Locally finite derivations and modular coinvariants

LOCALLY FINITE DERIVATIONS AND MODULAR

COINVARIANTS

by JONATHAN ELMER†

(Middlesex University, The Burroughs, Hendon, London NW4 4BT, UK) and MÜFIT SEZER‡

(Department of Mathematics, Bilkent University, Cankaya, Ankara 06800, Turkey) [Received 16 June 2016. Revised 5 February 2018]

Abstract

We consider a finite-dimensional G-module V of a p-group G over a field  of characteristic p. We describe a generating set for the corresponding Hilbert Ideal. In case G is cyclic, this yields that the algebra V[ ]Gof coinvariants is a free module over its subalgebra generated byG-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. Algebra 423 (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  be a field of positive characteristic p and V a finite-dimensional -vector space, and

£ ( )

G GL V a finite group. Then the induced action on V* extends to the symmetric algebra

[ ] ( )

 V ≔ S V * by the formula s( ) =f f ◦s-1 for s Î G and f Î [ ] V . The ring offixed points [ ]

 VG _{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, , which is defined to be the ideal of

[ ]

 V generated by invariants of positive degree, in other words

= [ ]₊ [ ]

  VG V .

In this article, we study the quotient[ ]VG ≔[ ]/V which is called the algebra of coinvariants. An equivalent definition is [ ]V G ≔[ ] ÄV [ ]VG , which shows that this object is, in a sense,

dual to [ ] VG_.

As V[ ]G is a finite-dimensional G-module, it is generally easier to handle than the ring of invariants. On the other hand, much information about V[ ]G _{is encoded in} V_{[ ]}

G. For example, Steinberg [13] famously showed thatdim( [ ] ) = V G ∣ ∣ if and only if (G G V, )is a complex re flec-tion group. Combined with the theorem of Chevalley [2], Shephard and Todd [11], this shows that †_{E-mail:j.elmer@mdx.ac.uk}

‡_{Corresponding author. E-mail:sezer@fen.bilkent.edu.tr}

1053

© The Author(s) 2018. Published by Oxford University Press. All rights reserved. For permissions, please e-mail: journals. permissions@oup.com

( [ ] ) = V G

dim G ∣ ∣ if and only if V[ ]Gis a polynomial ring. Smith [12] later generalized this by showing that dim( [ ] ) = VG ∣ ∣ if and only if G is a (pseudo)-reG flection group, where  is any field. Further, the polynomial property of [ ] V G_{is equivalent to the Poincaré duality property of}

[ ]

 VG, by Kane [6] and Steinberg [13].

Before we continue, wefix some terminology. Letx0,¼,xnbe a basis for V *. We will say xiis a terminal variable if the vector space spanned by the other variables is a G-submodule of V *. Note that if G is a p-group, thenVG¹0_{and there is a choice of a basis for V that contains afixed} 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 fÎ [ ] V, we define the norm



( ) =

Î

NG f h.

h G f·

For every terminal variable xi, we choose a polynomialN x( )i in V[ ]G which, when viewed as a polynomial in xi is monic of minimal positive degree. While N x( )i is not unique in general, its degree is well defined. Since NG( )x

i is monic of degree [G:Gxi], the degree ofN x( )i is bounded

above by this number. By‘degree of xi’, we understand degree of ( )N xi as a polynomial in xiand denote it bydeg( )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.

PROPOSITION1.1. Let G be a cyclic group of order p and V a G-module that containsk+1 non-trivial summands. Choose a basis x0,x1,¼,xn in which the variables x0,x1,¼,xk are the bottom variables of the respective Jordan blocks, and let A be the G-subalgebra of V gener-[ ] ated byxk+1,¼,xn. Denote the image of xiin [ ] VGby Xi. Then

(1) The Hilbert Ideal of [ ] VG_{is generated byN}G_{( )x ,N}G_{( ) ¼x , ,N}G_{( )x} k

0 1 , and polynomials in A.

(2)  V[ ]Ghas dimension divisible by pk+1.

(3)  V[ ]Gis free as a module over its subalgebra  generated byX0,X1,¼,Xk. (4) @ [ ¼ ]/( t , ,tk tp,¼,tk)

p

0 0 , wheret0,¼,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:

THEOREM1.2. Let G be afinite p-group and V a G-module that contains +k 1 non-trivial sum-mands. Choose a basis x0,x1,¼,xn in which the variables x0,x1,¼,xk coming from each sum-mand are terminal variables. Let di denote deg( )xi for 0£ £i k. Retain the notation in the proposition above, then

(1) There is a choice for polynomialsN x( )0 ,N x( ) ¼1 , ,N x( )k such that the Hilbert Ideal of V[ ]G is generated byN x( )0 ,N x( ) ¼1 , ,N x( )k, and polynomials in A.

(2)  V[ ]Ghas dimension divisible byi= d k

i 0 .

