Locally finite derivations and modular coinvariants

Locally finite derivations and modular coinvariants Abstract We consider a finite-dimensional kG-module V of a p-group G over a field k of characteristic p. We describe a generating set for the corresponding Hilbert Ideal. In case G is cyclic, this yields that the algebra k[V]G of coinvariants is a free module over its subalgebra generated by kG-module generators of V*. This subalgebra is a quotient of a polynomial ring by pure powers of its variables. The coinvariant ring was known to have this property only when G was cyclic of prime order [M. Sezer, Decomposing modular coinvariants, J. Algebra423 (2015), 87–92]. In addition, we show that if G is the Klein 4-group and V does not contain an indecomposable summand isomorphic to the regular module, then the Hilbert Ideal is a complete intersection, extending a result of the second author and Shank [M. Sezer and R. J. Shank, Rings of invariants for modular representations of the Klein four group, Trans. Amer. Math. Soc. 368 (2016), 5655–5673]. 1. Introduction Let k be a field of positive characteristic p and V a finite-dimensional k-vector space, and G≤GL(V) a finite group. Then the induced action on V* extends to the symmetric algebra k[V]≔S(V*) by the formula σ(f)=f◦σ−1 for σ∈G and f∈k[V]. The ring of fixed points k[V]G is called the ring of invariants, and is the central object of study in invariant theory. Another object which is often studied is the Hilbert Ideal, H, which is defined to be the ideal of k[V] generated by invariants of positive degree, in other words H=k[V]+Gk[V]. In this article, we study the quotient k[V]G≔k[V]/H which is called the algebra of coinvariants. An equivalent definition is k[V]G≔k[V]⊗k[V]Gk, which shows that this object is, in a sense, dual to k[V]G. As k[V]G is a finite-dimensional kG-module, it is generally easier to handle than the ring of invariants. On the other hand, much information about k[V]G is encoded in k[V]G. For example, Steinberg [13] famously showed that dim(C[V]G)=∣G∣ if and only if (G,V) is a complex reflection group. Combined with the theorem of Chevalley [2], Shephard and Todd [11], this shows that dim(C[V]G)=∣G∣ if and only if C[V]G is a polynomial ring. Smith [12] later generalized this by showing that dim(k[V]G)=∣G∣ if and only if G is a (pseudo)-reflection group, where k is any field. Further, the polynomial property of C[V]G is equivalent to the Poincaré duality property of C[V]G, by Kane [6] and Steinberg [13]. Before we continue, we fix some terminology. Let x0,…,xn be a basis for V*. We will say xi is a terminal variable if the vector space spanned by the other variables is a kG-submodule of V*. Note that if G is a p-group, then VG≠0 and there is a choice of a basis for V that contains a fixed point. Then the dual element corresponding to the fixed point is a terminal variable in the basis consisting of dual elements of this basis. For any f∈k[V], we define the norm NG(f)=∏h∈G·fh. For every terminal variable xi, we choose a polynomial N(xi) in k[V]G which, when viewed as a polynomial in xi is monic of minimal positive degree. While N(xi) is not unique in general, its degree is well defined. Since NG(xi) is monic of degree [G:Gxi], the degree of N(xi) is bounded above by this number. By ‘degree of xi’, we understand degree of N(xi) as a polynomial in xi and denote it by deg(xi). We will show that the degree of a terminal variable is always a p-power. The algebras of modular coinvariants for cyclic groups of order p were studied by the second author [8], and previously by the second author and Shank [9]. Note that there is a choice of basis such that an indecomposable representation of a p-group is afforded by an upper triangular matrix with 1s on the diagonal and the bottom variable is a terminal variable. In [8], the following was proven. Proposition 1.1. Let Gbe a cyclic group of order pand Va kG-module that contains k+1non-trivial summands. Choose a basis x0,x1,…,xnin which the variables x0,x1,…,xkare the bottom variables of the respective Jordan blocks, and let Abe the kG-subalgebra of k[V]generated by xk+1,…,xn. Denote the image of xi in k[V]G by Xi. Then The Hilbert Ideal of k[V]Gis generated by NG(x0),NG(x1),…,NG(xk), and polynomials in A. k[V]Ghas dimension divisible by pk+1. k[V]Gis free as a module over its subalgebra Tgenerated by X0,X1,…,Xk. T≅k[t0,…,tk]/(t0p,…,tkp), where t0,…,tkare independent variables. The goal of this article is to generalize the above, as far as possible, to the case of all finite p-groups. In particular, we show in section two: Theorem 1.2. Let Gbe a finite p-group and Va kG-module that contains k+1non-trivial summands. Choose a basis x0,x1,…,xnin which the variables x0,x1,…,xkcoming from each summand are terminal variables. Let didenote deg(xi)for 0≤i≤k. Retain the notation in the proposition above, then There is a choice for polynomials N(x0),N(x1),…,N(xk)such that the Hilbert Ideal of k[V]Gis generated by N(x0),N(x1),…,N(xk), and polynomials in A. k[V]Ghas dimension divisible by ∏i=0kdi.Suppose in addition that one has di=deg(NG(xi))for 0≤i≤k. Then we have: k[V]Gis free as a module over its subalgebra Tgenerated by X0,X1,…,Xk. T≅k[t0,…,tk]/(t0d0,…,tkdk), where t0,…,tkare independent variables. In Section 3, we describe the situation for a p-group, where the complete intersection property of the Hilbert Ideal corresponding to a module is inherited from the Hilbert Ideal of the indecomposable summands of the module. The final section is devoted to applications of our main results to cyclic p-groups and the Klein 4-group. It turns out that for a cyclic p-group, the bottom variables xi of Jordan blocks satisfy deg(xi)=deg(NG(xi)). Consequently, (3) and (4) above hold for a cyclic p-group. Additionally, for the Klein 4-group, we show that the Hilbert Ideal corresponding to a module is a complete intersection as long as the module does not contain the regular module as a summand. This generalizes a result of the second author and Shank [10], where the complete intersection property was established for indecomposable modules only. This article was composed during a visit of the second author to the University of Aberdeen, funded by the Edinburgh Mathematical Society’s Research Support Fund. We would like to thank the society for their support. 