] SL ( R ) [ Ra + Rb ] . The map ψ is obviously surjective. Hence itonly remains to show that ψ − ( { [ I ] } ) is equipotent with ̺ ( I ) × \ V ( I ) / SL ( R ) for every ideal class [ I ] such that µ ( I ) ≤ . We can assume that I ⊆ R .Moreover, every element in ψ − ( { [ I ] } ) , can be represented by a coset of theform [ a : b ] SL ( R ) with ( a, b ) ∈ V ( I ) . We claim that the map θ : [ a : b ] SL ( R ) → ̺ ( I ) × ( a, b ) SL ( R ) is a well-defined bijection from ψ − ( { [ I ] } ) onto ̺ ( I ) × \ V ( I ) / SL ( R ) . Indeed, the identity [ a : b ] SL ( R ) = [ a ′ : b ′ ] SL ( R ) holdsfor ( a, b ) , ( a ′ , b ′ ) ∈ V ( I ) if and only if there is u ∈ K ( R ) and σ ∈ SL ( R ) suchthat ( a ′ , b ′ ) = ( ua, ub ) σ . Under this assumption, we clearly have u ∈ ̺ ( I ) × ,thus θ is well-defined. Proving that θ is bijective is routine. (cid:3) Lemma 5.2.
Let I be an ideal of R and let u ∈ ̺ ( I ) × . Then det m ( u m ) =det m ′ ( u m ′ ) for every m , m ′ ∈ V ( I ) . QUIVALENT GENERATING PAIRS OF AN IDEAL OF A COMMUTATIVE RING 17
Proof.
Let m = ( a, b ) , m ′ = ( a ′ , b ′ ) ∈ V ( I ) . By Lemma 2.1. ii , we have det ( a ′ ,b ′ ) ( ua ′ , ub ′ ) = det ( a ′ ,b ′ ) ( a, b ) det ( a,b ) ( ua, ub ) det ( ua,ub ) ( ua ′ , ub ′ ) . It follows straightforwardly from Lemma 2.3 that det ( ua,ub ) ( ua ′ , ub ′ ) = det ( a,b ) ( a ′ , b ′ ) = det ( a ′ ,b ′ ) ( a, b ) − , hence the result. (cid:3) Definition 5.3.
For u ∈ ̺ ( I ) × , we denote by det( u ) the value of det m ( u m ) for any, and hence all generating pairs m of I . We denote by ǫ ( I ) the imageof ̺ ( I ) × by φ m (see Section 2.1) for any, and hence every m ∈ V ( I ) , that is ǫ ( I ) + { det( u ) | u ∈ ̺ ( I ) × } . By Lemma 2.3 , we have det(1) = 1 + Fitt ( I ) , det( uu ′ ) = det( u ) det( u ′ ) forevery u, u ′ ∈ ̺ ( I ) × and det( u ) = u + Fitt ( I ) for every u ∈ R × . Therefore ǫ ( I ) is a subgroup of ( R/ Fitt ( I )) × containing the image of ( R × ) . Proposition 5.4.
Let I be a two-generated ideal of R such that Fitt ( R ) = II − . Then the determinant map induces an injective map ̺ ( I ) × \ V ( R ) / SL ( R ) → ̺ ( I ) × \ V (det( I )) ≃ ( R/ Fitt ( I )) × /ǫ ( I ) . If I satisfies the hypotheses of Lemma 4.1, then the induced map is a bijection.Proof. Apply Theorem 3.5 and Lemma 5.2. (cid:3)
Proof of Corollary C.