Suppose in addition that one hasdi=deg(NG( ))xi for0£ £i k. Then we have: (3)  V[ ]Gis free as a module over its subalgebra  generated byX0,X1,¼,Xk. (4) @ [ ¼ ]/( t , ,tk td,¼,tk )

d

0 00 k , wheret0,¼,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 indecom-posable summands of the module. Thefinal 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 vari-ables xiof Jordan blocks satisfydeg( ) =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 correspond-ing 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,  a field of characteristic p and V a G-module, which may be decomposable. As trivial summands do not contribute to the coinvari-ants, we assume no direct summand of V is trivial. Let x0,x x1, 2,¼,xn be a basis of V * and assume that x0 is a terminal variable. Thenx x1, 2,¼,xn generate a G-subalgebra which we denote by A. We can define a nonlinear action of ( +), on [ ] V as follows:

= + ( )

t·x0 x0 t; 2.1

= > ( )

t·xi xi for any i 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 -linear maps Di: [ ]  [ ]V V_{, ³i 0} satis-fying the following properties:

(1) D =0 id[ ]V.

(2) For all >i 0 anda b, Î [ ] V, one has D (i ab) = å_{+ =}D ( )D ( )a b j k i

j k _.

(3) For allbÎ [ ]V , there exists ³i 0 such that D ( ) =i b 0_. (4) For all i j, , one hasDj D =i

( )

j i j

◦ .

The equivalence of the group action and the l.f.i.h.d. is given by the formula

å

See [3,14], for more details on l.f.i.h.d.’s.

Let fÎ [ ] VG_{be homogeneous of degree d in x}

0. We write = + _- - + + f f xd f x f , d d d 0 1 0 1 0 

where f_i ÎA. We have

å

That is to say that D ( )i f _{is the coefficient of t}i_{in the above expression. As the action of G} com-mutes with the action of , we see that D ( ) Îi f [ ]VG_{for all ³i 0.}

REMARK2.1. (1) Clearly D =1 _¶x¶

0. So the previous paragraph generalizes [

8, Lemma 1]. (2) Equation (2.4) gives thatD ( ) =j xi

( )

-j i j

0 0 providedi³j. Then, from Lucas’s theorem [5] on binomial coefficients in characteristic p, we see that we can think of Dpj

as‘Differentiation by x₀pj’: if the coefficient of pj_{in the base p expansion of m is a, then we have}

D ( ) =ìíïï îïï > = -x ax a a 0; 0 0. p m m p 0 0 j j

For later use, we also note the following consequence: for a homogeneous fÎ [ ] V, D ( )j f contains a non-zero constant if and only if the monomial x₀jappears in f .

(3) 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 D( - )e 1pr

.

We have the following statement generalizing [8, Lemma 2]:

LEMMA 2.2. Let f Î [ ]V be a homogeneous polynomial of degree d in x₀. Write

= + _- - + +

f f x_{d 0}d f_{d 1 0}xd 1  f₀, where f_i ÎA. Then we have

å

Proof. Write f= (f x0,x x1, 2,¼,xn). For any Î t , we have

= ( ¼ ) = ( + ¼ )

t·f f t·x0,t·x1, ,t·xn f x0 t x x, 1, 2, ,xn .

As this holds for all t it also holds when t is replaced by (- )x0 , and hence by Equation (2.3), we haveåi= (- )1 x D ( ) = (- )f x f= (f 0,x x, ,¼,x) = f

d i i i

n

0 0 0 · 1 2 0as required. □

We also note that the degree of a terminal variable is a p-power. LEMMA2.3. For any terminal variablex₀ÎV*,deg( )x₀ is a power of p.

Proof. Let d denote the degree of x0 and suppose fÎ [ ] VG is monic as a polynomial in x0 of degree d=d pr r +dr-1pr-1++d0 with0£di<panddr¹0. Ifdj¹0 for some j<r, then D ( ) Îpj f [ ]VG

has degreed-pj >0_{as a polynomial in x}

0 and its leading coefficient is

in . Similarly, ifdj=0for j<randdr>1, then D ( ) Îp f [ ]V G

r

has degreed-pr >0_{in x} 0 and its leading coefficient is in . Both cases violate the minimality of d. □ LEMMA2.4. Let d denote the degree of x₀. Then D ( ) Íj  for j<d .

Proof. Let fÎ [ ]V . From the second assertion of Remark2.1, we get that D ( )j f _{contains a} non-zero constant if and only if the monomial x₀j appears in f . Therefore, by the minimality of d, we have D ( [ ] ) Íj  V+G 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,x x1, 2,¼,xk are terminal variables coming from different summands, and A= [ xk+1,xk+2,¼,xn]. For each

= ¼

i 0, ,k letdi=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 x( )i depends only on variables that come from the summand that contains xi. We denote by Dithe l.f.i.h.d. associated to xi. We use reverse lexicographic order withxi>xjwhenever0£ £i kandk+1£ £j n. THEOREM 2.5.  is generated by N x( ) ¼0 , ,N x( )k and polynomials in A. Moreover, the lead term ideal of  is generated byxp ,xp ,¼,xk

p

0 1

r0 r1 rk

and monomials in A. Proof. Let fÎ [ ] VG_{. SinceN x( )}

0 is monic in x0, we may perform polynomial division and write

= ( ) +

f qN x0 rwhere r has x0-degree <pr0, and it is easily shown thatq r, Î [ ]V G. Then divid-ing r by N x( )1 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 for0£ £i k.

