PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volue 126, Nuber 3, March 1998, Pages 687 691 S 0002-9939(98)04229-4 NON-COMMUTATIVE GRÖBNER BASES FOR COMMUTATIVE ALGEBRAS DAVID EISENBUD, IRENA PEEVA, AND BERND STURMFELS (Counicated by Woler V. Vasconcelos) Abstract. An ideal I in the free associative algebra k X 1,...,X n over a field k is shown to have a finite Gröbner basis if the algebra defined by I is coutative; in characteristic 0 and generic coordinates the Gröbner basis ay even be constructed by lifting a coutative Gröbner basis and adding coutators. 1. Introduction Let k be a field and let k[x] =k[x 1,...,x n ] be the polynoial ring in n variables and k X = k X 1,...,X n the free associative algebra in n variables. Consider the natural ap γ : k X k[x] taking X i to x i. It is soeties useful to regard a coutative algebra k[x]/i through its non-coutative presentation k[x]/i = k X /J, wherej=γ 1 (I). This is especially true in the construction of free resolutions as in [An]. Non-coutative presentations have been exploited in [AR] and [PRS] to study hoology of coordinate rings of Grassannians and toric varieties. These applications all ake use of Gröbner bases for J (see [Mo] for non-coutative Gröbner bases). In this note we give an explicit description (Theore 2.1) of the inial Gröbner bases for J with respect to onoial orders on k X that are lexicographic extensions of onoial orders on k[x]. Non-coutative Gröbner bases are usually infinite; for exaple, if n = 3and I=(x 1 x 2 x 3 ) then γ 1 (I) does not have a finite Gröbner basis for any onoial order on k X. (There are only two ways of choosing leading ters for the three coutators, and both cases are easy to analyze by hand.) However, after a linear change of variables the ideal becoes I = (X 1 (X 1 + X 2 )(X 1 + X 3 )), and we shall see in Theore 2.1 that X 1 (X 1 + X 2 )(X 1 + X 3 ) and the three coutators X i X j X j X i are a Gröbner basis for γ 1 (I ) with respect to a suitable order. This situation is rather general: Theores 2.1 and 3.1 iply the following result: Corollary 1.1. Let k be an infinite field and I k[x] be an ideal. After a general linear change of variables, the ideal γ 1 (I) in k X has a finite Gröbner basis. In characteristic 0, if I is hoogeneous, such a basis can be found with degree at ost ax{2, regularity(i)}. Received by the editors Septeber 6, 1996. 1991 Matheatics Subject Classification. Priary 13P10, 16S15. The first and third authors are grateful to the NSF and the second and third authors are grateful to the David and Lucille Packard Foundation for partial support in preparing this paper. 687 c 1998 Aerican Matheatical Society
688 DAVID EISENBUD, IRENA PEEVA, AND BERND STURMFELS In characteristic 0 the Gröbner basis of γ 1 (I) in Corollary 1.1 ay be obtained by lifting the Gröbner basis of I, but this is not so in characteristic p; seeexaple 4.2. Furtherore, γ 1 (I) ight have no finite Gröbner basis at all if the field is finite; see Exaple 4.1. The behavior of γ 1 (I) is in sharp contrast to what happens for arbitrary ideals in k X. For exaple, the defining ideal in k X of the group algebra of a group with undecidable word proble has no finite Gröbner basis. Another exaple is Shearer s algebra k a, b /(ac ca, aba bc, b 2 a), which has irrational Hilbert series [Sh]. As any finitely generated onoial ideal defines an algebra with rational Hilbert series, the ideal (ac ca, aba bc, b 2 a) can have no finite Gröbner basis. (Other consequences of having a finite Gröbner basis are deducible fro [An] and [Ba]; these are well-known in the case of coutative algebras!) In the next section we present the basic coputation of the initial ideal and Gröbner basis for J = γ 1 (I). In 3 we give the application to finiteness and liftability of Gröbner bases. 