Combine Propositions 5.1 and 5.4, observing that thehypotheses of Lemma 4.1 are satisfied. Indeed, an order in a number field is aone-dimensional domain, hence almost of stable range . (cid:3) Equivalence of generating vectors in Bass rings
This section illustrates Theorem A with two examples of Bass rings. Sub-section 6.1 deals with arbitrary orders in quadratic number fields. Subsection6.2 deals with the coordinate ring of the plane curve y = x n where n ≥ isan odd rational integer.Recall that the normalization ˜ R of R is the integral closure of R in itstotal ring of quotients K ( R ) . A Bass ring is a Noetherian reduced ring whosenormalization ˜ R is a finitely generated R -module and whose ideals can begenerated by two elements. Bass domains were originally studied by Bass whoproved that their finitely generated torsion-free modules split into a directproduct of ideals and that the converse holds for a Noetherian domain R if ˜ R is a finitely generated R -module [Bas62, Theorem 1.7], see also [Lam00,Section 4] for historical remarks. For every example of Bass ring in this article, we observe that
Fitt ( I ) contains ( ˜ R : R ) if I is a faithful ideal of R . For each example, this impliesthat Fitt ( I ) takes only finitely many values because of Proposition 6.1. [LW85, Proposition 2.2]
Let R be a Bass ring. Then R/ ( R :˜ R ) is an Artinian principal ideal ring. Ideals in orders of quadratic fields.
In this section, we prove The-orem D. On our way, we compute
Fitt ( I ) and describe V ( I ) / SL ( I ) ≃ ( R/ Fitt ( I )) × explicitly for an arbitrary ideal I of an order in a quadraticnumber field.Let m be a square-free rational integer. We set K + Q ( √ m ) and denote by O the ring of integers of the quadratic field K . An order of K is a subringof finite index in O . Setting ω = (cid:26) √ m if m , √ m if m ≡ . , then we have O = Z + Z ω and any order of K is of the form O f + Z + Z f ω for somerational integer f > [IK14, Lemma 6.1]. Moreover, the inclusion O f ⊆ O f ′ holds true if and only if f ′ divides f . If I is an ideal of O f , then its ring ofmultipliers ̺ ( I ) + { r ∈ O | rI ⊆ I } is the smallest order O of K such that I is projective, equivalently invertible, as an ideal of O [LW85, Proposition 5.8].For the remainder of this section, we fix f > and set R + O f and ˜ R + O . Proposition 6.2.
Let I be an ideal of R and let f ′ > be such that ̺ ( I ) = O f ′ .Then we have ( i ) Fitt ( I ) = ( R : ̺ ( I )) , ( ii ) R/ Fitt ( I ) ≃ ̺ ( I ) /R as R -modules and R/ Fitt ( I ) ≃ Z / ( f /f ′ ) Z asrings.Proof of Proposition 6.2. ii . By assertion ( i ) , we have R/ Fitt ( I ) = R/ ( R : ̺ ( I )) . As e R/R ≃ Z /f Z , the R -submodule ̺ ( I ) /R of e R/R is cyclic so that R/ ( R : ̺ ( I )) ≃ ̺ ( I ) /R . Since the image of ̺ ( I ) /R in Z /f Z ≃ e R/R is f ′ ( Z /f Z ) , assertion ( ii ) is established. (cid:3) Our proof of Proposition 6.2. i relies on three lemmas revolving around the standard basis of an ideal, a definition that we introduce after the next lemma.We define the norm N( I ) of an ideal I of R as the index [ R : I ] = | R/I | of I in R . An ideal I of R is said to be primitive if it cannot be written as I = eJ some rational integer e and some ideal J of R . Lemma 6.3. [IK14, Lemma 6.2 and its proof]
Let I be a non-zero ideal of R = O f . Then there exist rational integers a, e > and d ≥ such that − a/ ≤ d < a/ , e divides both a and d and we have I = Z a + Z ( d + ef ω ) . QUIVALENT GENERATING PAIRS OF AN IDEAL OF A COMMUTATIVE RING 19
The integers a, d and e are uniquely determined by I . The integer ae is thenorm of I and we have Z ∩ I = Z a . The ideal I is primitive if and only if e = 1 Note that, since Z ∩ I = Z a , the rational integer a divides N( d + ef ω ) . Wecall the generating pairs ( a, d + ef ω ) the standard basis of I . Let us associateto I the binary quadratic form q I defined by q I ( x, y ) = N( xa + y ( d + ef ω ))N( I ) . Then we have eq I ( x, y ) = ax + bxy + cy with b = Tr( d + ef ω ) and c = N( d + ef ω ) a . We define the content of q I as the greatest common divisors of its coefficients,and we write content( q I ) + gcd( a, b, c ) e . Lemma 6.4.