2. Main results Throughout this section, we let G be a finite p-group, k a field of characteristic p and V a kG-module, which may be decomposable. As trivial summands do not contribute to the coinvariants, we assume no direct summand of V is trivial. Let x0,x1,x2,…,xn be a basis of V* and assume that x0 is a terminal variable. Then x1,x2,…,xn generate a G-subalgebra which we denote by A. We can define a nonlinear action of (k,+) on k[V] as follows: t·x0=x0+t; (2.1) t·xi=xiforanyi>0. (2.2) The terminality of x0 ensures this commutes with the action of G. It is well known that any action of the additive group of an infinite field of prime characteristic is determined by a locally finite iterative higher derivation. This is a family of k-linear maps Δi:k[V]→k[V], i≥0 satisfying the following properties: Δ0=idk[V]. For all i>0 and a,b∈k[V], one has Δi(ab)=∑j+k=iΔj(a)Δk(b). For all b∈k[V], there exists i≥0 such that Δi(b)=0. For all i,j, one has Δj◦Δi=(i+jj)Δi+j. The equivalence of the group action and the l.f.i.h.d. is given by the formula t·b=∑i≥0tiΔi(b). (2.3) See [3, 14], for more details on l.f.i.h.d.’s. Let f∈k[V]G be homogeneous of degree d in x0. We write f=fdx0d+fd−1x0d−1+⋯+f0, where fi∈A. We have t·f=fd(x0+t)d+fd−1(x0+t)d−1+⋯+f0=∑i≥0tiΔi(f). (2.4) That is to say that Δi(f) is the coefficient of ti in the above expression. As the action of G commutes with the action of k, we see that Δi(f)∈k[V]G for all i≥0. Remark 2.1. Clearly Δ1=∂∂x0. So the previous paragraph generalizes [8, Lemma 1]. Equation (2.4) gives that Δj(x0i)=(ij)x0i−j provided i≥j. Then, from Lucas’s theorem [5] on binomial coefficients in characteristic p, we see that we can think of Δpj as ‘Differentiation by x0pj’: if the coefficient of pj in the base p expansion of m is a, then we have Δpj(x0m)={ax0m−pja>0;0a=0. For later use, we also note the following consequence: for a homogeneous f∈k[V], Δj(f) contains a non-zero constant if and only if the monomial x0j appears in f. In [4], a G-equivariant map is constructed from polynomials whose x0-degree is at most epr (0<e<p) to polynomials whose x0-degree is at most pr. This map turns out to be a non-zero scalar multiple of Δ(e−1)pr. We have the following statement generalizing [8, Lemma 2]: Lemma 2.2. Let f∈k[V]be a homogeneous polynomial of degree din x0. Write f=fdx0d+fd−1x0d−1+⋯+f0, where fi∈A. Then we have ∑i=0d(−1)ix0iΔi(f)=f0. Proof Write f=f(x0,x1,x2,…,xn). For any t∈k, we have t·f=f(t·x0,t·x1,…,t·xn)=f(x0+t,x1,x2,…,xn). As this holds for all t it also holds when t is replaced by (−x0), and hence by Equation (2.3), we have ∑i=0d(−1)ix0iΔi(f)=(−x0)·f=f(0,x1,x2,…,xn)=f0 as required.□ We also note that the degree of a terminal variable is a p-power. Lemma 2.3. For any terminal variable x0∈V*, deg(x0)is a power of p. Proof Let d denote the degree of x0 and suppose f∈k[V]G is monic as a polynomial in x0 of degree d=drpr+dr−1pr−1+⋯+d0 with 0≤di<p and dr≠0. If dj≠0 for some j<r, then Δpj(f)∈k[V]G has degree d−pj>0 as a polynomial in x0 and its leading coefficient is in k. Similarly, if dj=0 for j<r and dr>1, then Δpr(f)∈k[V]G has degree d−pr>0 in x0 and its leading coefficient is in k. Both cases violate the minimality of d.□ Lemma 2.4. Let ddenote the degree of x0. Then Δj(H)⊆Hfor j<d. Proof Let f∈k[V]. From the second assertion of Remark 2.1, we get that Δj(f) contains a non-zero constant if and only if the monomial x0j appears in f. Therefore, by the minimality of d, we have Δj(k[V]+G)⊆H for j<d. Now the result follows from property (2) of l.f.i.h.d.’s.□ From this point on, we adopt the notation of the introduction. This means that x0,x1,x2,…,xk are terminal variables coming from different summands, and A=k[xk+1,xk+2,…,xn]. For each i=0,…,k let di=pri be the degree of xi. Since setting variables outside of a summand to zero sends invariants to invariants of the summand, we may also assume that N(xi) depends only on variables that come from the summand that contains xi. We denote by Δi the l.f.i.h.d. associated to xi. We use reverse lexicographic order with xi>xj whenever 0≤i≤k and k+1≤j≤n. Theorem 2.5. His generated by N(x0),…,N(xk)and polynomials in A. Moreover, the lead term ideal of His generated by x0pr0,x1pr1,…,xkprkand monomials in A. Proof Let f∈k[V]G. Since N(x0) is monic in x0, we may perform polynomial division and write f=qN(x0)+r where r has x0-degree <pr0, and it is easily shown that q,r∈k[V]G. Then dividing r by N(x1) yields another invariant remainder r′ that has x1-degree <pr1. Since x0-degree of N(x1) is zero, it follows that x0-degree of r′ is still <pr0. Thus, by repeating the process with each terminal variable, and replacing f with the final remainder we assume that xi-degree of f is <pri for 0≤i≤k. Let i be minimal such that f has non-zero degree d<pri in the terminal variable xi. We apply Lemma 2.2 with Δ=Δi to see