2. The Gröbner basis of γ 1 (I) Throughout this paper we fix an ideal I k[x] andj:= γ 1 (I) k X. We shall ake use of the lexicographic splitting of γ, which is defined as the k-linear ap δ : k[x] k X, x i2 X i1 X i2 X ir if i 1 i 2 i r. Fix a onoial order on k[x]. The lexicographic extension of to k X is defined for onoials M,N k X by { γ(m) γ(n) or M N if γ(m)=γ(n) and M is lexicographically saller than N. Thus, for exaple, X i X j X j X i if i<j. To describe the -initial ideal of J we use the following construction: Let L be any onoial ideal in k[x]. If = L and i 1 i r,denoteby U L () the set of all onoials u k[+1,..., 1] such that neither u nor lies in L. For instance, if L =(x 1 x 2 x 3,x d 2)thenU L (x 1 x 2 x 3 )={x j 2 j<d}. u Theore 2.1. The non-coutative initial ideal in (J) is inially generated by the set { X i X j j<i}together with the set {δ(u ) is a generator of in (I) and u U in (I)()}. In particular, a inial -Gröbner basis for J consists of {X i X j X j X i : j<i} together with the eleents δ(u f) for each polynoial f in a inial -Gröbner basis for I and each onoial u U in (I)(in (f)). Proof. We first argue that a non-coutative onoial M = X i1 X i2 X ir lies in in (J) if and only if its coutative iage γ(m) isin in (I) or i j >i j+1 for soe j. Indeed, if i j >i j+1 then M in (J) because X s X t X t X s J has initial ter X s X t with s>t. If on the contrary i 1 i r but γ(m) in (I), then there exists f I with in (f) =γ(m). The non-coutative polynoial F = δ(f) satisfies in (F ) = M. The opposite iplication follows because γ induces an isoorphis k[x]/i = k X /γ 1 (I).
NON-COMMUTATIVE GRÖBNER BASES FOR COMMUTATIVE ALGEBRAS 689 Now let = u, where = is a inial generator of in (I) with i 1 i r. We ust show that δ(u ) is a inial generator of in (J) if and only if u U in (I)(). For the only if direction, suppose that δ(u ) is a inial generator of in (J). Suppose that u contains the variable x j.weusthavej>i 1, since else, taking j inial, we would have δ(u ) =X j δ( u x j ). Siilarly j<i r.thus u k[+1,..., 1]. This iplies δ(u ) =X i1 δ(u ) = δ(u ) X ir. Therefore neither δ(u )norδ(u ) lies in in (J), and hence neither u nor u lies in in (I). For the if direction we reverse the last few iplications. If u U in (I)() then neither δ(u )norδ(u ) lies in in (J), and therefore δ(u ) is a inial generator of in (J). 3. Finiteness and lifting of non-coutative Gröbner bases We aintain the notation described above. Recall that for a prie nuber p the Gauss order on the natural nubers is described by ( ) t s p t if 0(odp). s We write 0 = for the usual order on the natural nubers. A onoial ideal L is called p-borel-fixed if it satisfies the following condition: For each onoial generator of L, ifis divisible by x t j but no higher power of x j,then(x i /x j ) s L for all i<jand s p t. Theore 3.1. With notation as in Section 2: (a) If in (I) is 0-Borel fixed, then a inial -Gröbner basis of J is obtained by applying δ to a inial -Gröbner basis of I and adding coutators. (b) If in (I) is p-borel-fixed for any p, thenjhas a finite -Gröbner basis. Proof. Suppose that the onoial ideal L := in (I) isp-borel-fixed for soe p. Let = be any generator of L, wherei 1 i r, and let x t i r be the highest power of dividing. Sincet p twe have x t l /xt i r L for each l<i r. This iplies x t l /x i r L for l<i r, and hence every onoial u U L () satisfies deg xl (u) <tfor i 1 <l<i r. We conclude that U L () is a finite set. If p =0then U L () consists of 1 alone, since x l / L for all l<i r. Theore 3.1 now follows fro Theore 2.1. Proof of Corollary 1.1. We apply Theore 3.1 together with the following results, due to Galligo, Bayer-Stillan and Pardue, which can be found in [Ei, Section 15.9]: if the field k is infinite, then after a generic change of variables, the initial ideal of I with respect to any order on k[x] is fixed under the Borel group of upper triangular atrices. This iplies that in (I) isp-borel-fixed in characteristic p 0 in the sense above. If the characteristic of k is 0 and I is hoogeneous then, taking the reverse lexicographic order in generic coordinates, we get a Gröbner basis whose axial degree equals the regularity of I. We call the onoial ideal L squeezed if U L () = {1} for all generators of L or if, equivalently, = L and i 1 i r iply x l L or x l L for every index l with i 1 <l<i r. Thus Theore 2.1 iplies that a inial -Gröbner basis of I lifts to a Gröbner basis of J if and only if the