Let I be a non-zero ideal of R with standard basis ( a, d + ef ω ) .Let f ′ > be such that ̺ ( I ) = O f ′ . Then we have gcd( a, d, ef ) = e content( q I ) = ef /f ′ . Proof of Lemma 6.4.
By Lemma 6.3, the fractional ideal J + e I is an integralideal of R with standard basis ( a/e, d/e + f ω ) . It is immediate to see that theresult holds for I if and only if it holds for J . Therefore, we can assume, withoutloss of generality, that e = 1 . Now the result can be inferred straightforwardlyfrom [IK14, Proof of Lemma 6.5]. (cid:3) We denote by σ the automorphism of K which maps √ m to −√ m . Lemma 6.5.
Let I be a non-zero ideal of R and let f ′ > be such that ̺ ( I ) = O f ′ . Then the two following hold. ( i ) I − = I ) σ ( I ) , ( ii ) Fitt ( I ) = II − = Z ff ′ + Z f ω . Assertion ( i ) of Lemma 6.5 is usually stated and proven in the case of an in-vertible ideal I of R . We provide a proof of the general case for the convenienceof the reader. Proof of Lemma 6.5.
We consider the standard basis ( a, d + ef ω ) of I . ByLemma 6.3, we can write I = eJ for some primitive ideal J of R . We claimthat if assertions ( i ) and ( ii ) are satisfied by J , then they are satisfied by I . Indeed, it is trivial to check that ( eJ ) − = e J − , N( eJ ) = e N( J ) and σ ( eJ ) = eσ ( J ) . These three facts settle the claim. Hence we can assume,without loss of generality, that e = 1 . ( i ) . By definition, we have I ) σ ( I ) = Z + Z d + fσ ( ω ) a and the inclusion I ) σ ( I ) ⊆ I − is evident. In order to prove the reverse inclusion, let ustake x ∈ I − . Since xa ∈ R , we can write x = u + vfωa for some u, v ∈ Z . Letus conveniently rewrite x as x = z − v d + fσ ( ω ) a with z + u + vf Tr( ω )+ vda . Since x ( d + f ω ) ∈ R , we deduce that z ( d + f ω ) ∈ R . The elements and f ω arelinearily independent over Q and freely generate the ring R as a Z -module.Therefore z lies into Z , which entails x ∈ Z + Z d + fσ ( ω ) a , as desired. ( ii ) . Using the formula of assertion ( i ) , we obtain II − = Z a + Z b + Z c + Z ( d + f ω ) . Applying Lemma 6.4, we get II − = Z ff ′ + Z ( d + f ω ) = Z ff ′ + Z f ω . (cid:3) We are now in position to complete the proof of Proposition 6.2.
Proof of Proposition 6.2. i . As R is a Bass domain, it follows from Lemma3.7 that Fitt ( I ) = II − . From the inclusion ̺ ( I ) II − ⊆ R , we infer that Fitt ( I ) ⊆ ( R : ̺ ( I )) . We have shown in the first part of the proof of 6.2,that the index of ( R : ̺ ( I )) in R is f /f ′ . The index of Fitt ( I ) is also f /f ′ byLemma 6.4. Therefore Fitt ( I ) = ( R : ̺ ( I )) . (cid:3) Proposition 6.6.
