The syzygy theorem for Bézout rings Maroua Gamanda* , Henri Lombardi** , Stefan Neuwirth** , and Ihsen Yengui* *

Département de mathématiques, Faculté des sciences de Sfax, Université de Sfax, 3000 Sfax, Tunisia, [email protected], [email protected]. ** Laboratoire de mathématiques de Besançon, Université Bourgogne Franche-Comté, 25030 Besançon Cedex, France, [email protected], [email protected].

10th February 2018

Contents Introduction

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1 Gröbner bases for modules over a discrete ring

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2 The algorithms The general context . . . . . . The division algorithm . . . . . The S-polynomial algorithm . . Buchberger’s algorithm . . . . The syzygy algorithm for terms Schreyer’s syzygy algorithm . .

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3 The algorithms in the case of a valuation ring

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4 Termination of Buchberger’s algorithm for a Bézout ring

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5 The syzygy theorem and Schreyer’s algorithm for a valuation ring

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6 The syzygy theorem and Schreyer’s algorithm for a Bézout ring 17 The case of the integers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 The case of Z/N Z . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 References

21 Abstract

We provide constructive versions of Hilbert’s syzygy theorem for Z and Z/N Z following Schreyer’s method. Moreover, we extend these results to arbitrary coherent strict Bézout rings with a divisibility test for the case of finitely generated modules whose module of leading terms is finitely generated.

MSC 2010: 13D02, 13P10, 13C10, 13P20, 14Q20. Keywords: Syzygy theorem, free resolution, monomial order, Schreyer’s monomial order, Schreyer’s syzygy algorithm, dynamical Gröbner basis, valuation ring, Gröbner ring, strict Bézout ring. 1

Introduction. This paper is written in the framework of Bishop style constructive mathematics (see [2, 3, 11, 12]). It can be seen as a sequel to the papers [10, 16]. The main goal is to obtain constructive versions of Hilbert’s syzygy theorem for Bézout domains of Krull dimension ≤ 1 with a divisibility test and for coherent zero-dimensional Bézout rings with a divisibility test (e.g. for Z and Z/N Z, see [11, 13, 18, 19]) following Schreyer’s method. These two cases are instances of Gröbner rings. Moreover, we extend these results to arbitrary coherent strict Bézout rings with a divisibility test for the case of finitely generated modules whose module of leading terms is finitely generated.

1

Gröbner bases for modules over a discrete ring

We start with recalling the following constructive definitions. Definition 1. • A ring is discrete if it is equipped with a zero test: equality is decidable. • A ring R is zero-dimensional and we write dim R ≤ 0 if ∀a ∈ R ∃n ∈ N ∃x ∈ R an (ax − 1) = 0. It is of Krull dimension ≤ 1 and we write dim R ≤ 1 if ∀a, b ∈ R ∃m, n ∈ N ∃x, y ∈ R bm (an (ax − 1) + by) = 0. • Let U be an R-module. The syzygy module of a list [v1 , . . . , vn ] ∈ U n is Syz(v1 , . . . , vn ) := { [b1 , . . . , bn ] ∈ Rn ; b1 v1 + · · · + bn vn = 0 }. The syzygy module of a single element v is the annihilator Ann(v) of v. • An R-module U is coherent if all syzygy modules with v1 , . . . , vn ∈ U are finitely generated.1 The ring R is coherent if it is coherent as an R-module. It is well known that a module is coherent iff on the one hand any intersection of two finitely generated submodules is finitely generated, and on the other hand the annihilator of every element is a finitely generated ideal. • A ring R is local if for every element x ∈ R, either x or 1 + x is invertible. The unit group of R is denoted by R× . • The (Jacobson) radical Rad(R) of a ring R is the ideal { x ∈ R ; 1 + xR ⊆ R× }. The residual field of a local ring R is the quotient R/ Rad(R). • A local ring R is residually discrete if its residual field is discrete. This means that x ∈ R× is decidable. A nontrivial local ring is residually discrete iff it is the disjoint union of R× and Rad(R). • A ring R is equipped with a divisibility test if, given a, b ∈ R, one can answer the question a ∈? hbi and, in the case of a positive answer, one can explicitly provide c ∈ R such that a = bc. • A ring is strongly discrete if it is equipped with a membership test for finitely generated ideals, i.e. if, given a, b1 , . . . , bn ∈ R, one can answer the question a ∈? hb1 , . . . , bn i and, in the case of a positive answer, one can explicitly provide c1 , . . . , cn ∈ R such that a = b1 c1 + · · · + bn cn . 1

In contradistinction to Bourbaki and to the Stacks project, we do not require U to be finitely generated.

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• A ring R is a valuation ring 2 if every two elements are comparable w.r.t. division, i.e. if, given a, b ∈ R, either a | b or b | a. A valuation ring is a local ring; it is coherent iff the annihilator of any element is principal. A valuation domain is coherent. A valuation ring is strongly discrete iff it is equipped with a divisibility test. • A ring R is a Bézout ring if every finitely generated ideal is principal, i.e. of the form hai = Ra with a ∈ R. A Bézout ring is strongly discrete iff it is equipped with a divisibility test; it is coherent iff the annihilator of any element is principal. A valuation ring is the same thing as a Bézout local ring (see [11, Lemma IV-7.1]). • A Bézout ring R is strict if for all a, b ∈ R we can find g, a1 , b1 , c, d ∈ R such that a = a1 g, b = b1 g, and ca1 + db1 = 1. Valuation rings and Bézout domains are strict Bézout rings; a quotient or a localisation of a strict Bézout ring is again a strict Bézout ring (see [11, Exercise IV-7, pp. 220–221, solution pp. 227–228]). A zero-dimensional Bézout ring is strict (because it is a “Smith ring”, see [7, Exercice XVI-9, p. 355, solution p. 526] and [11, Exercise IV-8, pp. 221-222, solution p. 228]). • A residually discrete valuation ring V is archimedean if ∀a, b ∈ Rad(V) \ {0} ∃n ∈ N a | bn . For a valuation domain V, we have constructively the classical equivalences V is archimedean ⇐⇒ the valuation group of V is archimedean ⇐⇒ dim V ≤ 1. Moreover, we know that a valuation ring V that contains a nonzero zerodivisor is archimedean if and only if dim V ≤ 0 (see [13, 18]). Remark 2. In some cases, e.g. Euclidean domains or polynomial rings over a discrete field, a strongly discrete ring is equipped with a division algorithm which, for arbitrary a, b1 , . . . , bn ∈ R, provides an expression a = b1 c1 + · · · + bn cn + r with quotients c1 , . . . , cn and a remainder r, where r = 0 iff a ∈ hb1 , . . . , bn i. When a strongly discrete ring is not equipped with a division algorithm, we shall consider that the division is trivial if a ∈ / hb1 , . . . , bn i: the quotients vanish and r = a. In the case of Bézout rings, dividing a by [b1 , . . . , bn ] amounts to dividing a by the gcd d of [b1 , . . . , bn ], since a = dc + r can be read as a = (cα1 )b1 + · · · + (cαn )bn + r, where d = α1 b1 + · · · + αn bn . Definition 3 (Monomial orders on finite-rank free R[X]-modules, see [1, 5]). Let R be a ring, n, m ≥ 1. Consider n indeterminates X1 , . . . , Xn and R[X] = R[X1 , . . . , Xn ]. Let Hm ' Am (R[X]) be a free R[X]-module with basis (e1 , . . . , em ). (1) A monomial in Hm is a vector of type X α ei (1 ≤ i ≤ m), where X α = X1α1 · · · Xnαn is a monomial in R[X]; the index i is the position of the monomial. The set of monomials in Hm 1 ∼ 3 is denoted by Mm n , with Mn = Mn (the set of monomials in R[X]). For example, X1 X2 e2 is a monomial in Hm , but 2X1 e3 , (X1 + X23 )e2 and X1 e2 + X23 e3 are not. If M = X α ei and N = X β ej , we say that M divides N if i = j and X α divides X β . For example, X1 e1 divides X1 X2 e1 , but does not divide X1 X2 e2 . Note that in the case that M divides N , there exists a monomial X γ in R[X] such that N = X γ M : in this case we define N/M := X γ ; for example, (X1 X2 e1 )/(X1 e1 ) = X2 . A term in Hm is a vector of type cM , where c ∈ R \ {0} and M ∈ Mm n . We say that a term cM 0 and M divides M 0 . divides a term c0 M 0 , with c, c0 ∈ R \ {0} and M, M 0 ∈ Mm , if c divides c n 2

Here we follow Kaplansky’s definition: R may have nonzero zerodivisors. In [19] it is added that a valuation ring be strongly discrete. We prefer to add this hypothesis when it is necessary, so as to insist on the fact that certain algorithms rely on the divisibility test.

