Unspecified Journal Volume 00, Number 0, Pages 000-000 S ????-????(XX)0000-0 ON THE HILBERT IDEAL MARA D. NEUSEL Abstract.We prove the Hilbert number conjecture. 1.Introduction Let F be a field, G a finite group. Consider a faithful representation ae : G ,! GL(n, F). It induces a G-action on the vector space V = Fn by matrix multiplication and t* *hus on the dual space V *. We extend this action additively and multiplicatively to* * the symmetric algebra on V *, denoted by F[V ]. Once we choose a basis x1, . .,.xn * *for V *we can identify F[V ] ~=F[x1, . .,.xn] as the ring of polynomial functions in n indeterminates over the field F. We are interested in the subalgebra F[V ]G ,! F[V ] of G-invariant polynomials. By a classical result due to E. Noether the ring of* * in- variants F[V ]G is finitely generated as an F-algebra. Furthermore, since our a* *ction is F-linear we can assume that the generators of F[V ]G are homogeneous polynomial* *s. We define fi(F[V ]G ) to be the maximal degree of an algebra generator in a min* *imal algebra generating set. This number is usually called the Noether number. If the group order, |G|, is invertible in F, the nonmodular case, then fi(F[V ]G ) |* *G|, independent of the representation, see Theorem 2.3.3 in [8]. In the modular cas* *e, i.e., |G| = 0 2 F, the value of fi(F[V ]G ) usually depends on the degree of th* *e rep- resentation or the order of the field and the group order, see [3] for an overv* *iew on degree bounds, see [4] and [5] for recent results. In this paper we consider not the ring of invariant polynomials but the Hilbe* *rt ideal ______ h(V, G) = (F[V ]G) F[V ] which is by definition generated by all homogeneous invariant polynomials of po* *si- tive degree. The Hilbert number, denoted by fi(h(()V, G)) is the maximal degree* * of an ideal generator in a minimal ideal generating set. In [1] it has been conjec* *tured that the Hilbert ideal is, in contrast to the ring of invariants, always genera* *ted by invariants of degree at most group order. Of course, in the nonmodular case this ____________ Received by the editors December 3, 2007. 2000 Mathematics Subject Classification. Primary 13A50 Commutative Rings. Key words and phrases. invariant theory of finite groups, degree bounds, Hil* *bert ideal, Hilbert number. Oc0000 (copyright holder) 1 2 MARA D. NEUSEL follows from Noether's bound. Furthermore, this conjecure was proven for permu- tation representations in [2] and for indecomposable representations of Z=p in * *[9]. In this paper we prove this conjecture for all representations. While typing t* *his manuscript I learned that the same conjecture has been proven simulteneously by Symonds, [10]. 2. The Proof Consider the induced FG-module FG F V and the canonical inclusion V ,! FG F V . This induces a G-equivariant projection jG : F[FG F V ] -! F[V ] of F-algebras. Upon restriction to the respective rings of invariants we obtain* * the classical Noether map jGG: F[FG F V ]G -! F[V ]G . This map remains surjective in the classical nonmodular case. However in the modular case it often fails to be surjective, see [6] and [7] for a thorough st* *udy of the Noether map. In the nonmodular case the ring of invariants and hence the Hilbert ideal is generated by polynomials of degree at most group order. Thus we restrict our attention to the modular case and obtain a commutative diagram as follows. jG F[FG F V ] i F[V ] [ [ F[V ]G [ jGG F[FG F V ]G i Im(jGG) Note that G acts by permutations on FG F V . Thus by Fleischmann's result, [2], we have fi(h(FG F V, G)) |G|. Therefore _______ (?) fi(Im (jGG)) |G|, _______ where Im (jGG) F[V ] denotes the ideal generated by the invariants of positive degree in the image of the Noether map. Furthermore, the kernel of jG is generated by linear forms. Moreover, Ker(jG ) F[FG V ] is a prime ideal of height