Let I be an ideal of R . Then glr ( I ) = 2 .Proof. As R is a Bass ring, we have µ ( I ) = 2 . Moreover R is a one-dimensionaldomain, hence it is almost of stable rank . Observing eventually that R is afree Z -module of rank , we can apply Corollary 4.5. (cid:3) We conclude this section with the proof of Theorem D and a closely re-lated result on the quotients ̺ ( I ) × \ V ( I ) / SL ( R ) and V ( I ) / GL ( R ) , namelyProposition 6.8 below. Both results use Lemma 6.7.
Let I be an ideal of R . Then ǫ ( I ) = { N K/ Q ( u ) + Fitt ( I ) | u ∈ ̺ ( I ) × } where N K/ Q ( u ) denotes the norm of u over Q . In particular, ǫ ( I ) hasat most two elements.Proof. Let m ∈ V ( I ) and let u ∈ ̺ ( I ) × . Let A ∈ GL ( Z ) be such that u m = m A . By the definitions of N K/ Q ( u ) and of det( u ) we have N K/ Q ( u ) = det( A ) and det( u ) = det( A ) + Fitt ( I ) , hence the result. (cid:3) Proof of Theorem D.
Let κ ( R ) be the number of cusps of R . By Propositions5.1 and 5.4, we have κ ( R ) = P [ I ] ∈ Cl( R ) [( R/ Fitt ( I )) × : ǫ ( I )] . We also have Fitt ( R ) = ( R : ̺ ( I )) by Proposition 6.2. This identity together with Defini-tion 5.3 imply that [( R/ Fitt ( R )) × : ǫ ( I )] depends only on ̺ ( I ) . Since ̺ ( I ) isthe smallest overring O f ′ of R such that I is an invertible ideal of O f ′ [LW85, QUIVALENT GENERATING PAIRS OF AN IDEAL OF A COMMUTATIVE RING 21
Proposition 5.8], we have κ ( R ) = P f ′ | f | Pic( O f ′ ) | [( R/ ( R : O f ′ )) × : U f ′ ] where U f ′ is described by Lemma 6.7. By Lemma 6.5, we have R/ ( R : O f ′ ) ≃ Z / ( f /f ′ ) Z . Since U f ′ has at most two elements and exactly two if O f ′ has aunit of norm − and f /f ′ > , we infer that [( R/ ( R : O f ′ )) × : U f ′ ] = ϕ ( f/f ′ )2 ǫ ( f,f ′ ) ,which completes the proof. (cid:3) Proposition 6.8.
Let I be a non-zero ideal of R and let f ′ be the positiverational integer such that ̺ ( I ) = O f ′ . Then the following hold. ( i ) | ̺ ( I ) × \ V ( I ) / SL ( R ) | = [( R/ Fitt ( I )) × : ǫ ( I )] = ϕ ( f/f ′ )2 ǫ ( f,f ′ ) with ǫ ( f, f ′ ) as in Theorem D. ( ii ) We have
Aut R ( I ) \ V ( I ) / GL ( R ) = V ( I ) / GL ( R ) . The determinantmap induces a bijection of V ( I ) / GL ( R ) onto the quotient of ( R/ Fitt ( I )) × by the image of R × through the natural map R → R/ Fitt ( I ) . ( iii ) If f /f ′ has at least three distinct odd rational prime divisors, then GL ( R ) doesn’t act transitively on V ( I ) .Proof. Assertion ( i ) has been already established in the course of the proof ofTheorem D. ( ii ) . Since I is an invertible ideal of ̺ ( I ) , every R -automorphism of I isinduced by the multiplication by a unit of ̺ ( I ) . Thus we haveAut R ( I ) \ V ( I ) / GL ( R ) = ̺ ( I ) × \ V ( I ) / GL ( R ) and the identity ̺ ( I ) × \ V ( I ) / GL ( R ) = V ( I ) / GL ( R ) follows from Lemma6.7. Consider now m ∈ V ( I ) . Then for every u ∈ R × we have det m ( m (cid:18) u