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(2) A monomial order on Hm is a relation > on Mm n such that (i) > is a total order on Mm n, α (ii) X α M > M for all M ∈ Mm n and X ∈ Mn \ {1}, α (iii) M > N =⇒ X α M > X α N for all M, N ∈ Mm n and X ∈ Mn .

Note that, when specialised to the case m = 1, this definition coincides with the definition of a monomial order on R[X]. When the ring R is discrete, any nonzero vector h ∈ Hm can be written as a sum of terms h = ct Mt + ct−1 Mt−1 + · · · + c1 M1 , with ci ∈ R \ {0}, Mi ∈ Mm n , and Mt > Mt−1 > · · · > M1 . We define the leading coefficient, leading monomial, and leading term of h as in the ring case: LC(h) = ct , LM(h) = Mt , LT(h) = ct Mt . Letting Mt = X α e` with X α ∈ Mm n and 1 ≤ ` ≤ m, we say that α is the multidegree of h and write mdeg(h) = α, and that the index ` is the leading position of h and write LPos(h) = `. We stipulate that LT(0) = 0 and mdeg(0) = −∞, but we do not define LPos(0). (3) A monomial order on R[X] gives rise to the following canonical monomial order on Hm : for monomials M = X α ei and N = X β ej ∈ Mm n , let us define that M >N

if

either X α > X β or both X α = X β and i < j.

This monomial order is called term over position (TOP) because it gives more importance to the monomial order on R[X] than to the vector position. For example, when X2 > X1 , we have X2 e1 > X2 e2 > X1 e1 > X1 e2 . Definition 4 (Gröbner bases and Schreyer’s monomial order). Let R be a discrete ring, n, m ≥ 1, and consider n indeterminates X1 , . . . , Xn . Let Hm be a free R[X]-module with basis (e1 , . . . , em ) and a monomial order >. Consider G = [g1 , . . . , gs ], gj ∈ Hm \ {0}, and the finitely generated submodule U = hg1 , . . . , gs i = R[X]g1 + · · · + R[X]gs of Hm . (1) The module of leading terms of U is LT(U ) := h LT(u) ; u ∈ U i. (2) G is a Gröbner basis for U if LT(U ) = hLT(G)i := hLT(g1 ), . . . , LT(gs )i. (3) Let (1 , . . . , s ) be the canonical basis of R[X]s . Schreyer’s monomial order induced by > and [g1 , . . . , gs ] on R[X]s is the order denoted by >g1 ,...,gs , or again by >, defined as follows: α

β

X i > X j

if

either LM(X α gi ) > LM(X β gj ) or both LM(X α gi ) = LM(X β gj ) and i < j.

Schreyer’s monomial order is defined on R[X]s in the same way as when R is a discrete field (see [8, p. 66]). Definition 5. Let R be a discrete ring. We say that R is a Gröbner ring if for every n ∈ N and every finitely generated ideal I of R[X1 , . . . , Xn ] endowed with the lexicographic monomial order, the module LT(I) is finitely generated as well. The ring R is Gröbner if and only if every finitely generated ideal of R[X] has a Gröbner basis w.r.t. the lexicographic monomial order. A coherent valuation ring with a divisibility test is Gröbner if and only if it is archimedean (see [19, Theorem 272]). 4

2

The algorithms

The general context Let us now present the algorithms to be discussed in this article in a form that adapts as well to the case where R is a coherent valuation ring with a divisibility test as to the case where R is a coherent strict Bézout ring with a divisibility test. Note that the latter case includes the former because a valuation ring is nothing but a local Bézout ring that is strict. This is achieved by appeals to “find . . . such that . . .” commands that will adapt to the corresponding framework. The following general context is needed for the algorithms, except that coherence and strictness is not used in the division algorithm and that the divisibility test is not used for the computation of S-polynomials. Context 6. We consider a coherent strict Bézout ring R with a divisibility test. In the local case, R is a valuation ring. We take n indeterminates X1 , . . . , Xn and consider a free R[X]-module Hm with basis (e1 , . . . , em ). We consider a monomial order > on Hm .

The division algorithm This algorithm takes place in Context 6 for R; note however that coherence and strictness are not used here. Like the classical division algorithm for F[X]m with F a discrete field (see [19, Algorithm 211]), this algorithm has the following goal. Input h ∈ Hm , h1 , . . . , hs ∈ Hm \ {0}. Output  q1 , . . . , qs ∈ R[X] and r ∈ Hm such that   h = q1 h1 + · · · + qs hs + r,

LM(h) ≥ LM(q h ) whenever q h 6= 0,

` ` ` `    LT(h ) - f, . . . , LT(h ) - f for each term f of r. 1 s

Definition and notation 7. The vector r is called a remainder of h on division by H = [h1 , . . . , hs ] H and is denoted by r = h . This notation would gain in precision if it included the dependence of the remainder on the algorithm mentioned in Remark 2. Division algorithm 8. 1 2 3 4 5 6 7 8 9 10 11 12 13 14

l o c a l v a r i a b l e s i : {1, . . . , s} , D : subset of {1, . . . , s} , a, c, d, bi : R , p : Hm ; q1 ← 0 ; . . . ; qs ← 0 ; r ← 0 ; p ← h ; w h i l e p 6= 0 do D ← { i ; LM(hi ) | LM(p) } ; f i n d d, bi (i ∈ D) such t h a t P d = gcd(LC(hi ))i∈D = i∈D bi LC(hi ) ; f i n d a, c such t h a t LC(p) = ad + c (with c = 0 iff d divides LC(p), see Remark 2) ; f o r i i n D do qi ← qi + abi (LM(p)/ LM(hi )) od ; r ← r + c LM(p) ; P p ← p − i∈D abi (LM(p)/ LM(hi ))hi − c LM(p) od

5

By convention, if D is empty, then d = 0. At each step of the algorithm, the equality h = q1 h1 + · · · + qs hs + p + r holds while mdeg(p) decreases. Note that in the case of a valuation ring, the gcd d is an LC(hi0 ) dividing all the LC(hi ), and the Bézout identity may be given by setting bi0 = 1 and bi = 0 for i 6= i0 : see Algorithm 15.

The S-polynomial algorithm This algorithm takes also place in Context 6 for R. Note however that the divisibility test is not used here; only the zero test is used. This algorithm is a key tool for constructing a Gröbner basis and has been introduced by Buchberger [4] for the case where the base ring is a discrete field. It has the following goal. Input f, g ∈ Hm \ {0}. Output the S-polynomial given by bX β and aX α as S(f, g) = bX β f − aX α g: if f = g, then S(f, f ) = bf with b a generator of Ann(LC(f )); otherwise, + + if LM(f ) = X µ ei and LM(g) = X ν ei , then S(f, g) = bX (ν−µ) f − aX (µ−ν) g with b LC(f ) = a LC(g), gcd(a, b) = 1; otherwise, S(f, g) = 0. Here α+ = (max(α1 , 0), . . . , max(αn , 0)) is the positive part of α ∈ Zn . S-polynomial algorithm 9. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

l o c a l v a r i a b l e s a, b : R , µ, ν : Nn ; i f f = g then f i n d b such t h a t Ann(LC(f )) = hbi ; S(f, f ) ← bf else i f LPos(f ) 6= LPos(g) then S(f, g) ← 0 else µ ← mdeg(f ) ; ν ← mdeg(g) ; f i n d a, b such t h a t gcd(a, b) = 1, a gcd(LC(f ), LC(g)) = LC(f ), b gcd(LC(f ), LC(g)) = LC(g) ; + + S(f, g) ← bX (ν−µ) f − aX (µ−ν) g fi fi Note the following important properties of S(f, g): • If LM(f ) = X µ ei and LM(g) = X ν ei , then either S(f, g) = 0 or LM(S(f, g)) < X sup(µ,ν) ei ; otherwise, S(f, g) = 0; • S(X δ f, X δ g) = X δ S(f, g) for all δ ∈ Nn . S(f, f ) is called the auto-S-polynomial of f . It is designed to produce cancellation of the leading term of f by multiplying f with a generator of the annihilator of LC(f ). If the leading coefficient of f is regular, then S(f, f ) = 0 as in the discrete field case. In case R is a domain, this algorithm is not supposed to compute auto-S-polynomials and we can remove lines 2–5 and 16: if nevertheless executed with f = g, it yields S(f, f ) = 0. 6

The S-polynomial S(f, g) is designed to produce cancellation of the leading terms of f and g. It is worth pointing out that S(f, g) is not uniquely determined (up to a unit) when R has nonzero zerodivisors. Also S(g, f ) is generally not equal (up to a unit) to S(f, g) (in the discrete field case, this ambiguity is taken care of by making the S-polynomial monic). These issues are repaired through the consideration of the auto-S-polynomials S(f, f ) and S(g, g). Note that in the case of a valuation ring, the computation of the coefficients a, b is particularly easy: see Algorithm 16.