n|G|-|G|. Thus we obtain the following chai* *ns of ideals jG F[FG F V ] i F[V ] [_______ ___[___ Ker(jG ) j-1G(Im (jGG)) Im (jGG) [ h(FG V, G) ON THE HILBERT IDEAL 3 Let I h(V, G) F[V ] be the ideal generated by all G-invariant polynomials of positive degree at most group order. We want to show that I = h(V, G). By Inequality (?) we have _______ Im (jGG) I h(V, G) F[V ]. Thus we can extend the above diagram to the following jG F[FG F V ] i F[V ] [ [ j-1G(h(V, G)) h(V, G) [ [ j-1G(I) I [_______ ___[___ Ker(jG ) j-1G(Im (jGG)) Im (jGG) [ h(FG V, G) We collect some properties of the ideals involved in the following lemma. Lemma 2.1. The ideals I h(V, G) F[V ] as well as the ideals j-1G(I) j-1G(h(V, G)) F[FG V ] are closed under the action of the group G. Proof.The first two ideals are by definition generated by invariant polynomials. Thus they are closed under the G-action. Since the map jG is G-equivariant, the latter two are also closed under the G-action. Next we present a series a reduction arguments. The first one shows that it is enough to prove that the inverse images of I and h(V, G) in F[FG V ] are equa* *l. Lemma 2.2. If j-1G(h(V, G)) = j-1G(I) then h(V, G) = I. Proof.By construction we have I h(V, G). To prove the reverse inclusion let f 2 h(V, G). Then there exists an element F 2 j-1G(h(V, G)) = j-1G(I) with f = jG (F ) 2 jG (j-1G(h(V, G))) = jG (j-1G(I)) = I as desired. The third lemma shows that it is enough to consider elements that map onto G-invariant polynomials under the Noether map. Lemma 2.3. If F 2 j-1G(I) for all F 2 F[FG V ] with jG (F ) = f 2 F[V ]G , th* *en j-1G(I) = j-1G(h(V, G)). Proof.The inclusion is true by construction. To prove the reverse inclusion l* *et H 2 j-1G(h(V, G)) be an arbitrary element. Then X jG (H) = ffifi2 h(V, G), where fi 2 F[V ]G have positive degree and ffi 2 F[V ]. By assumption for each i there exists an Fi2 j-1G(I) such that jG (Fi) = fi. Thus X H = AiFi+ K 2 j-1G(I), for some jG (Ai) = ffi and K 2 Ker(jG ). 4 MARA D. NEUSEL We write (x1g1 . . .x1g|G|) x2g1 . . .x2g|G| FG V = spanF .. . . .. xng1 . . .xng|G| for some enumeration of the group elements G = {g1, . .,.g|G|}. Withoutloss of generality we assume that g1 = 1 2 G is the identity element. Then the group G acts on the first index, and the Noether map is given by jG (xigj) = gjxi2 F[V ]. We choose the reverse lexicographic orderPx1g1< . .<.x1g|G|< x2g1< . .<.xng|G|. We note that for any polynomial f = aixi11. .x.inn2 F[V ], where ai2 F, the* *re exists an inverse image X F = j-1G(f) = aixi111.x.i.nn12 F[FG V ] whose terms are monomials in the elements of the first column of (xigj)igj. Lemma 2.4. Let f 2 h(V, G), and assume that the inverse image F of f consisting of terms in the first column is an element in j-1G(I). Then all inverse images * *of f lie in j-1G(I). Proof.Since two elements F and F 0of j-1G(f) differ by an element in the kernel* * of jG which in turn is contained in j-1G(I), the result follows. We want to prove our statement degree-wise by induction on the term order. The next result constitutes the induction start for every degree. Proposition 2.5. Let F 2 j-1G(h(V, G)) have degree d. Let F be minimal with re- spect to the reverse lexicographic order. Assume that the ideals j-1G(I) and j-* *1G(h(V, G)) coincide in degree less than d. Then F 2 j-1G(I). Proof.The unique minimal monomial in F[FG V ] of degree d is xd11. If G is a p-group we assume without loss of generality that G fixes x1, then x1 2 I and t* *hus x112 j-1G(I). Now, let G be an arbitrary group with p-Sylow subgroup P fixing x1. Order the elements of G such that g2 2 P is a nontrivial element. Then jG (x1g2) = g2x1 =* * x1. We have to consider two cases: CASE: xd1162 j-1G(h(V, G)) In this case we have F = xd-111(x11- x1g2) 2 Ker(jG ) j-1G(h(V, G)). Its leading term is xd-111x1g2and is thus the smallest leading