00 1 (cid:19) ) = u + Fitt ( I ) . Since the determinant map induces a bijection from V ( R ) / SL ( R ) onto thegroup ( R/ Fitt ( I )) × by Corollary 4.4, it induces a bijection from V ( I ) / GL ( R ) onto the quotient group Q + ( R/ Fitt ( I )) × / ( R × + Fitt ( I )) . ( iii ) . Since ( R/ Fitt ( I )) × ≃ ( Z / ( f /f ′ ) Z ) × , it follows easily from the Chi-nese Remainder Theorem that ( R/ Fitt ( I )) × surjects onto ( Z / Z ) . Hence ( R/ Fitt ( I )) × cannot be generated by less than three elements. As R × is gen-erated by − and a power of the fundamental unit of K , the group Q definedin the proof of ( ii ) is not trivial. (cid:3) Ideals in the coordinate ring of the curve x = y n . This section isdedicated to the proof of Proposition E which illustrates Theorem A in thecoordinate ring R of the curve x = y n over K with n ≥ an odd integer and K an arbitrary field. The coordinate ring of an algebraic curve is a Bass domainif and only if the only singularities of the curve are double points [Gre82,Theorem 2.1]. This is well-known to hold for R . We identify R with K [ x , x n ] , a notation which stays in effect throughout this section. The integral closureof R in its field of fractions is K [ x ] and we have R = K [ x ] + K [ x ] x n − = K [ x ] + K [ x ] x n and ( R : K [ x ]) = K [ x ] x n − , i.e., the conductor of R in K [ x ] is K [ x ] x n − .We compute first Fitt ( I ) for I a non-zero ideal of R . This is achieved bymeans of the next two lemmas. Lemma 6.9.
Let I be a non-zero ideal of R and let d ( x ) ∈ K [ x ] be a generatorof K [ x ] I . Then there is an even integer ν satisfying ≤ ν ≤ n − and twocoprime polynomials p ( x ) = P i p i x i , q ( x ) = P i q i x i of K [ x ] such that thefollowing hold: ( i ) Rp ( x ) + Rq ( x ) = d ( x ) I , ( ii ) p q ν +1 − p q ν = 0 if ν < n − , ( iii ) p = 0 and q i = 0 for every i < ν . The x -valuation of a polynomial p ( x ) ∈ K [ x ] is the greatest integer v ≥ such that x v divides p ( x ) . Proof.
Let p ( x ) , p ′ ( x ) be the quotients of two generators of I by d ( x ) . Then p ( x ) and p ′ ( x ) are coprime in K [ x ] . Swapping them if need be, we can supposethat p (0) = 0 . Let q ( x ) ∈ p ( x ) + Rp ′ ( x ) be a polynomial with maximal x -valuation. If q ( x ) = 0 , the result is obvious. Thus we can assume that q ( x ) = 0 . Let m be the greatest even integer such that q m or q m +1 is non-zero.Then we have p q m +1 − p q m = 0 since otherwise q ( x ) − λx m p ( x ) would havean x -valuation greater than the x -valuation of q ( x ) for some λ ∈ K . Setting ν = min( m, n − , we easily check that ν, p ( x ) and q ( x ) satisfy the conditions ( i ) , ( ii ) and ( iii ) . (cid:3) Lemma 6.10.