Buchberger’s algorithm This algorithm takes place in Context 6 for R. Here coherence, strictness, and the divisibility test are used. The termination is guaranteed if R is an archimedean valuation ring or a Bézout domain of Krull dimension ≤ 1 (see [19, Theorem 272 and Section 3.3.11]). See also the generalisations given in Theorems 22 and 24. This algorithm has the following goal. Input g1 , . . . , gs ∈ Hm \ {0}. Output a Gröbner basis [g1 , . . . , gs , . . . , gt ] for hg1 , . . . , gs i. Buchberger’s algorithm 10. 1 2 3 4 5 6

l o c a l v a r i a b l e s S : Hm , i, j, u : N ; t ← s; repeat u ← t; f o r i from 1 to u do f o r j from i to u do [g1 ,...,gu ]

7 8 9 10 11 12 13 14

S ← S(gi , gj ) i f S 6= 0 then t ← t + 1; gt ← S fi

by A l g o r i t h m s 9 and 8 ;

od od until t = u This algorithm is almost the same algorithm as in the case where the base ring is a discrete field. The modifications are in the definition of S-polynomials, in the consideration of the auto-Spolynomials, and in the division of terms (see Item (1) of Definition 3). In line 7, the algorithm may be sped up by computing the remainder w.r.t. [g1 , . . . , gt ] instead of [g1 , . . . , gu ] only. Remark 11. If the algorithm terminates, then we can transform the obtained Gröbner basis into a Gröbner basis [h1 , . . . , hr ] such that no term of an element h` lies in h LT(hk ) ; k 6= ` i by replacing each element of the Gröbner basis with a remainder of it on division by the other nonzero elements and by repeating this process until it stabilises. Such a Gröbner basis is called a pseudo-reduced Gröbner basis.

7

The syzygy algorithm for terms This algorithm takes also place in Context 6 for R. Note however that the divisibility test is not used here; only the zero test is used. It has the following goal. Input terms T1 , . . . , Ts ∈ Hm . Output a generating system [Si,j ]1≤i≤j≤s,LPos(Tj )=LPos(Ti ) for Syz(T1 , . . . , Ts ). In this algorithm, (1 , . . . , s ) is the canonical basis of R[X]s . Syzygy algorithm for terms 12. 1 2 3 4 5 6 7 8

l o c a l v a r i a b l e s i, j : {1, . . . , s} , J : subset of {1, . . . , s} , a, b : R , α, β : Nn ; f o r i from 1 to s do J ← { j ≥ i ; LPos(Tj ) = LPos(Ti ) } ; f o r j i n J do compute bX β , aX α such t h a t S(Ti , Tj ) = bX β Ti − aX α Tj by A l g o r i t h m 9 ; Si,j ← bX β i − aX α j od od

Schreyer’s syzygy algorithm This algorithm takes also place in Context 6 for R. It has the following goal. Input a Gröbner basis [g1 , . . . , gs ] for a submodule of Hm . Output a Gröbner basis [ui,j ]1≤i≤j≤s,LPos(gj )=LPos(gi ) for Syz(g1 , . . . , gs ) w.r.t. Schreyer’s monomial order induced by > and [g1 , . . . , gs ]. In this algorithm, (1 , . . . , s ) is the canonical basis of R[X]s . Schreyer’s syzygy algorithm 13. 1 2 3 4 5 6 7 8 9 10 11

l o c a l v a r i a b l e s i, j : {1, . . . , s} , J : subset of {1, . . . , s} , a, b : R , α, β : Nn , q` : R[X] ; f o r i from 1 to s do J ← { j ≥ i ; LPos(gj ) = LPos(gi ) } ; f o r j i n J do compute bX β , aX α such t h a t S(gi , gj ) = bX β gi − aX α gj by A l g o r i t h m 9 ; compute q1 , . . . , qs such t h a t S(gi , gj ) = q1 g1 + · · · + qs gs by A l g o r i t h m 8 (note that LM(S(gi , gj )) ≥ LM(q` g` ) whenever q` g` 6= 0) ; ui,j ← bX β i − aX α j − q1 1 − · · · − qs s od od The polynomials q1 , . . . , qs of lines 6–8 may have been computed while constructing the Gröbner basis. Remark 14. For an arbitrary system of generators [h1 , . . . , hr ] for a submodule U of Hm , the syzygy module of [h1 , . . . , hr ] is easily obtained from the syzygy module of a Gröbner basis for U (see [19, Theorem 296]).

8

3

The algorithms in the case of a valuation ring

This is the case of local Bézout rings. We consider a coherent valuation ring V with a divisibility test. In this case, we get simplified versions of the algorithms given in Section 2. We recover the algorithms given in [16, 19], but for modules instead of ideals. In particular, we generalise Buchberger’s algorithm to convenient valuation rings and modules. Note that the algorithm given in [16] contains a bug which is corrected in the corrigendum [17] to the papers [10, 16]. Division algorithm 15 (see [19, Definition 226]). Let V be a valuation ring with a divisibility test, Hm a free V[X]-module with basis (e1 , . . . , em ), and > a monomial order on Hm . In the Division algorithm 8, instead of defining the set D and finding the gcd d, one may look out for the first LT(hi ) such that LT(hi ) divides LT(p); in case of success, the algorithm proceeds with this index i, and the Bézout identity of line 6 is not needed. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

l o c a l v a r i a b l e s i : {1, . . . , s} , a : R , p : Hm , n o t d i v : b o o l e a n ; q1 ← 0 ; . . . ; qs ← 0 ; r ← 0 ; p ← h ; w h i l e p 6= 0 do i ← 1; n o t d i v ← true ; w h i l e i ≤ s and n o t d i v do i f LT(hi ) | LT(p) then f i n d a such t h a t a LC(hi ) = LC(p) ; qi ← qi + a(LM(p)/ LM(hi )) ; p ← p − a(LM(p)/ LM(hi ))hi ; notdiv ← false else i←i+1 fi od ; i f n o t d i v then r ← r + LT(p) ; p ← p − LT(p) fi od S-polynomial algorithm 16 (see [19, Definition 229]). Let V be a coherent valuation ring, Hm a free V[X]-module with basis (e1 , . . . , em ), and > a monomial order on Hm . We define the S-polynomial of two nonzero vectors in Hm by the S-polynomial algorithm 9. In this algorithm, the finding of a, b in lines 10-13 will take the following simple form, typical for valuation rings: f i n d a, b such t h a t a LC(g) = b LC(f ) with a = 1 or b = 1 This does not use the divisibility test: the explicit disjunction “a divides b or b divides a” is sufficient. When we have a divisibility test, the following expression arises for S(f, g) with f 6= g, LPos(f ) = LPos(g), mdeg(f ) = µ, mdeg(g) = ν: (

S(f, g) =

+

+

X (ν−µ) f − aX (µ−ν) g + + bX (ν−µ) f − X (µ−ν) g

if LC(g) | LC(f ), where LC(f ) = a LC(g) otherwise, where b LC(f ) = LC(g).

Note also that the annihilator Ann(LC(f )) appearing in the computation of the auto-S-polynomial is principal because V is a coherent valuation ring: there is a b such that Ann(LC(f )) = bV (b being defined up to a unit, see [11, Exercise IX-7]). 9

Example 17 (S-polynomials over F2 [Y ]/hY r i, r ≥ 2, a generalisation of [19, Example 231]). The ring V := F2 [Y ]/hY r i = F2 [y] (where y = Y¯ ) is a zero-dimensional coherent valuation ring with nonzero zerodivisors (because Ann(y k ) = hy r−k i). Each nonzero element a of this ring may be written in a unique way as y k (1 + yb) with k = 0, . . . , r − 1 and 1 + yb a unit. Let f 6= g ∈ V[X] \ {0}, and > a monomial order. Let µ = mdeg(f ), ν = mdeg(g). If LC(g) = y k (1 + yb) and LC(f ) = y ` (1 + yc), then ( + + X (ν−µ) f − (1 + yc)(1 + yb)−1 y `−k X (µ−ν) g if k ≤ ` S(f, g) = + + (1 + yb)(1 + yc)−1 y k−` X (ν−µ) f − X (µ−ν) g if k > ` +

(

=

up to a unit

+

(1 + yb)X (ν−µ) f − (1 + yc)y `−k X (µ−ν) g + + (1 + yb)y k−` X (ν−µ) f − (1 + yc)X (µ−ν) g

if k ≤ ` if k > `.