term of an eleme* *nt of degree d in j-1G(h(V, G)). Furthermore, j-1G(I) contains the kernel of jG a* *nd hence it contains xd-111(x11- x1g2). Finally, note that F is the unique polynom* *ial in j-1G(h(V, G)) with this leading term. CASE: xd112 j-1G(h(V, G)) Then jG (xd11) = xd12 h(V, G). Thus if d |G| it follows that xd12 I by defi* *nition of I, and hence x112 j-1G(I). Otherwise we have X xd1= ffifi i ON THE HILBERT IDEAL 5 for some G-invariant polynomials fi. If the degree of the ffi's is positive, we* * have fi2 I by assumption, and thus X xd11= AiFi+ K 2 j-1G(I) i for jG (Fi) = fi, jG (Ai) = ffi, and an element K in the kernel. Finally assume* * we have X xd1= f0 + ffifi. i Then the relative transfer gives X TrGP(xd1) = |G : P |f0 + TrGP(ffi)fi. i Since d > |G| we can apply Lemma 2.2 in [2] and obtain TrGP(xd1) 2 I. Hence X f0 = __1___|G(:TPr|GP(xd1) - TrGP(ffi)fi) 2 I. i So, finally we get X xd112 j-1G(xd1) = j-1G(f0 + ffifi) j-1G(I) i as desired. We need one more preparation: Lemma 2.6. Let M, M0 2 F[FG V ] be monomials of the same degree. Assume that M consists of elements in the first column, and assume that M > M0 in the reverse lexicographic order. Then for all g 2 G we have that gM > gM0. Proof.M is a monomial in the first column, say M = xi11,1.x.i.nn,1. Thus gM = xi11,g.x.i.nn,g is a monomial in the gth column. We turn to the monomial M0. Since M0 is smaller it must look like M0 = xinn,1.x.i.j+1j+1,1xkjj,1N where N is a monomial avoiding the columns j through n and kj < ij. Thus gM0 looks like gM0 = xinn,g.x.i.j+1j+1,gxkjj,ggN where gN still avoids columns j through n. Thus gM0 < gM. The following result proves the Hilbert ideal conjecture for p-groups in char* *ac- teristic p. Theorem 2.7. Let P be a p-group, and assume that ae(P ) GL(n, F) is in lower triangular form, i.e., gxi2 spanF{x1, . .,.xn}. Then h(V, P ) = I. 6 MARA D. NEUSEL Proof.By Lemma 2.2 it is enough to show that j-1G(h(()V, G)) = j-1G(I). The inclusion " " is valid by construction. To show the reverse inclusion we proc* *eed by double induction on degree and term order. Let F 2 j-1P(h(V, P )) be an element of degree d. If d |G| there is nothing* * to show. Thus we assume that the degree of F is strictly larger than the group ord* *er. By Lemma 2.4 we can assume that F lives in the first column and by Lemma 2.3 we can assume that jP (F ) is invariant. Since the induction start is proven in Proposition 2.5 we assume that for all elements of smaller degree or same degree and lower in term order we have shown that it is in j-1G(I). We write F as a sum of monomials F = M0 + M1 + . .+.Mk and without loss of generality we assume that M0 > M1 > . .>.Mk. Let r = max{1, . .,.n} such that the variable xr1 appears in F . CASE: M0 = xdr1 We consider the top orbit Chern class of xr ctop(xr) 2 F[V ]P which is a polynomial of degree, say, t with leading term xdr. Note that t cann* *ot exceed the group order. We find an inverse image C 2 j-1P(ctop(xr)) in the first column C 2 F[x11, x21, . .,.xr1]. We note that this polynomial is by construction in j-1P(I). Then F - Cd-t lies in j-1P(h(V, P )) and has lower leading term than F . Thus by induction F - Cd-t 2 j-1P(I) and thus F 2 j-1P(I) as desired. CASE: Assume that M0 = xir1N0, where N0 has positive degree and xr1 does not divide N0, M1, . .,.Mk. Since jP (F ) is invariant we have that gF - F 2 Ker(jP ) j-1P(I) for all g* * 2 P . Since xr1appears solely in M0, we have that xr|jP (M0) = m0 but it does not div* *ide any of the other jP (Mi) for i = 1, . .,.k. Since jP (gF - F ) = gjP (F ) - jP * *(F ) = 0 the terms with the same xr-degree must cancel. Thus jP (gM0 - M0) = g(xr)ign0 - xirn0 = xir(gn0 - n0) + R, where the remainder R has lower xr-degree and jG (N0) = n0. Thus gn0 - n0 = 0 and n0 is an invariant of strictly smaller degree than m0. Thus N0 2 j-1P(I), therefore M0 2 j-1P(I). Hence F - M0 2 j-1P(h(V, P )) has strictly smaller term-order, and is by induction in j-1P(I). Thus F 2 j-1P(* *I). CASE: xr1 divides M0, . .,.Mj but none of the other terms of F Write M0 = xi1r1N0, . .,.Mj = xijr1Nj If i1 > max{i2, . .,.ij}, then the preceding argument goes through without chan* *ge. Thus we assume that i1 = i2 = . .