that f=f0−(∑j=1d(−1)jxijΔij(f)), where f0 is the ‘constant term’ of f, that is, f0∈k[xi+1,…,xn]. So from the previous lemma, we get that f0∈H since d<pri. Moreover, since Δi decreases xi-degrees and does not increase degrees in any other variable, the xi-degree of each Δij(f) in the expression above is strictly less than d, and the xl-degree for every i<l≤k remains strictly less than prl. Thus, by induction on degree, f can be expressed as a k[V]-combination of elements of H whose degrees in the terminal variables x0,…,xi are all zero, and degrees in the remaining terminal variables xl for i<l≤k are strictly less than prl, respectively. Repeating the same argument with the remaining terminal variables gives us that f can be written as a k[V]-combination of elements of H∩A together with N(x1),…,N(xk) as required. The first assertion of the theorem follows. Note that the leading monomial of N(xi) is xipri for 0≤i≤k. So it remains to show that all other monomials in the lead term ideal of H lie in A. Recall that by Buchberger’s algorithm a Gröbner basis is obtained by reduction of S-polynomials of a generating set by polynomial division, see [1, Section 1.7]. By the first part, H has a generating set consisting of N(xi) for 0≤i≤k and polynomials in A. But the S-polynomial of two polynomials in A is also in A, and via polynomials in A, it also reduces to a polynomial in A. Finally, the S-polynomial of N(xi) and a polynomial in A and the S-polynomial of a pair N(xi) and N(xj) with 0≤i≠j≤k reduce to zero since their leading monomials are pairwise relatively prime.□ Corollary 2.6. The vector space dimension of k[V]Gis divisible by ∏0≤i≤kdi=p∑i=0kri. Proof The set of monomials that are not in the lead term ideal of H form a vector space basis for k[V]G. Let Λ denote this set of monomials. By the previous theorem, a monomial M∈A lies in Λ if and only if Mx0a0,…,xkak lies in Λ for 0≤ai<pri and 0≤i≤k. It follows that the size of the set Λ is divisible by p∑i=0kri.□ The following generalizes the content of [8, Theorem 5] partially for a p-group. Theorem 2.7. Let xibe a terminal variable of degree d, and write N(xi)=xid+∑j=0d−1fjxij, where xi-degree of fjis zero for 0≤j≤d−1. Then xid+f0∈H. Proof Consider N¯=N(xi)−xid. This is a polynomial of degree e<d in xi. By Lemma 2.2, ∑j=0e(−1)jxijΔij(N¯)=f0, since f0 is the constant term of N¯. Now recall that Δij(xid) is the coefficient of tj in (xi+t)d=xid+td (note that d is a p-power by Lemma 2.3). Thus, Δj(xid)=0 for all 0<j<d. As Δij is a linear map for all j it follows that Δij(N(xi))=Δij(N¯) for all 0<j<d. Therefore, ∑j=1e(−1)jxijΔj(N(xi))=f0−N¯. As Δij(N(xi))∈H for all j<d by Lemma 2.4, we get that f0−N¯∈H. Therefore xid+f0=N(xi)−N¯+f0∈H as required.□ Lemma 2.8. Suppose that for each i=0,…,kwe have xidi∈H. Then k[V]Gis free as a module over its subalgebra Tgenerated by X0,X1,…,Xk, and T≅k[t0,…,tk]/(t0d0,…,tkdk), where t0,…,tkare independent variables. Proof The hypothesis on the xi is equivalent to Xidi=0 in k[V]G. Let t0,…,tk be independent variables and consider the natural surjective ring homomorphism from k[t0,…,tk] to k[X0,…,Xk]. Since Xidi=0, the kernel of this map contains (t0d0,…,tkdk). If this ideal is not all the kernel, then H must contain a polynomial in x0,…,xk such that no monomial in this polynomial is divisible by xidi for 0≤i≤k. This is a contradiction with the description of the lead term ideal in Theorem 2.5. Secondly, let Λ denote the set of monomials in k[V] that are not in the lead term ideal of H. Then the set of images of monomials in Λ′=Λ∪A generate k[V]G over T. Further, they generate freely because Mx0a0,…,xkak∈Λ for all M∈Λ′ and 0≤ai<di and 0≤i≤k, and the images of monomials in Λ form a vector space basis for k[V]G.□ Proof of Theorem 1.2 The first two assertions of the theorem are contained in Theorem 2.5 and its corollary. Next assume that di=deg(NG(xi)) for 0≤i≤k. So we can take N(xi)=NG(xi). Then from Theorem 2.7, it follows that xidi∈H for 0≤i≤k since the constant term of NG(xi) (as a polynomial in xi) is zero. Now the third and the fourth assertions follow from Lemma 2.8.□ 3. Complete intersection property of  In this section, we show that if the Hilbert Ideals of two modules are generated by fixed points and powers of terminal variables, then so is the Hilbert Ideal of the direct sum. As an incidental result, we prove that the degree of a terminal variable does not change after taking direct sums. We continue with the notation and the convention of the previous section. Let V1 and V2 be arbitrary kG-modules. We choose a basis x1,1,…,xn1,1,y1,1,…,ym1,1 for V1* and x1,2,…,xn2,2,y1,2,…,ym2,2 for V2* such that x1,1,…,xn1,1,x1,2,…,xn2,2 are fixed points. Note that both k[V1] and k[V2] are subrings of k[V1⊕V2], and we identify k[V1⊕V2]=k[x1,1,…,xn1,1,x1,2,…,xn2,2,y1,1,…,ym1,1,y1,2,…,ym2,2]. Note that if yi,j is a terminal variable in Vj* for some 1≤i≤mj, 1≤j≤2, then it is also a terminal variable in V1*⊕V2*. Lemma 3.1. Assume the notation of the previous paragraph. Let yi,j∈Vj*be a terminal variable. Then the degrees of yi,j in Vj*and V1*⊕V2*are equal. Proof Since k[Vj]G⊆k[V1⊕V2]G, we have that the degree of yi,j in Vj* is bigger than its degree in V1*⊕V2*. On the other hand, the restriction map k[V1⊕V2]G→k[Vj]G given f→f∣Vj preserves any power of the form yi,jd. This gives the reverse inequality.