For the computation of the auto-S-polynomial S(f, f ), two cases may arise: • If LC(f ) is a unit, then S(f, f ) = 0. • If LC(f ) is y k (k > 0) up to a unit, then S(f, f ) = y r−k f . For example, with r = 2, using the lexicographic order for which X2 > X1 and considering the polynomials f1 = yX2 + X1 and f2 = yX1 + y, we have: S(f1 , f2 ) = X1 f1 − X2 f2 = X12 + yX2 , S(f1 , f1 ) = yf1 = yX1 , S(f2 , f2 ) = yf2 = 0.

4

Termination of Buchberger’s algorithm for a Bézout ring

The following lemma provides a necessary and sufficient condition for a term to belong to a module generated by terms over a coherent strict Bézout ring with a divisibility test. Lemma 18 (term modules, see [19, Lemma 227]). Let R be a coherent strict Bézout ring with a divisibility test, Hm a free R[X]-module with basis e1 , . . . , em , and > a monomial order on Hm . Let U be a submodule of Hm generated by a finite collection of terms aα X α eiα with α ∈ A. A term bX β er lies in U iff there is a nonempty subset A0 of A such that X α eiα divides X β er for every α ∈ A0 (i.e. iα = r and X α | X β ) and gcdα∈A0 (aα ) divides b. Proof. The condition is clearly sufficient. For the necessity, write bX β er = α∈A˜ cα aα X γα X α eiα P with A˜ ⊆ A, cα ∈ R \ {0}, and X γα ∈ Mn . Then b = α∈A0 cα aα , where A0 is the set of those α such that γα + α = β and iα = r. For each α ∈ A0 , X α divides X β . Since the gcd of the aα ’s with α ∈ A0 divides every aα , it also divides b. P

The following lemma is a key result for the characterisation of Gröbner bases by means of S-polynomials: see [6, Chapter 2, §6, Lemma 5] and, for valuation rings, [19, Lemma 233, adding the hypothesis of coherence]. Lemma 19. Let R be a coherent strict Bézout ring, Hm a free R[X]-module with basis e1 , . . . , em , > a monomial order, and f1 , . . . , fs ∈ Hm \{0} with the same leading monomial M . Let c1 , . . . , cs ∈ R. If c1 f1 +· · ·+cs fs vanishes or has leading monomial < M , then c1 f1 +· · ·+cs fs is a linear combination with coefficients in R of the S-polynomials S(fi , fj ) with 1 ≤ i ≤ j ≤ s. Proof. Let us write, for j 6= i, LC(fj ) = di,j ai,j with di,j = gcd(LC(fi ), LC(fj )), so that gcd(ai,j , aj,i ) = 1 and S(fi , fj ) = ai,j fi − aj,i fj . For each permutation i1 , . . . , is of 1, . . . , s, we shall transform the sum ai1 ,i2 · · · ais−1 ,is (c1 f1 + · · · + cs fs ) by replacing successively ai1 ,i2 fi1 .. . ais−1 ,is fis−1

by S(fi1 , fi2 ) + ai2 ,i1 fi2 , .. . by S(fis−1 , fis ) + ais ,is−1 fis . 10

At the end, the sum will be a linear combination of S(fi1 , fi2 ), S(fi2 , fi3 ), . . . , S(fis−1 , fis ), and fis ; let z be the coefficient of fis in this combination. The sum as well as each of the S-polynomials vanish or have leading monomial < M , so that the hypothesis yields z LC(fis ) = 0; therefore zfis is a multiple of S(fis , fis ). It remains to obtain a Bézout identity w.r.t. the products ai1 ,i2 · · · ais−1 ,is , because it yields an expression of c1 f1 + · · · + cs fs as
a linear combination of the required form. For this, it is enough to develop the product of the 2s Bézout identities w.r.t. ai,j and aj,i , 1 ≤ i < j ≤ s: this yields a
sum of products of 2s terms, each of which is either ai,j or aj,i , 1 ≤ i < j ≤ s, so that it is indexed by the tournaments on the vertices 1, . . . , s; every such product contains a product of the above form ai1 ,i2 · · · ais−1 ,is because every tournament contains a hamiltonian path (see [14]). Remark 20. The above proof results from an analysis of the following proof in the case where R is local and m = 1, which entails in fact the general case. Since R is a valuation ring, we may consider a permutation i1 , . . . , is of 1, . . . , s such that LC(fis ) | LC(fis−1 ) | · · · | LC(fi1 ). Thus S(fi1 , fi2 ) = fi1 − ai2 ,i1 fi2 , . . . , S(fis−1 , fis ) = fis−1 − ais ,is−1 fis for some ai2 ,i1 , . . . , ais ,is−1 . Then, by replacing successively fik by S(fik , fik+1 ) + aik+1 ,ik fik+1 , the linear combination c1 f1 + · · · + cs fs may be rewritten as a linear combination of S(fi1 , fi2 ), . . . , S(fis−1 , fis ), and fis , with the coefficient of fis turning out to lie in Ann(LC(fis )). Lemma 19 enables us to generalise some classical results about the existence and characterisation of Gröbner bases to the case of coherent strict Bézout rings with a divisibility test. See [19, Theorem 234] for the case of valuation rings and ideals. Theorem 21 (Buchberger’s criterion for Gröbner bases). Let R be a coherent strict Bézout ring with a divisibility test, U = hg1 , . . . , gs i a submodule of a free R[X]-module Hm , and > a monomial order on Hm . Then G = [g1 , . . . , gs ] is a Gröbner basis for U if and only if the remainder of S(gi , gj ) on division by G vanishes for all pairs i ≤ j. Theorem 21 entails that Buchberger’s Algorithm 10 constructs a Gröbner basis for coherent valuation rings with a divisibility test when such a basis exists (compare [19, Algorithm 235]). Note that the precise reason why Buchberger’s algorithm 10 terminates at every run is that the valuation ring is archimedean, or equivalently, that it is either a valuation domain of Krull dimension ≤ 1 or a zero-dimensional valuation ring containing nonzero zerodivisors. The proof follows the lines of [19, Theorem 272], the module case being analogous to the ideal case. In this proof, the archimedean property is used only for obtaining that the module of leading terms is finitely generated: this is the gist of Theorem 22 below, which is a new key result. Theorem 22. Let V be a coherent valuation ring with a divisibility test, I = hf1 , . . . , fs i a nonzero ideal of V[X], and > a monomial order on V[X]. If LT(I) is finitely generated, then Buchberger’s Algorithm 10 computes a finite Gröbner basis for I. Proof. Let LT(I) = hLT(g1 ), . . . , LT(gr )i with gi ∈ I \ {0}. Let 1 ≤ k ≤ r. As gk ∈ I, there exist h1 , . . . , hs ∈ V[X] such that Xs gk = hi fi (1) i=1

with mdeg(gk ) ≤ sup1≤i≤s (mdeg(Mi Ni )) =: γ1 (we call it the multidegree of the expression (1) of gk w.r.t. the generating set {f1 , . . . , fs } of I), where Mi = LM(hi ) and Ni = LM(fi ). Case 1: mdeg(gk ) = γ1 , say mdeg(gk ) = mdeg(Mi0 Ni0 ) = γ1 for some i0 ∈ {1, . . . , s}. As the leading coefficients of the hi fi ’s with mdeg(gk ) = mdeg(Mi Ni ) are comparable w.r.t. division, we can suppose that all of them are divisible by the leading coefficient of hi0 fi0 . It follows that LT(gk ) ∈ hLT(fi0 )i ⊆ hLT(f1 ), . . . , LT(fs )i. 11