690 DAVID EISENBUD, IRENA PEEVA, AND BERND STURMFELS initial ideal in (I) is squeezed. Monoial ideals that are 0-Borel-fixed, and ore generally stable ideals (in the sense of [EK]), are squeezed. Squeezed ideals appear naturally in algebraic cobinatorics: Proposition 3.2. A square-free onoial ideal L is squeezed if and only if the siplicial coplex associated with L is the coplex of chains in a poset. Proof. This follows fro Lea 3.1 in [PRS]. 4. Exaples in characteristic p Over a finite field Corollary 1.1 fails even for very siple ideals: Exaple 4.1. Let k be a finite field and n =3. IfIis the principal ideal generated by the product of all linear fors in k[x 1,x 2,x 3 ], then γ 1 (I) has no finite Gröbner basis, even after a linear change of variables. Proof. The ideal I is invariant under all linear changes of variables. The -Gröbner basis for J is coputed by Theore 2.1, and is infinite. That no other onoial order on k X yields a finite Gröbner basis can be shown by direct coputation as in the exaple in the second paragraph of the introduction. Soeties in characteristic p>0nogröbner basis for a coutative algebra can be lifted to a non-coutative Gröbner basis, even after a change of variables: Exaple 4.2. Let k be an infinite field of characteristic p>0, and consider the Frobenius power L := ( (x 1,x 2, x 3 ) 3) [p] k[x 1,x 2,x 3 ] of the cube of the axial ideal in 3 variables. No inial Gröbner basis of L lifts to a Gröbner basis of γ 1 (L), and this is true even after any linear change of variables. Proof. The ideal L is invariant under linear changes of variable, so it suffices to consider L itself. Since L is a onoial ideal, it is its own initial ideal, so by Corollary 3.2 it suffices to show that L is not squeezed, that is, that neither x p 1 1 x p+1 2 x p 3 nor x p 1 xp+1 2 x p 1 3 is in L. This is obvious, since the power of each variable occurring in a generator of L is divisible by p and has total degree 3p. References [An] D. Anick, On the hoology of associative algebras, Transactions Aer. Math. Soc. 296 (1986), 641-659. MR 87i:16046 [AR] D. Anick, G.-C. Rota, Higher-order syzygies for the bracket algebra and for the ring of coordinates of the Grassannian, Proc. Nat. Acad. Sci. U.S.A. 88 (1991), 8087 8090. MR 92k:15058 [Ba] J. Backelin, On the rates of growth of the hoologies of Veronese subrings., Algebra, algebraic topology and their interactions (Stockhol, 1983), Lecture Notes in Math. 1183, Springer-Verlag, NY, 1986, p. 79 100. MR 87k:13042 [Ei] D. Eisenbud, Coutative Algebra With a View Toward Algebraic Geoetry, Springer- Verlag, NY, 1995. MR 97a:13001 [EK] S. Eliahou, M. Kervaire, Minial resolutions of soe onoial ideals, Journal of Algebra 129 (1990) 1 25. MR 91b:13019 [Mo] T. Mora, An introduction to coutative and non-coutative Gröbner bases, Theoretical Coputer Science 134 (1994), 131 173. MR 95i:13027
NON-COMMUTATIVE GRÖBNER BASES FOR COMMUTATIVE ALGEBRAS 691 [PRS] I. Peeva, V. Reiner and B. Sturfels, How to shell a onoid, preprint, 1996. [Sh] J. B. Shearer, A graded algebra with a nonrational Hilbert series, J. Alg. 62 (1980), 228 231. MR 81b:16002 MSRI, 1000 Centennial Dr., Berkeley, California 94720 E-ail address: de@sri.org Departent of Matheatics, Massachusetts Institute of Technology, Cabridge, Massachusetts 02139 E-ail address: irena@ath.it.edu Departent of Matheatics, University of California, Berkeley, California 94720 E-ail address: bernd@ath.berkeley.edu