□ We denote the Hilbert Ideals k[V1⊕V2]+Gk[V1⊕V2], k[V1]+Gk[V1] and k[V2]+Gk[V2] with H, H1 and H2, respectively. Theorem 3.2. Assume that H1and H2are generated by the powers of the variables in V1*and V2*, respectively, and that the variables y1,1,…,ym1,1,y1,2,…,ym2,2are terminal variables. Then His generated by the union of the generating sets for H1and H2. Proof Assume that H1 is generated by x1,1,…,xn1,1,y1,1d1,1,…,ym1,1dm,1 and H2 is generated by x1,2,…,xn2,2,y1,2d1,2,…,ym2,2dm,2. We show that di,j is equal to the degree of the variable yi,j for 1≤i≤mj and 1≤j≤2. For simplicity, we set i=j=1 and denote the degree of y1,1 with d. Since H1 is generated by monomials, each monomial in a polynomial in H1 is divisible by one of its monomial generators. So we get d1,1≤d. On the other hand, since y1,1d1,1 is a member of H1 there is a positive degree invariant with a monomial that divides y1,1d1,1. So by the minimality of d, we get d≤d1,1 as well. By Lemma 3.1, di,j is also equal to the degree of yi,j in k[V1⊕V2]G. We claim that the union of the generating sets for H1 and H2 generate H. Otherwise, there exists a polynomial f in H that contains a non-constant monomial ∏1≤i≤mj,1≤j≤2yi,jei,j with 0≤ei,j<di,j. Let Δi,j denote the derivation with respect to the terminal variable yi,j. Then applying Δi,jei,j successively to f for 1≤i≤mj,1≤j≤2 yields an invariant with a non-zero constant. This is a contradiction by Lemma 2.4 since ei,j<di,j.□ We end this section with an example which shows that the degree of a terminal variable may be strictly less than the degree of its norm: Example 3.3. Let H=⟨σ,τ⟩ be the Klein 4-group, k a field of characteristic 2 and m≥2. Let Ω−m(k) be a vector space of dimension m=2n+1 over k. Choose a basis {x1,x2,…,xm,y1,y2,…,ym+1} of V*. One can define an action of H on V in such a way that its action on V* is given by σ(yj)=yj+xj,σ(xj)=xj,τ(yj)=yj+xj−1,τ(xj)=xj using the convention that x0=xm+1=0. The variables y1,y2,…,ym+1 are terminal. One can readily check that y22+x2y2+x1y2+x2y1+x1y3+y1x3 is invariant under H (note the last term is zero if m=2), so y2 has degree 2. On the other hand, y2 is not fixed by either σ or τ, which means NH(y2) has degree 4. It is interesting to note that x1y2+x2y1+x1y3+y1x3∈H, so we still have y22∈H. 4. Cyclic p-groups and the Klein 4-group In this section, we apply the results of the previous sections to cyclic p-groups and the Klein 4-group. Let G=Zpr denote a cyclic group of order pr. Fix a generator σ of G. There are pr indecomposable kG-modules V1,…,Vpr over k, and each indecomposable module Vi is afforded by σ−1 acting via a Jordan block of dimension i with ones on the diagonal. For an arbitrary kG-module V, we write V=⨁i=0kVni(with1≤ni≤prforalli), where each Vni is spanned as a vector space by e1,i,…,eni,i. Then the action of σ−1 is given by σ−1(ej,i)=ej,i+ej+1,i for 1≤j<ni and σ−1(eni,i)=eni,i. Note that the fixed point space VG is k-linearly spanned by en1,0,…,enk,k. The dual Vni* is isomorphic to Vni. Let x1,i,…,xni,i denote the corresponding dual basis, then we have k[V]=k[xj,i∣1≤j≤ni,0≤i≤k], and the action of σ is given by σ(xj,i)=xj,i+xj−1,i for 1<j≤ni and σ(x1,i)=x1,i for 0≤i≤k. Notice that the variables xni,i for 0≤i≤k are terminal variables. We follow the notation of Section 2 and denote xni,i with xi. We show that Theorem 1.2 applies completely to G by computing deg(xi) explicitly for 0≤i≤k. For each 0≤i≤k, let ai denote the largest integer such that ni>pai−1. Lemma 4.1. We have deg(xi)=pai. In particular, we may take N(xi)=NG(xi). Proof From [7, Lemma 3], we get that deg(xi) is at least pai. On the other hand, since pai≥ni>pai−1, a Jordan block of size ni has order pai. That is, this block affords a faithful module of the subgroup of G of size pai. It follows that the orbit of xi has pai elements and so that the orbit product NG(xi) is a monic polynomial that is of degree pai in xi.□ Applying Theorem 1.2, we obtain the following. Proposition 4.2. Assume the notation of Theorem1.2with specialization G=Zpr. We have an isomorphism k[X0,…,Xk]≅k[t0,…,tk]/(t0pa0,…,tkpak).Moreover, k[V]Gis free as a module over k[X0,…,Xk].□ Now let H denote the Klein 4-group and p=2. For each indecomposable kH-module V, there exists a basis of V* with one of the terminal variables xi satisfying deg(xi)=[H:Hxi], see [10]. In this source, it is also proven that with the exception of the regular module, each basis consists of fixed points and the terminal variables, and the Hilbert Ideal of every such module is generated by fixed points and the powers of the terminal variables. So we have by Theorems 1.2 and 3.2: Proposition 4.3. Let Vbe a kH-module containing k+1indecomposable summands. There is a basis {x0,x1,…,xn}of V*in which x0,x1,…,xkare terminal variables, each coming from one summand, such that k[V]His free as a module over its subalgebra Tgenerated by the images X0,X1,…,Xkof the terminal variables. Moreover, T≅k[t0,…,tk]/(t0a0,…,tkak), where t0,…,tkare independent variables, and for each i, we have ai=2or 4. Proposition 4.4. Let Vbe a kH-module such that Vdoes not contain the regular module kHas a summand. Then there exists a basis of V*such that k[V]+Hk[V]is generated by powers of basis elements. In particular, k[V]+Hk[V]is a complete intersection. Funding The second author is supported by a grant from TÜBITAK:114F427 References 1 W. W. Adams and P. Loustaunau , An introduction to Gröbner bases, vol. 3 of Graduate Studies in Mathematics , American Mathematical Society , Providence, RI , 1994 . 2 C. Chevalley , Invariants of finite groups generated by reflections , Amer. J. Math. 77 ( 1955 ), 778 – 782 . Google Scholar CrossRef Search ADS 3 E. Dufresne and A. 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Locally finite derivations and modular coinvariants