Case 2: mdeg(gk ) < γ1 . We have X

gk =

X

hi fi

mdeg(Mi Ni )=γ1

mdeg(Mi Ni ) be a TOP monomial order on K[X, Y ]2 for which Y > X, K being a field, let e1 = (1, 0) and e2 = (0, 1), and consider the free submodule U of K[X, Y ]2 generated by u1 = (Y, X) and u2 = (X, 0). Then LT(u1 ) = Y e1 , LT(u2 ) = Xe1 , S(u1 , u2 ) = Xu1 − Y u2 = X 2 e2 =: u3 , and S(u1 , u3 ) = S(u2 , u3 ) = 0. It follows that (u1 , u2 , u3 ) is a Gröbner basis for U , and LT(U ) = hY, Xie1 ⊕ hX 2 ie2 . One can see that hY, Xi is not principal and LT(U ) is not free, while U is free. So we content ourselves with the following observation. L Remark 26. Let > be a monomial order on the free S-module F = m S = V[X] and j=1 Sej , where L V is a valuation domain. Let U be a submodule of F and suppose that LT(U ) = m j=1 Ij ej , where Ij is a principal ideal for j = 1, . . . , m. Then LT(U ) and U are free S-modules. (Of course, this is not true anymore if V is a valuation ring with nonzero zerodivisors. For example, consider the ideal U = h8X + 2i in (Z/16Z)[X]: we have LT(U ) = h2i (so that it is principal), but U is not free since 8U = (0).) We shall need the following proposition, which generalises [19, Theorem 291] to the case of modules. Proposition 27 (Generating set for the syzygy module of a list of terms for a coherent valuation ring). Let V be a coherent valuation ring, Hm a free V[X]-module with basis (e1 , . . . , em ), and terms T1 , . . . , Ts in Hm . Considering the canonical basis (1 , . . . , s ) of V[X]s , the syzygy module Syz(T1 , . . . , Ts ) is generated by the Si,j ∈ V[X]s with 1 ≤ i ≤ j ≤ s and LPos(Ti ) = LPos(Tj ), as computed by the Syzygy algorithm for terms 12. Note that in the Syzygy algorithm for terms 12, the a, b will be found as in the S-polynomial algorithm 16, so that we get

Si,j =

   bi

β

α

X  − aX 

i j   bX β  − X α  i j

if i = j, where hbi = Ann(LC(Ti )), if i < j and LC(Tj ) | LC(Ti ), where LC(Ti ) = a LC(Tj ), otherwise, where b LC(Ti ) = LC(Tj ). 13

(2)

Here β = (mdeg(Tj ) − mdeg(Ti ))+ and α = (mdeg(Ti ) − mdeg(Tj ))+ . Now we shall follow closely Schreyer’s ingenious proof [15] of Hilbert’s syzygy theorem via Gröbner bases, but with a valuation ring instead of a field. Schreyer’s proof is very well explained in [8, §§ 4.4.1–4.4.3]. Theorem 28 (Schreyer’s algorithm for a coherent valuation ring with a divisibility test). Let V be a coherent valuation ring with a divisibility test, Hm a free V[X]-module with basis (e1 , . . . , em ), and > a monomial order on Hm . Let U be a submodule of Hm with Gröbner basis [g1 , . . . , gs ]. Then the relations ui,j computed by Schreyer’s syzygy algorithm 13 form a Gröbner basis for the syzygy module Syz(g1 , . . . , gs ) w.r.t. Schreyer’s monomial order induced by > and [g1 , . . . , gs ]. Moreover, for 1 ≤ i ≤ j ≤ s such that LPos(gi ) = LPos(gj ),

LT(ui,j ) =

  bi 

X

(mdeg(gj )−mdeg(gi ))+

if i = j, where hbi = Ann(LC(gi )), if i < j and LC(gj ) | LC(gi ), otherwise, where b LC(gi ) = LC(gj ).



i   bX (mdeg(gj )−mdeg(gi ))+ 

i

(3)

Proof (a slight modification of the proof of [8, Theorem 4.16]). Let us use the notation of Schreyer’s syzygy algorithm 13. Let 1 ≤ i = j ≤ s. As LM(q` g` ) ≤ LM(S(gi , gi )) < LM(gi ) whenever q` g` 6= 0, we infer that LT(ui,i ) = bi with hbi = Ann(LC(gi )). Let 1 ≤ i < j ≤ s such that LPos(gi ) = LPos(gj ). Suppose that LC(gi ) = a LC(gj ) for some a: as LM(X β gi ) = LM(aX α gj ) and i < j, we have LT(X β i − aX α j ) = X β i , and because LM(q` g` ) ≤ LM(S(gi , gj )) < LM(X β gi ) whenever q` g` 6= 0, we infer that LT(ui,j ) = X β i ; otherwise, with b such that b LC(gi ) = LC(gj ), we obtain similarly LT(ui,j ) = bX β i . Thus, considering Equation (2) with T` = LT(g` ), LT(ui,j ) = LT(Si,j ) holds for all 1 ≤ i ≤ j ≤ s. Let us show now that the relations ui,j form a Gröbner basis for the syzygy module Syz(g1 , . . . , gs ). P For this, let v = s`=1 v` ` ∈ Syz(g1 , . . . , gs ) and let us show that there exist 1 ≤ i ≤ j ≤ s with LPos(gi ) = LPos(gj ) such that LT(ui,j ) divides LT(v). Let us write LM(v` ` ) = N` ` and P LC(v` ` ) = c` for 1 ≤ ` ≤ s. Then LM(v) = Ni i for some 1 ≤ i ≤ s. Now let v 0 = `∈S c` N` ` , where S is the set of those ` for which N` LM(g` ) = Ni LM(gi ). By definition of Schreyer’s monomial order, we have ` ≥ i for all ` ∈ S. Substituting each ` in v 0 by T` , the sum becomes zero. Therefore v 0 is a relation of the terms T` with ` ∈ S. By virtue of Proposition 27, v 0 is a linear combination of elements of the form S`,k with ` ≤ k in S. Since ` > i for all ` ∈ S with ` 6= i, we infer, by virtue of Lemma 18, that LT(v 0 ) is a multiple of LT(Si,j ) for some j ∈ S. The desired result follows since LT(v) = LT(v 0 ). As a consequence of Theorem 28, we obtain the following constructive versions of Hilbert’s syzygy theorem for a valuation domain. Theorem 29 (Syzygy theorem for a valuation domain with a divisibility test). Let M = Hm /U be a finitely presented V[X]-module, where V is a valuation domain with a divisibility test. Assume that, w.r.t. the TOP lexicographic monomial order, LT(U ) is finitely generated. Then M admits a free V[X]-resolution 0 → Fp → Fp−1 → · · · → F1 → F0 → M → 0 of length p ≤ n + 1. Proof. It suffices to prove that U has a free V[X]-resolution of length p ≤ n. Let us use the lexicographic monomial order with Xn > Xn−1 > · · · > X1 on V[X]. Let (g1 , . . . , gs ) be a Gröbner basis for U w.r.t. the corresponding TOP order. We can w.l.o.g. suppose that whenever LM(gi ) and LM(gj ) involve the same basis element for some i < j, say LM(gi ) = Ni k and LM(gj ) = Nj k , 14