Let I be an ideal of R and let p ( x ) , q ( x ) and ν be as in Lemma6.9. Then Fitt ( I ) = K [ x ] x n − ν − + K [ x ] x n − . In particular ν is uniquely determined by I .Proof. As p ( x ) and q ( x ) are coprime elements of K [ x ] , it easily follows fromLemma 2.2 that Fitt ( I ) is generated by the elements of the form h ( x ) p ( x ) and h ( x ) q ( x ) with h ( x ) ∈ K [ x ] such that(1) h ( x ) p ( x ) ∈ R, h ( x ) q ( x ) ∈ R. Hence
Fitt ( I ) contains the conductor K [ x ] x n − of R in K [ x ] . Since R/K [ x ] x n − ≃ K [ x ] /K [ x ] x n − is a local principal ideal ring whose maximal ideal is generated by the image of x , we infer that Fitt ( I ) = K [ x ] x ν ′ + K [ x ] x n − for some even integer ν ′ ≥ . QUIVALENT GENERATING PAIRS OF AN IDEAL OF A COMMUTATIVE RING 23
We claim that every polynomial h ( x ) satisfying (1) has an x -valuation at least n − ν − and there is such an h ( x ) with an x -valuation equal to n − ν − . Itimmediately follows from the claim that ν ′ = n − ν − , hence the result.In order to prove the claim, we consider a polynomial h ( x ) = P i h i x i ∈ K [ x ] .and let h = ( h , h , . . . , h n − ) . Then h ( x ) satisfies (1) if and only if A h = 0 where A is the ( n − × ( n − matrix obtained by appending the last ν rowsof the matrix P = p p p p p p ... ... ... ... p n − p n − p n − p n − · · · p p to Q = q ν +1 q ν p p q ν +3 q ν +2 q ν +1 q ν p p p p ... ... ... ... q n − q n − q n − q n − · · · q ν +1 q ν p n − ν − p n − ν − p n − ν − p n − ν − · · · p p where every unspecified coefficient is zero.Let h ( x ) be such that A h = 0 . If ν < n − , then p q ν +1 − p q ν = 0 , so that Q is invertible. Otherwise Q is the × matrix and hence is invertible as well.As a result, we have ( h , . . . , h n − ν − ) = 0 , i.e., the x -valuation of h ( x ) is atleast n − ν − . The equation A h = 0 is equivalent to ( h , . . . , h n − ν − ) = 0 and A ′ h ′ = 0 where h ′ = ( h n − ν − , . . . , h n − ) and A ′ consists in the first ν rows of P . Since the latter equation clearly admits a solution h ′ with h n − ν − = 0 , wecan find a polynomial h ( x ) which satisfies (1) and whose x -valuation is exactly n − ν − . (cid:3) Lemma 6.11.
Let I be an ideal of R and let ν be as in Lemma 6.10. Thenwe have ̺ ( I ) = K [ x ] + K [ x ] x ν and Fitt ( I ) = ( ̺ ( I ) : R ) .Proof. Let us prove first that every proper R -submodule of K [ x ] containing R is of the form K [ x ] + K [ x ] x ν ′ for some even integer ν ′ > . Clearly, thereare exactly n − modules of the latter form. Hence it suffices to show that K [ x ] /R has length n − . This follows from the isomorphisms K [ x ] /R ≃ R/ ( R : K [ x ]) ≃ K [ x ] /K [ x ] x n − . The claim is therefore established and we have ̺ ( I ) = K [ x ] + K [ x ] x ν ′ for some even ν ′ ≥ . Clearly ν ′ + 1 is the smallest oddinteger i > such that x i I ⊂ I . To complete the proof, we show that ν + 1 enjoys the same property. Let B be the K -basis of K [ x ] /K [ x ] x n − consistingof the images of , x, . . . , x n − . We shall consider an ( n − × ( n − matrix A over K such that the two following are equivalent for every odd integer i > : • x i p ( x ) ∈ I/d ( x ) , • the linear system A h = b i has a solution h ∈ K n − where b i is thecomponent vector of the image of x i p ( x ) in K [ x ] /K [ x ] x n − with respectto B .Let A be the