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Abstract

Abstract We consider a finite-dimensional kG-module V of a p-group G over a field k of characteristic p. We describe a generating set for the corresponding Hilbert Ideal. In case G is cyclic, this yields that the algebra k[V]G of coinvariants is a free module over its subalgebra generated by kG-module generators of V*. This subalgebra is a quotient of a polynomial ring by pure powers of its variables. The coinvariant ring was known to have this property only when G was cyclic of prime order [M. Sezer, Decomposing modular coinvariants, J. Algebra423 (2015), 87–92]. In addition, we show that if G is the Klein 4-group and V does not contain an indecomposable summand isomorphic to the regular module, then the Hilbert Ideal is a complete intersection, extending a result of the second author and Shank [M. Sezer and R. J. Shank, Rings of invariants for modular representations of the Klein four group, Trans. Amer. Math. Soc. 368 (2016), 5655–5673]. 1. Introduction Let k be a field of positive characteristic p and V a finite-dimensional k-vector space, and G≤GL(V) a finite group. Then the induced action on V* extends to the symmetric algebra k[V]≔S(V*) by the formula σ(f)=f◦σ−1 for σ∈G and f∈k[V]. The ring of fixed points k[V]G is called the ring of invariants, and is the central object of study in invariant theory. Another object which is often studied is the Hilbert Ideal, H, which is defined to be the ideal of k[V] generated by invariants of positive degree, in other words H=k[V]+Gk[V]. In this article, we study the quotient k[V]G≔k[V]/H which is called the algebra of coinvariants. An equivalent definition is k[V]G≔k[V]⊗k[V]Gk, which shows that this object is, in a sense, dual to k[V]G. As k[V]G is a finite-dimensional kG-module, it is generally easier to handle than the ring of invariants. On the other hand, much information about k[V]G is encoded in k[V]G. For example, Steinberg [13] famously showed that dim(C[V]G)=∣G∣ if and only if (G,V) is a complex reflection group. Combined with the theorem of Chevalley [2], Shephard and Todd [11], this shows that dim(C[V]G)=∣G∣ if and only if C[V]G is a polynomial ring. Smith [12] later generalized this by showing that dim(k[V]G)=∣G∣ if and only if G is a (pseudo)-reflection group, where k is any field. Further, the polynomial property of C[V]G is equivalent to the Poincaré duality property of C[V]G, by Kane [6] and Steinberg [13]. Before we continue, we fix some terminology. Let x0,…,xn be a basis for V*. We will say xi is a terminal variable if the vector space spanned by the other variables is a kG-submodule of V*. Note that if G is a p-group, then VG≠0 and there is a choice of a basis for V that contains a fixed point. Then the dual element corresponding to the fixed point is a terminal variable in the basis consisting of dual elements of this basis. For any f∈k[V], we define the norm NG(f)=∏h∈G·fh. For every terminal variable xi, we choose a polynomial N(xi) in k[V]G which, when viewed as a polynomial in xi is monic of minimal positive degree. While N(xi) is not unique in general, its degree is well defined. Since NG(xi) is monic of degree [G:Gxi], the degree of N(xi) is bounded above by this number. By ‘degree of xi’, we understand degree of N(xi) as a polynomial in xi and denote it by deg(xi). We will show that the degree of a terminal variable is always a p-power. The algebras of modular coinvariants for cyclic groups of order p were studied by the second author [8], and previously by the second author and Shank [9]. Note that there is a choice of basis such that an indecomposable representation of a p-group is afforded by an upper triangular matrix with 1s on the diagonal and the bottom variable is a terminal variable. In [8], the following was proven. Proposition 1.1. Let Gbe a cyclic group of order pand Va kG-module that contains k+1non-trivial summands. Choose a basis x0,x1,…,xnin which the variables x0,x1,…,xkare the bottom variables of the respective Jordan blocks, and let Abe the kG-subalgebra of k[V]generated by xk+1,…,xn. Denote the image of xi in k[V]G by Xi. Then The Hilbert Ideal of k[V]Gis generated by NG(x0),NG(x1),…,NG(xk), and polynomials in A. k[V]Ghas dimension divisible by pk+1. k[V]Gis free as a module over its subalgebra Tgenerated by X0,X1,…,Xk. T≅k[t0,…,tk]/(t0p,…,tkp), where t0,…,tkare independent variables. The goal of this article is to generalize the above, as far as possible, to the case of all finite p-groups. In particular, we show in section two: Theorem 1.2. Let Gbe a finite p-group and Va kG-module that contains k+1non-trivial summands. Choose a basis x0,x1,…,xnin which the variables x0,x1,…,xkcoming from each summand are terminal variables. Let didenote deg(xi)for 0≤i≤k. Retain the notation in the proposition above, then There is a choice for polynomials N(x0),N(x1),…,N(xk)such that the Hilbert Ideal of k[V]Gis generated by N(x0),N(x1),…,N(xk), and polynomials in A. k[V]Ghas dimension divisible by ∏i=0kdi.Suppose in addition that one has di=deg(NG(xi))for 0≤i≤k. Then we have: k[V]Gis free as a module over its subalgebra Tgenerated by X0,X1,…,Xk. T≅k[t0,…,tk]/(t0d0,…,tkdk), where t0,…,tkare independent variables. In Section 3, we describe the situation for a p-group, where the complete intersection property of the Hilbert Ideal corresponding to a module is inherited from the Hilbert Ideal of the indecomposable summands of the module. The final section is devoted to applications of our main results to cyclic p-groups and the Klein 4-group. It turns out that for a cyclic p-group, the bottom variables xi of Jordan blocks satisfy deg(xi)=deg(NG(xi)). Consequently, (3) and (4) above hold for a cyclic p-group. Additionally, for the Klein 4-group, we show that the Hilbert Ideal corresponding to a module is a complete intersection as long as the module does not contain the regular module as a summand. This generalizes a result of the second author and Shank [10], where the complete intersection property was established for indecomposable modules only. This article was composed during a visit of the second author to the University of Aberdeen, funded by the Edinburgh Mathematical Society’s Research Support Fund. We would like to thank the society for their support. 