=.il for some 1 < l j. Since M0 > M1 > ON THE HILBERT IDEAL 7 . .>.Ml there exists a largest index s, 1 s r - 1, such that the monomials differ. Hence we can write M0 = xirr1xir-1r-1,1.x.i.s+1,1s+1,1xis,1s1N0, . .,.Ml= xirr1xir-1r-1,1.x.i.* *s+1,1s+1,1xjs,1s1Nl. Similar to the above we obtain that xis,1s1N0 + . .+.xjs,1s1Nl maps under jP to an invariant. By construction it has lower degree than F and is thus in j-1P(I). Here we can apply the argument of the preceding case to find t* *hat N0 2 j-1P(I). Finally, we are prepared to prove the Hilbert number conjecture for all groups and representations. Theorem 2.8. Let ae : G ,! GL(n, F) be a faithful representation of a finite gr* *oup over an arbitrary field F. Then h(V, G) = I and hence the Hilbert number is bounded above by the group order fi(h(V, G)) = fi(I) |G|. Proof.By Lemma 2.2 it is enough to show that j-1G(h(()V, G)) = j-1G(I). The inclusion " " is valid by construction. To show the reverse inclusion we proc* *eed as above by double induction on degree and term order. Let F 2 j-1G(h(V, G)).If degree if F is at most group order then F 2 j-1G(I) * *by construction. Thus we assume that deg(F ) = d > |G|. The polynomial of minimal leading term of degree d in j-1G(h(V, G)) is contai* *ned in j-1G(I) by Proposition 2.5. Thus assume that the leading term of F is not minimal, and all polynomials of degree less than d or of degree d with smaller leading term than F are in j-1G(* *I). We denote the leading term of F by LT(F ). We note that by Lemmata 2.3 and 2.4 we may assume that F is a polynomial in the first column mapping under jG onto an invariant polynomial. We note also that the leading term of gF is strictly larger than F for all g * *6= 1, because F lives in the first column, while gF lives in the gth column. Since F maps to an invariant polynomial under jG we have that gF - F 2 Ker(jG ) j-1G(I). Thus LT (gF - F ) = LT(gF ) 2 LT(Ker(jG )) LT(j-1G(I)), where LT(I) denotes the ideal of leading terms of the ideal I. Hence there exists an element H 2 j-1G(I) such that LT(H) = LT(gF ). If H = gF we are done by Lemma 2.3. Otherwise, consider gF - H 2 j-1G(h(V, G)) with LT(gF - H) < LT(gF ). If LT(gF - H) < LT(F ), then gF - H 2 j-1G(I) by induction on the term order, and thus gF 2 j-1G(I). Hence F 2 j-1G(I). Otherwise, we consider the polynomial (gF - F ) - H which is an element in j-1G(I). If LT(gF -F -H) > LT(F ), we pick an element K1 2 j-1G(I) such that LT(K1) = LT(gF - F - H). Then LT(gF - F - H - K1) < LT(gF - F - H) 8 MARA D. NEUSEL and we can proceed inductively and find K1, . .,.Kl2 j-1G(I) such that LT(gF - F - H - K1 - . .-.Kl) LT(F ). Thus we may assume without loss of generality that LT(gF - F - H) LT(F ). If there exists an element H such that LT(gF - F - H) = LT(F ), we have LT(gF - F - H - F ) < LT(gF - F - H) = LT(F ). By construction gF - F - H - F 2 j-1G(h(V, G)) and thus by induction on the term order in j-1G(I). Since H as well as gF - F are in j-1G(I) we conclude that F 2 j-1G(I). Finally we have to take care of the case LT(gF - F - H) < LT (F ) for all H 2 j-1G(I) and all g 2 G. Recall that F is a polynomial in the first column and that jG (F ) = f is inv* *ariant. We write it as a sum of monomials F = M0 + M1 + . .+.Mk and assume without loss of generality that M0 > M1 > . .>.Mk in the term-order. Then gF = gM0 + gM1 + . .+.gMk has leading term gM0 by Lemma 2.6. Since the leading term of gF - F - H is strictly smaller than the leading term of F we have X H = gM0 - M0 + Ki2 j-1G(I), Ki