matrix whose columns are the component vectors of the imagesin K [ x ] /K [ x ] x n − of the polynomials p ( x ) , x p ( x ) , . . . , x ν − p ( x ) ,x ν p ( x ) , q ( x ) , x ν +2 p ( x ) , x q ( x ) , . . . , x n − p ( x ) , x n − − ν q ( x ) with respect to B . Since I/d ( x ) contains K [ x ] x n − , the matrix A has thedesired property. Moreover A is of the form (cid:18) A ′ ∗ Q (cid:19) with A ′ = p p p p p p ... ... . . . . . . p ν − p ν − p ν − · · · · · · p p ν − p ν − p ν − · · · · · · p and where Q is a lower block-trigonal matrix with diagonal blocks all equal tothe invertible matrix (cid:18) p q ν p q ν +1 (cid:19) . As Q is invertible, the system A h = b with h = ( h , . . . , h n − ) , b = ( b , . . . , b n − ) has a solution if and only if A ′ h ′ = b ′ has a solution where h ′ = ( h , . . . , h ν − ) and b ′ = ( b , . . . , b ν − ) . Let i > bean odd integer. It is easy to check that the system A ′ h ′ = b ′ i has a solutionif and only if i ≥ ν + 1 . Let c ν +1 be the component vector of the imageof x ν +1 q ( x ) . It is also easy to check that A h = c ν +1 has a solution, whichmeans that x ν +1 q ( x ) ∈ I/d ( x ) . Therefore x i I ⊂ I holds true if and only if i ≥ ν + 1 . (cid:3) Proposition 6.12.
Let I be an ideal of R . Then glr ( I ) = 2 .Proof. Since R is a Bass ring, we have µ ( I ) ≤ . As R is a one-dimensional, itis almost of stable rank . Since R is moreover an integral domain which is afree K [ x ] -module of rank , Corollary 4.5 applies. (cid:3) References [Bas62] H. Bass. Torsion free and projective modules.
Trans. Amer. Math. Soc. , 102:319–327, 1962. 17[Bas64] H. Bass. K -theory and stable algebra. Inst. Hautes Études Sci. Publ. Math. ,(22):5–60, 1964. 7
QUIVALENT GENERATING PAIRS OF AN IDEAL OF A COMMUTATIVE RING 25 [BS66] A. Borevich and I. Shafarevich.
Number theory . Translated from the Russian byNewcomb Greenleaf. Pure and Applied Mathematics, Vol. 20. Academic Press,New York-London, 1966. 3[Che20] J. Chen. Surjections of unit groups and semi-inverses. To appear in J. Commut.Algebra, Forthcoming (2020). 11[CLQ04] T. Coquand, H. Lombardi, and C. Quitté. Generating non-Noetherian modulesconstructively.
Manuscripta Math. , 115(4):513–520, 2004. 7[Cou07] F. Couchot. Finitely presented modules over semihereditary rings.
Comm. Algebra ,35(9):2685–2692, 2007. 12[DTZ18] M. Das, S. Tikader, and M. Zinna. Orbit spaces of unimodular rows over smoothreal affine algebras.
Invent. Math. , 212(1):133–159, 2018. 2[Eis95] D. Eisenbud.
Commutative algebra , volume 150 of
Graduate Texts in Mathematics .Springer-Verlag, New York, 1995. With a view toward algebraic geometry. 6, 10,11, 13, 14, 15[EO67] D. Estes and J. Ohm. Stable range in commutative rings.
J. Algebra , 7:343–362,1967. 8, 11[Fas11] J. Fasel. Some remarks on orbit sets of unimodular rows.
Comment. Math. Helv. ,86(1):13–39, 2011. 2[FS01] L. Fuchs and L. Salce.
Modules over non-Noetherian domains , volume 84 of
Math-ematical Surveys and Monographs . American Mathematical Society, Providence,RI, 2001. 11[Fur59] Y. Furuta. Norm of units of quadratic fields.
J. Math. Soc. Japan , 11:139–145,1959. 4[Gre82] C. Greither. On the two generator problem for the ideals of a one-dimensionalring.
J. Pure Appl. Algebra , 24(3):265–276, 1982. 15, 21[Guy17] L. Guyot. On finitely generated modules over quasi-euclidean rings.