2. Main results Throughout this section, we let G be a finite p-group, k a field of characteristic p and V a kG-module, which may be decomposable. As trivial summands do not contribute to the coinvariants, we assume no direct summand of V is trivial. Let x0,x1,x2,…,xn be a basis of V* and assume that x0 is a terminal variable. Then x1,x2,…,xn generate a G-subalgebra which we denote by A. We can define a nonlinear action of (k,+) on k[V] as follows: t·x0=x0+t; (2.1) t·xi=xiforanyi>0. (2.2) The terminality of x0 ensures this commutes with the action of G. It is well known that any action of the additive group of an infinite field of prime characteristic is determined by a locally finite iterative higher derivation. This is a family of k-linear maps Δi:k[V]→k[V], i≥0 satisfying the following properties: Δ0=idk[V]. For all i>0 and a,b∈k[V], one has Δi(ab)=∑j+k=iΔj(a)Δk(b). For all b∈k[V], there exists i≥0 such that Δi(b)=0. For all i,j, one has Δj◦Δi=(i+jj)Δi+j. The equivalence of the group action and the l.f.i.h.d. is given by the formula t·b=∑i≥0tiΔi(b). (2.3) See [3, 14], for more details on l.f.i.h.d.’s. Let f∈k[V]G be homogeneous of degree d in x0. We write f=fdx0d+fd−1x0d−1+⋯+f0, where fi∈A. We have t·f=fd(x0+t)d+fd−1(x0+t)d−1+⋯+f0=∑i≥0tiΔi(f). (2.4) That is to say that Δi(f) is the coefficient of ti in the above expression. As the action of G commutes with the action of k, we see that Δi(f)∈k[V]G for all i≥0. Remark 2.1. Clearly Δ1=∂∂x0. So the previous paragraph generalizes [8, Lemma 1]. Equation (2.4) gives that Δj(x0i)=(ij)x0i−j provided i≥j. Then, from Lucas’s theorem [5] on binomial coefficients in characteristic p, we see that we can think of Δpj as ‘Differentiation by x0pj’: if the coefficient of pj in the base p expansion of m is a, then we have Δpj(x0m)={ax0m−pja>0;0a=0. For later use, we also note the following consequence: for a homogeneous f∈k[V], Δj(f) contains a non-zero constant if and only if the monomial x0j appears in f. In [4], a G-equivariant map is constructed from polynomials whose x0-degree is at most epr (0<e<p) to polynomials whose x0-degree is at most pr. This map turns out to be a non-zero scalar multiple of Δ(e−1)pr. We have the following statement generalizing [8, Lemma 2]: Lemma 2.2. Let f∈k[V]be a homogeneous polynomial of degree din x0. Write f=fdx0d+fd−1x0d−1+⋯+f0, where fi∈A. Then we have ∑i=0d(−1)ix0iΔi(f)=f0. Proof Write f=f(x0,x1,x2,…,xn). For any t∈k, we have t·f=f(t·x0,t·x1,…,t·xn)=f(x0+t,x1,x2,…,xn). As this holds for all t it also holds when t is replaced by (−x0), and hence by Equation (2.3), we have ∑i=0d(−1)ix0iΔi(f)=(−x0)·f=f(0,x1,x2,…,xn)=f0 as required.□ We also note that the degree of a terminal variable is a p-power. Lemma 2.3. For any terminal variable x0∈V*, deg(x0)is a power of p. Proof Let d denote the degree of x0 and suppose f∈k[V]G is monic as a polynomial in x0 of degree d=drpr+dr−1pr−1+⋯+d0 with 0≤di<p and dr≠0. If dj≠0 for some j<r, then Δpj(f)∈k[V]G has degree d−pj>0 as a polynomial in x0 and its leading coefficient is in k. Similarly, if dj=0 for j<r and dr>1, then Δpr(f)∈k[V]G has degree d−pr>0 in x0 and its leading coefficient is in k. Both cases violate the minimality of d.□ Lemma 2.4. Let ddenote the degree of x0. Then Δj(H)⊆Hfor j<d. Proof Let f∈k[V]. From the second assertion of Remark 2.1, we get that Δj(f) contains a non-zero constant if and only if the monomial x0j appears in f. Therefore, by the minimality of d, we have Δj(k[V]+G)⊆H for j<d. Now the result follows from property (2) of l.f.i.h.d.’s.□ From this point on, we adopt the notation of the introduction. This means that x0,x1,x2,…,xk are terminal variables coming from different summands, and A=k[xk+1,xk+2,…,xn]. For each i=0,…,k let di=pri be the degree of xi. Since setting variables outside of a summand to zero sends invariants to invariants of the summand, we may also assume that N(xi) depends only on variables that come from the summand that contains xi. We denote by Δi the l.f.i.h.d. associated to xi. We use reverse lexicographic order with xi>xj whenever 0≤i≤k and k+1≤j≤n. Theorem 2.5. His generated by N(x0),…,N(xk)and polynomials in A. Moreover, the lead term ideal of His generated by x0pr0,x1pr1,…,xkprkand monomials in A. Proof Let f∈k[V]G. Since N(x0) is monic in x0, we may perform polynomial division and write f=qN(x0)+r where r has x0-degree <pr0, and it is easily shown that q,r∈k[V]G. Then dividing r by N(x1) yields another invariant remainder r′ that has x1-degree <pr1. Since x0-degree of N(x1) is zero, it follows that x0-degree of r′ is still <pr0. Thus, by repeating the process with each terminal variable, and replacing f with the final remainder we assume that xi-degree of f is <pri for 0≤i≤k. Let i be minimal such that f has non-zero degree d<pri in the terminal variable xi. We apply Lemma 2.2 with Δ=Δi to see that f=f0−(∑j=1d(−1)jxijΔij(f)), where f0 is the ‘constant term’ of f, that is, f0∈k[xi+1,…,xn]. So from the previous lemma, we get that f0∈H since d<pri. Moreover, since Δi decreases xi-degrees and does not increase degrees in any other variable, the xi-degree of each Δij(f) in the expression above is strictly less than d, and the xl-degree for every i<l≤k remains strictly less than prl. Thus, by induction on degree, f can be expressed as a k[V]-combination of elements of H whose degrees in the terminal variables x0,…,xi are all zero, and degrees in the remaining terminal variables xl for i<l≤k are strictly less than prl, respectively. Repeating the same argument with the remaining terminal variables gives us that f can be written as a k[V]-combination of elements of H∩A together with N(x1),…,N(xk) as required. The first assertion of the theorem follows. Note that the leading monomial of N(xi) is xipri for 0≤i≤k. So it remains to show that all other monomials in the lead term ideal of H lie in A. Recall that by Buchberger’s algorithm a Gröbner basis is obtained by reduction of S-polynomials of a generating set by polynomial division, see [1, Section 1.7]. By the first part, H has a generating set consisting of N(xi) for 0≤i≤k and polynomials in A. But the S-polynomial of two polynomials in A is also in A, and via polynomials in A, it also reduces to a polynomial in A. Finally, the S-polynomial of N(xi) and a polynomial in A and the S-polynomial of a pair N(xi) and N(xj) with 0≤i≠j≤k reduce to zero since their leading monomials are pairwise relatively prime.□ Corollary 2.6. The vector space dimension of k[V]Gis divisible by ∏0≤i≤kdi=p∑i=0kri. Proof The set of monomials that are not in the lead term ideal of H form a vector space basis for k[V]G. Let Λ denote this set of monomials. By the previous theorem, a monomial M∈A lies in Λ if and only if Mx0a0,…,xkak lies in Λ for 0≤ai<pri and 0≤i≤k. It follows that the size of the set Λ is divisible by p∑i=0kri.□ The following generalizes the content of [8, Theorem 5] partially for a p-group. Theorem 2.7. Let xibe a terminal variable of degree d, and write N(xi)=xid+∑j=0d−1fjxij, where xi-degree of fjis zero for 0≤j≤d−1. Then xid+f0∈H. Proof Consider N¯=N(xi)−xid. This is a polynomial of degree e<d in xi. By Lemma 2.2, ∑j=0e(−1)jxijΔij(N¯)=f0, since f0 is the constant term of N¯. Now recall that Δij(xid) is the coefficient of tj in (xi+t)d=xid+td (note that d is a p-power by Lemma 2.3). Thus, Δj(xid)=0 for all 0<j<d. As Δij is a linear map for all j it follows that Δij(N(xi))=Δij(N¯) for all 0<j<d. Therefore, ∑j=1e(−1)jxijΔj(N(xi))=f0−N¯. As Δij(N(xi))∈H for all j<d by Lemma 2.4, we get that f0−N¯∈H. Therefore xid+f0=N(xi)−N¯+f0∈H as required.□ Lemma 2.8. Suppose that for each i=0,…,kwe have xidi∈H. Then k[V]Gis free as a module over its subalgebra Tgenerated by X0,X1,…,Xk, and T≅k[t0,…,tk]/(t0d0,…,tkdk), where t0,…,tkare independent variables. Proof The hypothesis on the xi is equivalent to Xidi=0 in k[V]G. Let t0,…,tk be independent variables and consider the natural surjective ring homomorphism from k[t0,…,tk] to k[X0,…,Xk]. Since Xidi=0, the kernel of this map contains (t0d0,…,tkdk). If this ideal is not all the kernel, then H must contain a polynomial in x0,…,xk such that no monomial in this polynomial is divisible by xidi for 0≤i≤k. This is a contradiction with the description of the lead term ideal in Theorem 2.5. Secondly, let Λ denote the set of monomials in k[V] that are not in the lead term ideal of H. Then the set of images of monomials in Λ′=Λ∪A generate k[V]G over T. Further, they generate freely because Mx0a0,…,xkak∈Λ for all M∈Λ′ and 0≤ai<di and 0≤i≤k, and the images of monomials in Λ form a vector space basis for k[V]G.□ Proof of Theorem 1.2 The first two assertions of the theorem are contained in Theorem 2.5 and its corollary. Next assume that di=deg(NG(xi)) for 0≤i≤k. So we can take N(xi)=NG(xi). Then from Theorem 2.7, it follows that xidi∈H for 0≤i≤k since the constant term of NG(xi) (as a polynomial in xi) is zero. Now the third and the fourth assertions follow from Lemma 2.8.□ 3. Complete intersection property of  In this section, we show that if the Hilbert Ideals of two modules are generated by fixed points and powers of terminal variables, then so is the Hilbert Ideal of the direct sum. As an incidental result, we prove that the degree of a terminal variable does not change after taking direct sums. We continue with the notation and the convention of the previous section. Let V1 and V2 be arbitrary kG-modules. We choose a basis x1,1,…,xn1,1,y1,1,…,ym1,1 for V1* and x1,2,…,xn2,2,y1,2,…,ym2,2 for V2* such that x1,1,…,xn1,1,x1,2,…,xn2,2 are fixed points. Note that both k[V1] and k[V2] are subrings of k[V1⊕V2], and we identify k[V1⊕V2]=k[x1,1,…,xn1,1,x1,2,…,xn2,2,y1,1,…,ym1,1,y1,2,…,ym2,2]. Note that if yi,j is a terminal variable in Vj* for some 1≤i≤mj, 1≤j≤2, then it is also a terminal variable in V1*⊕V2*. Lemma 3.1. Assume the notation of the previous paragraph. Let yi,j∈Vj*be a terminal variable. Then the degrees of yi,j in Vj*and V1*⊕V2*are equal. Proof Since k[Vj]G⊆k[V1⊕V2]G, we have that the degree of yi,j in Vj* is bigger than its degree in V1*⊕V2*. On the other hand, the restriction map k[V1⊕V2]G→k[Vj]G given f→f∣Vj preserves any power of the form yi,jd. This gives the reverse inequality.