then Ni > Nj . More precisely, whenever Ni = Nj , one of LC(gi ) and LC(gj ) divides the other, say LC(gj ) = b LC(gi ), and the corresponding gj may be reduced into gj − bgi . In a nutshell, all the possible reductions between the LT(gk )’s are being exhausted. Now, since we have used the lexicographic order with Xn > Xn−1 > · · · > X1 , it turns out that the indeterminate Xn cannot appear in the leading terms of the ui,j ’s in (3). Thus, after at most n computations of the iterated syzygies, we reach a situation where none of the indeterminates Xn , . . . , X1 appears in the leading terms of the computed Gröbner basis for the iterated syzygy module. This implies that the iterated syzygy module is free (as noted in Remark 26). Remark 30. In the proof of this theorem, we need to work with the TOP lexicographic monomial order. We do not know what happens for other monomial orders. This applies also for Theorems 33 and 37. Corollary 31 (Syzygy theorem for a one-dimensional valuation domain with a divisibility test). Let M = Hm /U be a finitely presented V[X]-module, where V is a one-dimensional valuation domain with a divisibility test. Then M admits a finite free V[X]-resolution 0 → Fp → Fp−1 → · · · → F1 → F0 → M → 0 of length p ≤ n + 1 as described in the previous theorem. Example 32. Let g1 = Y 4 − Y, g2 = 2Y, g3 = X 3 − 1 ∈ Z2Z [X, Y ], and let us use the lexicographic order >1 for which Y >1 X. We have S(g1 , g2 ) = 2g1 − Y 3 g2 = −2Y = −g2 , S(g1 , g3 ) = X 3 g1 − Y 4 g3 = Y 4 − Y X 3 = g1 − Y g3 , S(g2 , g3 ) = X 3 g2 − 2Y g3 = 2Y = g2 . Thus (g1 , g2 , g3 ) is a (pseudo-reduced) Gröbner basis for I = hg1 , g2 , g3 i and LT(I) = hY 4 , 2Y, X 3 i. By Theorem 28, u1,3 = [X 3 − 1, 0, −Y 4 + Y ], u1,2 = [2, −Y 3 + 1, 0], u2,3 = [0, X 3 − 1, −2Y ] form a (pseudo-reduced) Gröbner basis for the syzygy module Syz(g1 , g2 , g3 ) w.r.t. Schreyer’s monomial order >2 induced by >1 and [g1 , g2 , g3 ]. In particular, LT(Syz(g1 , g2 , g3 )) = hLT(u1,3 ), LT(u1,2 ), LT(u2,3 )i = hX 3 e1 , 2e1 , X 3 e2 i = h2, X 3 ie1 ⊕ hX 3 ie2 , where (e1 , e2 , e3 ) stands for the canonical basis of Z2Z [X, Y ]3 . We have S(u1,3 , u1,2 ) = 2u1,3 − X 3 u1,2 = (−2, Y 3 X 3 − X 3 , −2Y 4 + 2Y ) = −u1,2 + (Y 3 − 1)u2,3 , S(u1,3 , u2,3 ) = S(u1,2 , u2,3 ) = 0. We recover that [u1,3 , u1,2 , u2,3 ] is a Gröbner basis for Syz(g1 , g2 , g3 ). By Theorem 28, the elements u1,3;1,2 = [2, −X 3 + 1, −Y 3 + 1] forms a (pseudo-reduced) Gröbner basis for the syzygy module Syz(u1,3 , u1,2 , u2,3 ) w.r.t. Schreyer’s monomial order >3 induced by >2 and [u1,3 , u1,2 , u2,3 ]. In particular, LT(Syz(u1,3 , u1,2 , u2,3 )) = hLT(u1,3;1,2 )i = h2i1 , where (1 , 2 , 3 ) stands for the canonical basis of Z2Z [X, Y ]3 . By Remark 26, Syz(u1,3 , u1,2 , u2,3 ) is free. We conclude that I admits the following length 2 free Z2Z [X, Y ]-resolution: u

1,3

u1,3;1,2

0 −→ Z2Z [X, Y ] −−−−→ Z2Z [X, Y

]3

u1,2 u2,3



−−−−−→ Z2Z [X, Y

15

 g1 

]3

g2 g3

−−−−→ I → 0.

It follows that Z2Z [X, Y ]/I admits the following length 3 free Z2Z [X, Y ]-resolution: π

0 → Z2Z [X, Y ] → Z2Z [X, Y ]3 → Z2Z [X, Y ]3 → Z2Z [X, Y ] → Z2Z [X, Y ]/I → 0. Another consequence of Theorem 28 is the following result. Theorem 33. Let M = Hm /U be a finitely presented V[X]-module, where V is a coherent valuation ring with nonzero zerodivisors. Assume that, w.r.t. the TOP lexicographic monomial order, LT(U ) is finitely generated. Then M admits a resolution by finite free V[X]-modules ϕp+3

ϕp+2

ϕp+1

ϕp−1

ϕp

ϕ2

ϕ1

ϕ0

· · · −→ Fp −→ Fp −→ Fp −→ Fp−1 −→ · · · −→ F1 −→ F0 −→ M −→ 0 such that for some p ≤ n + 1, • LT(Ker(ϕp )) =

Lmp

j=1 hbj iej

• LT(Ker(ϕp+2k−1 )) = • LT(Ker(ϕp+2k )) =

with b1 , . . . , bmp ∈ V and (e1 , . . . , emp ) a basis for Fp ,

Lmp

j=1 Ann(bj )ej

Lmp

for k ≥ 1,

j=1 Ann(Ann(bj ))ej

for k ≥ 1,

and at each step where indeterminates remain present, the considered monomial order is Schreyer’s monomial order (as in the proof of Theorem 29). Proof. The part ϕp

ϕp−1

ϕ2

ϕ1

ϕ0

Fp −→ Fp−1 −→ · · · −→ F1 −→ F0 −→ M −→ 0 Lm

p of the free V[X]-resolution with p ≤ n + 1 and LT(Ker(ϕp )) = j=1 hbj iej follows from the proof of Theorem 29. W.l.o.g., the bj ’s are 6= 0. Let us denote by [g1 , . . . , gmp ] a Gröbner basis for Ker(ϕp ) such that LT(gj ) = bj ej for 1 ≤ j ≤ mp . So S(gi , gj ) = 0 for i < j. Thus the Lmp Lmp fact that LT(Ker(ϕp+1 )) = j=1 Ann(bj )ej , LT(Ker(ϕp+2 )) = j=1 Ann(Ann(bj ))ej , etc. follows immediately from Theorem 28. Finally, let us recall the equality Ann(Ann(Ann(I))) = Ann(I) for an ideal I.

Let us point out that this shows that the free resolution is in general not a finite one. Corollary 34. Let M = Hm /U be a finitely presented V[X]-module, where V is a zero-dimensional coherent valuation ring3 . Then M admits a free V[X]-resolution as described in Theorem 33. Example 35. Let g1 = Y 4 −Y, g2 = 2Y, g3 = X 3 −1 ∈ (Z/4Z)[X, Y ], and let us use the lexicographic order >1 for which Y >1 X. We have S(g1 , g1 ) = 0g1 = 0, S(g1 , g2 ) = 2g1 − Y 3 g2 = −2Y = −g2 , S(g2 , g2 ) = 2g2 = 0, S(g2 , g3 ) = X 3 g2 − 2Y g3 = 2Y = g2 , S(g3 , g3 ) = 0g3 = 0, S(g1 , g3 ) = X 3 g1 − Y 4 g3 = Y 4 − Y X 3 = g1 − Y g3 . Thus (g1 , g2 , g3 ) is a (pseudo-reduced) Gröbner basis for I = hg1 , g2 , g3 i, and LT(I) = hY 4 , 2Y, X 3 i. By Theorem 28, u1,3 = (X 3 − 1, 0, −Y 4 + Y ), u1,2 = (2, −Y 3 + 1, 0), u2,3 = (0, X 3 − 1, −2Y ), u2,2 = (0, 2, 0) form a (pseudo-reduced) Gröbner basis for the syzygy module Syz(g1 , g2 , g3 ) w.r.t. Schreyer’s monomial order >2 induced by >1 and [g1 , g2 , g3 ]. In particular, LT(Syz(g1 , g2 , g3 )) = hLT(u1,3 ), . . . , LT(u2,2 )i = hX 3 e1 , 2e1 , X 3 e2 , 2e2 i = h2, X 3 ie1 ⊕ h2, X 3 ie2 , 3

Note that a zero-dimensional ring without nonzero zerodivisors is a discrete field.

16

where (e1 , e2 , e3 ) stands for the canonical basis of (Z/4Z)[X, Y ]3 . We have S(u1,3 , u1,3 ) = 0u1,3 = 0,

S(u1,2 , u2,3 ) = S(u1,2 , u2,2 ) = 0, 3

S(u1,3 , u1,2 ) = 2u1,3 − X u1,2 3

3

S(u2,3 , u2,3 ) = 0u2,3 = 0, 3

4

S(u2,3 , u2,2 ) = 2u2,3 − X 3 u2,2

= (−2, Y X − X , −2Y + 2Y ) = −u1,2 + (Y 3 − 1)u2,3 ,

= (0, −2, 0Y ) = (0, −2, 0) = −u2,2 ,

S(u1,3 , u2,3 ) = S(u1,3 , u2,2 ) = 0, S(u1,2 , u1,2 ) = 2u1,2 = (0, −2Y 3 + 2, 0) = (−Y 3 + 1)u2,2 ,

S(u2,2 , u2,2 ) = 2u2,2 = 0.