Arch. Math.(Basel). , 2017. 2[Guy20a] L. Guyot. Equivalent generating vectors of finitely generated modules over a com-mutative ring.
Preprint , 2020. 8[Guy20b] L. Guyot. Nielsen equivalence in Z ⋊ Z . 2020. In preparation. 4[Hei84] R. Heitmann. Generating non-Noetherian modules efficiently. Michigan Math. J. ,31(2):167–180, 1984. 7[HSG86] J. Hermida and T. Sánchez-Giralda. Linear equations over commutative rings anddeterminantal ideals.
J. Algebra , 99(1):72–79, 1986. 8[IK14] T. Ibukiyama and M. Kaneko.
Quadratic Forms and Ideal Theory of QuadraticFields. In: Bernoulli Numbers and Zeta Functions. , pages 75–93. Springer Mono-graphs in Mathematics, Tokyo, 2014. 18, 19[Jen62] C. Jensen. On the solvability of a certain class of non-Pellian equations.
Math.Scand. , 10:71–84, 1962. 4[Lam00] T. Lam. Bass’s work in ring theory and projective modules. arXiv:0002217v1[math.RA] 25 Feb 2000, February 2000. 17[Lam06] T. Lam.
Serre’s problem on projective modules . Springer Monographs in Mathe-matics. Springer-Verlag, Berlin, 2006. 2[Lev85] L. Levy. Z G n -modules, G n cyclic of square-free order n . J. Algebra , 93(2):354–375,1985. 15 [LQ15] H. Lombardi and C. Quitté.
Commutative algebra: constructive methods , vol-ume 20 of
Algebra and Applications . Springer, Dordrecht, revised edition, 2015.Finite projective modules, Translated from the French by Tania K. Roblot. 11[LW85] L. Levy and R. Wiegand. Dedekind-like behavior of rings with -generated ideals. J. Pure Appl. Algebra , 37(1):41–58, 1985. 5, 15, 18, 21[Mat83] E. Matlis. The minimal prime spectrum of a reduced ring.
Illinois J. Math. ,27(3):353–391, 1983. 13[McG08] W. McGovern. Bézout rings with almost stable range 1.
J. Pure Appl. Algebra ,212(2):340–348, 2008. 12[MR87] J. McConnell and J. Robson.
Noncommutative Noetherian rings . Pure and AppliedMathematics (New York). John Wiley & Sons, Ltd., Chichester, 1987. With thecooperation of L. W. Small, A Wiley-Interscience Publication. 7[Mur03] M. Murthy. Generators of a general ideal. In
A tribute to C. S. Seshadri (Chennai,2002) , Trends Math., pages 379–384. Birkhäuser, Basel, 2003. 2, 11[Neu99] J. Neukirch.
Algebraic number theory , volume 322 of
Grundlehren der Mathe-matischen Wissenschaften [Fundamental Principles of Mathematical Sciences] .Springer-Verlag, Berlin, 1999. Translated from the 1992 German original and witha note by Norbert Schappacher, With a foreword by G. Harder. 4[R´53] L. Rédei. Die -Ringklassengruppe des quadratischen Zahlkörpers und die Theorieder Pellschen Gleichung. Acta Math. Acad. Sci. Hungar. , 4:31–87, 1953. 4[Shi63] H. Shimizu. On discontinuous groups operating on the product of the upper halfplanes.
Ann. of Math. (2) , 77:33–71, 1963. 3[vdG88] G. van der Geer.
Hilbert modular surfaces , volume 16 of
Ergebnisse der Mathe-matik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)] .Springer-Verlag, Berlin, 1988. 3, 8[vdK83] W. van der Kallen. A group structure on certain orbit sets of unimodular rows.
J. Algebra , 82(2):363–397, 1983. 2
EPFL ENT CBS BBP/HBP. Campus Biotech. B1 Building, Chemin des mines,9, Geneva 1202, Switzerland
Email address ::