□ We denote the Hilbert Ideals k[V1⊕V2]+Gk[V1⊕V2], k[V1]+Gk[V1] and k[V2]+Gk[V2] with H, H1 and H2, respectively. Theorem 3.2. Assume that H1and H2are generated by the powers of the variables in V1*and V2*, respectively, and that the variables y1,1,…,ym1,1,y1,2,…,ym2,2are terminal variables. Then His generated by the union of the generating sets for H1and H2. Proof Assume that H1 is generated by x1,1,…,xn1,1,y1,1d1,1,…,ym1,1dm,1 and H2 is generated by x1,2,…,xn2,2,y1,2d1,2,…,ym2,2dm,2. We show that di,j is equal to the degree of the variable yi,j for 1≤i≤mj and 1≤j≤2. For simplicity, we set i=j=1 and denote the degree of y1,1 with d. Since H1 is generated by monomials, each monomial in a polynomial in H1 is divisible by one of its monomial generators. So we get d1,1≤d. On the other hand, since y1,1d1,1 is a member of H1 there is a positive degree invariant with a monomial that divides y1,1d1,1. So by the minimality of d, we get d≤d1,1 as well. By Lemma 3.1, di,j is also equal to the degree of yi,j in k[V1⊕V2]G. We claim that the union of the generating sets for H1 and H2 generate H. Otherwise, there exists a polynomial f in H that contains a non-constant monomial ∏1≤i≤mj,1≤j≤2yi,jei,j with 0≤ei,j<di,j. Let Δi,j denote the derivation with respect to the terminal variable yi,j. Then applying Δi,jei,j successively to f for 1≤i≤mj,1≤j≤2 yields an invariant with a non-zero constant. This is a contradiction by Lemma 2.4 since ei,j<di,j.□ We end this section with an example which shows that the degree of a terminal variable may be strictly less than the degree of its norm: Example 3.3. Let H=⟨σ,τ⟩ be the Klein 4-group, k a field of characteristic 2 and m≥2. Let Ω−m(k) be a vector space of dimension m=2n+1 over k. Choose a basis {x1,x2,…,xm,y1,y2,…,ym+1} of V*. One can define an action of H on V in such a way that its action on V* is given by σ(yj)=yj+xj,σ(xj)=xj,τ(yj)=yj+xj−1,τ(xj)=xj using the convention that x0=xm+1=0. The variables y1,y2,…,ym+1 are terminal. One can readily check that y22+x2y2+x1y2+x2y1+x1y3+y1x3 is invariant under H (note the last term is zero if m=2), so y2 has degree 2. On the other hand, y2 is not fixed by either σ or τ, which means NH(y2) has degree 4. It is interesting to note that x1y2+x2y1+x1y3+y1x3∈H, so we still have y22∈H. 4. Cyclic p-groups and the Klein 4-group In this section, we apply the results of the previous sections to cyclic p-groups and the Klein 4-group. Let G=Zpr denote a cyclic group of order pr. Fix a generator σ of G. There are pr indecomposable kG-modules V1,…,Vpr over k, and each indecomposable module Vi is afforded by σ−1 acting via a Jordan block of dimension i with ones on the diagonal. For an arbitrary kG-module V, we write V=⨁i=0kVni(with1≤ni≤prforalli), where each Vni is spanned as a vector space by e1,i,…,eni,i. Then the action of σ−1 is given by σ−1(ej,i)=ej,i+ej+1,i for 1≤j<ni and σ−1(eni,i)=eni,i. Note that the fixed point space VG is k-linearly spanned by en1,0,…,enk,k. The dual Vni* is isomorphic to Vni. Let x1,i,…,xni,i denote the corresponding dual basis, then we have k[V]=k[xj,i∣1≤j≤ni,0≤i≤k], and the action of σ is given by σ(xj,i)=xj,i+xj−1,i for 1<j≤ni and σ(x1,i)=x1,i for 0≤i≤k. Notice that the variables xni,i for 0≤i≤k are terminal variables. We follow the notation of Section 2 and denote xni,i with xi. We show that Theorem 1.2 applies completely to G by computing deg(xi) explicitly for 0≤i≤k. For each 0≤i≤k, let ai denote the largest integer such that ni>pai−1. Lemma 4.1. We have deg(xi)=pai. In particular, we may take N(xi)=NG(xi). Proof From [7, Lemma 3], we get that deg(xi) is at least pai. On the other hand, since pai≥ni>pai−1, a Jordan block of size ni has order pai. That is, this block affords a faithful module of the subgroup of G of size pai. It follows that the orbit of xi has pai elements and so that the orbit product NG(xi) is a monic polynomial that is of degree pai in xi.□ Applying Theorem 1.2, we obtain the following. Proposition 4.2. Assume the notation of Theorem1.2with specialization G=Zpr. We have an isomorphism k[X0,…,Xk]≅k[t0,…,tk]/(t0pa0,…,tkpak).Moreover, k[V]Gis free as a module over k[X0,…,Xk].□ Now let H denote the Klein 4-group and p=2. For each indecomposable kH-module V, there exists a basis of V* with one of the terminal variables xi satisfying deg(xi)=[H:Hxi], see [10]. In this source, it is also proven that with the exception of the regular module, each basis consists of fixed points and the terminal variables, and the Hilbert Ideal of every such module is generated by fixed points and the powers of the terminal variables. So we have by Theorems 1.2 and 3.2: Proposition 4.3. Let Vbe a kH-module containing k+1indecomposable summands. There is a basis {x0,x1,…,xn}of V*in which x0,x1,…,xkare terminal variables, each coming from one summand, such that k[V]His free as a module over its subalgebra Tgenerated by the images X0,X1,…,Xkof the terminal variables. Moreover, T≅k[t0,…,tk]/(t0a0,…,tkak), where t0,…,tkare independent variables, and for each i, we have ai=2or 4. Proposition 4.4. Let Vbe a kH-module such that Vdoes not contain the regular module kHas a summand. Then there exists a basis of V*such that k[V]+Hk[V]is generated by powers of basis elements. In particular, k[V]+Hk[V]is a complete intersection. Funding The second author is supported by a grant from TÜBITAK:114F427 References 1 W. W. Adams and P. Loustaunau , An introduction to Gröbner bases, vol. 3 of Graduate Studies in Mathematics , American Mathematical Society , Providence, RI , 1994 . 2 C. Chevalley , Invariants of finite groups generated by reflections , Amer. J. Math. 77 ( 1955 ), 778 – 782 . Google Scholar CrossRef Search ADS 3 E. Dufresne and A. 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The Quarterly Journal of MathematicsOxford University Press

Published: Mar 15, 2018

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