We recover that (u1,3 , u1,2 , u2,3 , u2,2 ) is a Gröbner basis for Syz(g1 , g2 , g3 ). By Theorem 28, u1,3;1,2 = (2, −X 3 +1, −Y 3 +1, 0), u1,2;1,2 = (0, 2, 0, Y 3 −1), u2,3;2,2 = (0, 0, 2, −X 3 +1), u2,2;2,2 = (0, 0, 0, 2) form a (pseudo-reduced) Gröbner basis for the syzygy module Syz(u1,3 , u1,2 , u2,3 , u2,2 ) w.r.t. Schreyer’s monomial order >3 induced by >2 and [u1,3 , u1,2 , u2,3 , u2,2 ]. In particular, LT(Syz(u1,3 , u1,2 , u2,3 , u2,2 )) = hLT(u1,3;1,2 ), . . . , LT(u2,2;2,2 )i = h21 , . . . , 24 i = h2i1 ⊕ h2i2 ⊕ h2i3 ⊕ h2i4 , where (1 , . . . , 4 ) stands for the canonical basis of (Z/4Z)[X, Y ]4 . By Theorem 28, we find four vectors u(1,3;1,2),(1,3;1,2) , . . . , u(2,2;2,2),(2,2;2,2) ∈ (Z/4Z)[X, Y ]4 forming a (pseudo-reduced) Gröbner basis for the syzygy module Syz(u1,3;1,2 , . . . , u2,2;2,2 ) w.r.t. Schreyer’s monomial order >4 induced by >3 and [u1,3;1,2 , . . . , u2,2;2,2 ]. In particular, LT(Syz(u1,3;1,2 , . . . , u2,2;2,2 )) = hLT(u(1,3;1,2),(1,3;1,2) ), . . . , LT(u(2,2;2,2),(2,2;2,2) )i = h2i1 ⊕ h2i2 ⊕ h2i3 ⊕ h2i4 , etc. We conclude that I admits the free (Z/4Z)[X, Y ]-resolution ϕ3

ϕ2

ϕ1

ϕ0

· · · −→ (Z/4Z)[X, Y ]4 −→ (Z/4Z)[X, Y ]4 −→ (Z/4Z)[X, Y ]3 −→ I −→ 0 such that LT(Ker(ϕi )) = h2i1 ⊕ h2i2 ⊕ h2i3 ⊕ h2i4 for i ≥ 2.

6

The syzygy theorem and Schreyer’s algorithm for a Bézout ring

As explained in the proof of Theorem 24, one can avoid branching when computing a dynamical Gröbner basis (see [10, 16, 19]) for a Bézout domain of Krull dimension ≤ 1 (e.g. Z and the ring of all algebraic integers—note that the last one is not a PID) or a zero-dimensional coherent Bézout ring. Note that this is not√possible for Prüfer domains of Krull dimension ≤ 1 which are not Bézout domains (for example, Z[ −5], see [10, Section 4]). Let us now generalise the results of Section 5 to the case of coherent strict Bézout rings. Theorem 36 (Schreyer’s algorithm for Bézout rings). We consider a coherent strict Bézout ring R with a divisibility test. Let Hm be a free R[X]-module with basis (e1 , . . . , em ) and > a monomial order on Hm . Let U ⊆ Hm be a submodule of Hm with Gröbner basis [g1 , . . . , gs ]. Then the relations ui,j computed by Algorithm 13 form a Gröbner basis for the syzygy module Syz(g1 , . . . , gs ) w.r.t. Schreyer’s monomial order induced by > and [g1 , . . . , gs ]. Proof. This follows directly from the local case given by Theorem 28: see the proof of Theorem 24 for an explanation.

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Theorem 37 (Syzygy theorem for a Bézout domain with a divisibility test). Let M = Hm /U be a finitely presented R[X]-module, where R is a Bézout domain with a divisibility test and Hm a free R[X]-module. Assume that, w.r.t. the TOP lexicographic monomial order, LT(U ) is finitely generated. Then M admits a finite free R[X]-resolution 0 → Fp → Fp−1 → · · · → F1 → F0 → M → 0 of length p ≤ n + 1. Proof. This follows directly from the local case. Corollary 38 (Syzygy theorem for a one-dimensional Bézout domain with a divisibility test). Let M = Hm /U be a finitely presented R[X]-module, where R is a one-dimensional Bézout domain with a divisibility test. Then M admits a finite free R[X]-resolution 0 → Fp → Fp−1 → · · · → F1 → F0 → M → 0 of length p ≤ n + 1 as described in the previous theorem. Let us now treat the case of zero-dimensional coherent Bézout rings. Theorem 39 (Syzygy theorem for a zero-dimensional Bézout ring with a divisibility test). Let M = Hm /U be a finitely presented R[X]-module, where R is a coherent zero-dimensional Bézout ring with a divisibility test and Hm a free R[X]-module. Then M admits a free R[X]-resolution ϕp+3

ϕp+2

ϕp+1

ϕp−1

ϕp

ϕ2

ϕ1

ϕ0

· · · −→ Fp −→ Fp −→ Fp −→ Fp−1 −→ · · · −→ F1 −→ F0 −→ M −→ 0 such that for some p ≤ n + 1, • LT(Ker(ϕp )) =

Lmp

j=1 hbj iej

• LT(Ker(ϕp+2k−1 )) = • LT(Ker(ϕp+2k )) =

with b1 , . . . , bmp ∈ R and (e1 , . . . , emp ) a basis for Fp ,

Lmp

j=1 Ann(bj )ej

Lmp

for k ≥ 1,

j=1 Ann(Ann(bj ))ej

for k ≥ 1,

and at each step where indeterminates remain present, the considered monomial order is Schreyer’s monomial order. Proof. Follows directly from the local case.

The case of the integers The following theorems are particular cases of Theorem 36 and Corollary 38. Theorem 40 (Schreyer’s algorithm for the integers). Let Hm be a free Z[X]-module with basis (e1 , . . . , em ) and > a monomial order on Hm . Let U ⊆ Hm be a submodule of Hm with Gröbner basis [g1 , . . . , gs ]. Then the relations ui,j computed by Algorithm 13 form a Gröbner basis for the syzygy module Syz(g1 , . . . , gs ) w.r.t. Schreyer’s monomial order induced by > and [g1 , . . . , gs ]. Moreover, for 1 ≤ i < j ≤ s such that LPos(gi ) = LPos(gj ), we have LT(ui,j ) =

LC(gj ) (mdeg(gj )−mdeg(gi ))+ i . gcd(LC(gi ),LC(gj )) X

Theorem 41 (Syzygy theorem for the integers). Let M be a finitely generated Z[X]-module. Then M admits a free Z[X]-resolution 0 → Fp → Fp−1 → · · · → F1 → F0 → M → 0 of length p ≤ n + 1. 18

Example 42. Let g1 = Y 2 − X + 3, g2 = 4X 2 − 4, g3 = 6X + 6 ∈ Z[X, Y ], and let us use the lexicographic order >1 for which Y >1 X. We have: S(g1 , g2 ) = 4X 2 g1 − Y 2 g2 = 4g1 + (−X + 3)g2 , S(g1 , g3 ) = 6Xg1 − Y 2 g3 = −6g1 + (−X + 3)g3 , S(g2 , g3 ) = 3g2 − 2Xg3 = −2g3 . Thus (g1 , g2 , g3 ) is a Gröbner basis for I = hg1 , g2 , g3 i, and LT(I) = hY 2 , 4X 2 , 6Xi. By Theorem 40, u1,2 = (4X 2 − 4, −Y 2 + X − 3, 0), u1,3 = (6X + 6, 0, −Y 2 + X − 3), u2,3 = (0, 3, −2X + 2) form a Gröbner basis for the syzygy module Syz(g1 , g2 , g3 ) w.r.t. Schreyer’s monomial order >2 induced by >1 and [g1 , g2 , g3 ]. In particular, LT(Syz(g1 , g2 , g3 )) = hLT(u1,2 ), LT(u1,3 ), LT(u2,3 )i = h4X 2 e1 , 6Xe1 , 3e2 i = h4X 2 , 6Xie1 ⊕ h3ie2 = 2h2X 2 , 3Xie1 ⊕ h3ie2 = 2hX 2 , 3Xie1 ⊕ h3ie2 , where (e1 , e2 , e3 ) stands for the canonical basis of Z[X, Y ]3 . Thus u01,2 = Xu1,3 − u1,2 = (2X 2 + 6X + 4, Y 2 − X + 3, −Y 2 X + X 2 − 3X), u1,3 , u2,3 form a reduced Gröbner basis for Syz(g1 , g2 , g3 ). We have: S(u01,2 , u1,3 ) = 3u01,2 − Xu1,3 = 2u1,3 + (Y 2 − X + 3)u2,3 , S(u01,2 , u2,3 ) = S(u1,3 , u2,3 ) = 0. We recover that (u01,2 , u1,3 , u2,3 ) is a Gröbner basis for Syz(g1 , g2 , g3 ). By Theorem 40, u1,2;1,3 = (3, −X − 2, −Y 2 + X − 3) forms a (pseudo-reduced) Gröbner basis for the syzygy module Syz(u01,2 , u1,3 , u2,3 ) w.r.t. Schreyer’s monomial order >3 induced by >2 and [u01,2 , u1,3 , u2,3 ]. In particular, LT(Syz(u01,2 , u1,3 , u2,3 )) = hLT(u1,2;1,3 )i = h3i1 where (1 , 2 , 3 ) stands for the canonical basis of Z[X, Y ]3 . It follows that Syz(u01,2 , u1,3 , u2,3 ) is free. We conclude that I admits the following length 2 free Z[X, Y ]-resolution: u

1,2

u1,2;1,3

0 → Z[X, Y ] −−−−→ Z[X, Y

]3

u1,3 u2,3



−−−−−→ Z[X, Y

 g1 

]3

g2 g3

−−−−→ I → 0.

The case of Z/N Z The elements of Z/N Z are simply written as integers (their representatives in [[0, N − 1]]). When talking about the gcd of two nonzero elements in Z/N Z we mean the gcd of their representatives in N [[1, N − 1]]. For a nonzero element a in Z/N Z, letting b = gcd(a, N ), the class of in Z/N Z will b be denoted by ann(a); it generates Ann(a). • The Division algorithm 8 attains its goal: the gcd and the Bézout identity to be found in line 6 will be computed by finding d, b, bi (i ∈ D) in Z such that d = gcd(N, gcd{ LC(hi ) ; P i ∈ D }) = bN + i∈D bi LC(hi ); the euclidean division in line 6 will be performed in Z; • The S-polynomial algorithm 9 attains its goal: note that in this case, the generator of the annihilator of LC(f ) to be found on line 3 may be taken to be ann(LC(f )), so that the auto-Spolynomial of f is S(f, f ) = ann(LC(f ))f ; • Buchberger’s algorithm 10 attains its goal. 19

The following theorems are particular cases of Theorems 36 and 39. Theorem 43 (Schreyer’s algorithm for Z/N Z). Let Hm be a free (Z/N Z)[X]-module with basis (e1 , . . . , em ) and > a monomial order on Hm . Let U ⊆ Hm be a submodule of Hm with Gröbner basis [g1 , . . . , gs ]. Then the relations ui,j computed by Algorithm 13 form a Gröbner basis for the syzygy module Syz(g1 , . . . , gs ) w.r.t. Schreyer’s monomial order induced by > and [g1 , . . . , gs ]. Moreover, for all 1 ≤ i ≤ j ≤ s such that LPos(gi ) = LPos(gj ), we have LT(ui,j ) =

 ann(LC(gi ))i 

if i = j,

LC(gj ) (mdeg(gj )−mdeg(gi gcd(LC(gi ),LC(gj )) X

))+

i

otherwise.

Theorem 44 (Syzygy theorem for Z/N Z). Let M be a finitely presented (Z/N Z)[X]-module. Then M admits a free (Z/N Z)[X]-resolution ϕp+3

ϕp+2

ϕp+1

ϕp

ϕp−1

ϕ2

ϕ1

ϕ0

· · · −→ Fp −→ Fp −→ Fp −→ Fp−1 −→ · · · −→ F1 −→ F0 −→ M −→ 0 such that for some p ≤ n + 1, LT(Ker(ϕp )) =

mp M

hbj iej , LT(Ker(ϕp+1 )) =

j=1 mp

LT(Ker(ϕp+2 )) =

M

hbj iej , LT(Ker(ϕp+3 )) =

j=1

mp M

N ej , gcd(N, bj ) j=1

mp M

N ej , etc., gcd(N, bj ) j=1

where (e1 , . . . , emp ) is a basis for Fp , b1 , . . . , bmp ∈ Z/N Z, and the considered monomial order is Schreyer’s monomial order. Example 45. Let g1 = Y + 1, g2 = X 3 + X 2 + 6, g3 = 3X 2 , g4 = 9 ∈ (Z/12Z)[X, Y ], and let us use the lexicographic order >1 for which Y >1 X. We have S(g2 , g3 ) = 3g2 − Xg3 = g3 + 2g4 ,

S(g1 , g1 ) = 0g1 = 0, 3

2

S(g1 , g2 ) = X g1 − Y g2 = (−X − 6)g1 + g2 , S(g2 , g4 ) = 9g2 − X 3 g3 = (X 2 + 6)g4 , S(g1 , g3 ) = 3X 2 g1 − Y g3 = g3 ,

S(g3 , g3 ) = 4g3 = 0,

S(g1 , g4 ) = 9g1 − Y g4 = g4 ,

S(g3 , g4 ) = 3g3 − X 2 g4 = 0,

S(g2 , g2 ) = 0g2 = 0,

S(g4 , g4 ) = 4g4 = 0.

Thus (g1 , g2 , g3 , g4 ) is a (pseudo-reduced) Gröbner basis for I = hg1 , g2 , g3 , g4 i, and LT(I) = hY, X 3 , 3X 2 , 9i. By Theorem 43, u1,2 = (X 3 + X 2 + 6, −Y − 1, 0, 0), u1,3 = (3X 2 , 0, −Y − 1, 0), u1,4 = (9, 0, 0, −Y − 1), u2,3 = (0, 3, −X − 1, −2), u2,4 = (0, 9, −X 3 , −X 2 − 6), u3,3 = (0, 0, 4, 0), u3,4 = (0, 0, 3, −X 2 ), u4,4 = (0, 0, 0, 4) form a Gröbner basis for the syzygy module Syz(g1 , g2 , g3 , g4 ) w.r.t. Schreyer’s monomial order >2 induced by >1 and [g1 , g2 , g3 , g4 ]. In particular, LT(Syz(g1 , g2 , g3 , g4 )) = hLT(u1,2 ), . . . , LT(u4,4 )i = hX 3 , 3X 2 , 9ie1 ⊕ h3, 9ie2 ⊕ h4, 3ie3 ⊕ h4ie4 = hX 3 , 3ie1 ⊕ h3ie2 ⊕ h1ie3 ⊕ h4ie4 , where (e1 , e2 , e3 , e4 ) stands for the canonical basis of (Z/12Z)[X, Y ]4 . Thus u1,2 , u01,4 = −u1,4 = (3, 0, 0, Y + 1), u2,3 , u03,3 = u3,3 − u3,4 = (0, 0, 1, X 2 ), u4,4 form a reduced Gröbner basis for

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Syz(g1 , g2 , g3 , g4 ). We have S(u1,2 , u01,4 ) = 3u1,2 − X 3 u01,4 = (X 2 + 2)u01,4 + (3Y + 3)u2,3 + (3Y X + 3Y + 3X + 3)u03,3 +(2Y X 3 + 2Y X 2 + 2X 3 + 2X 2 + Y + 1)u4,4 , S(u01,4 , u01,4 ) = 4u01,4 = (Y + 1)u4,4 , S(u2,3 , u2,3 ) = 4u2,3 = (8X + 8)u03,3 + (X 3 + X 2 + 1)u4,4 , S(u4,4 , u4,4 ) = 3u4,4 = 0. By Theorem 43, the elements u1,2;1,4 = (3, −X 3 − X 2 − 2, −3Y − 3, −3Y X − 3Y − 3X − 3, −2Y X 3 − 2Y X 2 − Y − 2X 3 − 2X 2 − 1), u1,4;1,4 = (0, 4, 0, 0, −Y − 1), u2,3;2,3 = (0, 0, 4, −8X − 8, −X 3 − X 2 − 1), u4,4;4,4 = (0, 0, 0, 0, 3) form a (pseudo-reduced) Gröbner basis for the syzygy module Syz(u1,2 , u01,4 , u2,3 , u03,3 , u4,4 ) w.r.t. Schreyer’s monomial order >3 induced by >2 and [u1,2 , u01,4 , u2,3 , u03,3 , u4,4 ]. In particular, LT(Syz(u1,2 , u01,4 , u2,3 , u03,3 , u4,4 )) = h3i1 ⊕ h4i2 ⊕ h4i3 ⊕ h3i5 , where (1 , . . . , 5 ) stands for the canonical basis of (Z/12Z)[X, Y ]5 . We conclude that I admits the free (Z/12Z)[X, Y ]-resolution ϕ4

ϕ3

ϕ2

ϕ1

ϕ0

· · · → (Z/12Z)[X, Y ]4 → (Z/12Z)[X, Y ]4 → (Z/12Z)[X, Y ]5 → (Z/12Z)[X, Y ]4 → I → 0 with LT(Ker(ϕ2i )) = h4iω1 ⊕h3iω2 ⊕h3iω3 ⊕h4iω4 and LT(Ker(ϕ2i+1 )) = h3iω1 ⊕h4iω2 ⊕h4iω3 ⊕h3iω4 for i ≥ 1, where (ω1 , . . . , ω4 ) stands for the canonical basis of (Z/12Z)[X, Y ]4 .

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