Let i be minimal such that f has non-zero degreed<pr_{i in the terminal variable x}

i. We apply Lemma2.2with D = Dito see that

(

å

)

where f₀ is the‘constant term’ of f , that is, f₀ Î [ xi+1,¼,xn]. So from the previous lemma, we get that f₀ Î  since d<pr_{i. Moreover, since D}

i decreases xi-degrees and does not increase degrees in any other variable, the xi-degree of each D ( )i f

j

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 [ ] V -combination of elements of  whose degrees in the terminal variablesx0,¼,xiare 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}

vari-ables gives us that f can be written as a V -combination of elements of[ ] ÇA together with

( ) ¼ ( )

N x1, ,N xk as required. Thefirst assertion of the theorem follows. Note that the leading monomial of N x( )i is xi

pri

for0£ £i k. So it remains to show that all other monomials in the lead term ideal of  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 div-ision, see [1, Section 1.7]. By the first part,  has a generating set consisting of N x( )i for

£ £i k

0 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 ofN x( )i and

a polynomial in A and the S-polynomial of a pairN x( )i andN x( )j with0£ ¹ £i j k reduce to

zero since their leading monomials are pairwise relatively prime. □

COROLLARY2.6. The vector space dimension of V[ ]Gis divisible by₀£ £i kdi=påi= r k

i

0 . Proof. The set of monomials that are not in the lead term ideal of  form a vector space basis for

[ ]

 VG. Let L denote this set of monomials. By the previous theorem, a monomialMÎAlies in L if and only ifMxa,¼,xk

a

00 k lies in L for0£ai<pri and0£ £i k. It follows that the size of the

set L is divisible by_påik=0ri. □

The following generalizes the content of [8, Theorem 5] partially for a p-group.

THEOREM 2.7. Let xi be a terminal variable of degree d , and write N x( ) =i xid + åj=- f x d j i j 0 1 , where xi-degree of fjis zero for0£ £j d-1. Thenxid +f0 Î .

Proof. ConsiderN =N x( ) -i xid. This is a polynomial of degreee<d in xi. By Lemma2.2,

å

since f₀ is the constant term of N¯ . Now recall that D (_ij xid)is the coefficient of tj in (xi+ ) =t d +

xid td (note that d is a p-power by Lemma2.3). Thus, D (j xid) =0for all0< <j d. As Di j

is a linear map for all j it follows that D ( ( )) = D ( )_ij N xi i N

j _{¯ for all < <} j d 0 . Therefore,

å

As D ( ( )) Î _ij N xi for all j<d by Lemma 2.4, we get that f₀ -N¯ Î . Therefore

+ = ( ) - + Î 

xid f₀ N xi N¯ f₀ as required. □

LEMMA2.8. Suppose that for each =i 0,¼,k we havex_idiÎ _{. Then} V[ ]

Gis free as a module over its subalgebra  generated byX0,X1,¼,Xk, and@ [ ¼ ]/( t, ,tk td,¼,tk )

d

0 00 k , wheret0,¼,tk are independent variables.

Proof. The hypothesis on the xiis equivalent toXi =0 di _in V[ ]

G. Lett0,¼,tkbe independent vari-ables and consider the natural surjective ring homomorphism from t[ ¼ ]0, ,tk to X[ 0,¼,Xk]. Since

= Xi 0

d_i

, the kernel of this map contains (td,¼,tk ) d

00 k . If this ideal is not all the kernel, then  must contain a polynomial in x0,¼,xk such that no monomial in this polynomial is divisible by xi

di _for

£ £i k

0 . This is a contradiction with the description of the lead term ideal in Theorem2.5. Secondly, let L denote the set of monomials in V that are not in the lead term ideal of .[ ] Then the set of images of monomials in L¢ = L È A generate [ ] VGover  . Further, they generate freely becauseMxa,¼,xk Î L

a

00 k for allMÎ L¢and0£ai<diand0£ £i k, and the images of

monomials in L form a vector space basis for [ ] VG. □

Proof of Theorem 1.2. Thefirst two assertions of the theorem are contained in Theorem2.5and its corollary. Next assume that di=deg(NG( ))xi for 0£ £i k. So we can take N x( ) =i NG( )xi . Then from Theorem2.7, it follows thatx_idi Î _for0£ £_{i ksince the constant term ofN}G( )_x

i (as a polynomial in xi) is zero. Now the third and the fourth assertions follow from Lemma2.8. □

3. Complete intersection property of 

In this section, we show that if the Hilbert Ideals of two modules are generated byfixed 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 con-tinue with the notation and the convention of the previous section. Let V1 and V2 be arbitrary G-modules. We choose a basisx1,1,¼,xn1,1,y1,1,¼,ym1,1for V1* and x1,2,¼,xn2,2,y1,2,¼,ym2,2for V₂* such that x1,1,¼,xn1,1,x1,2,¼,xn2,2arefixed points. Note that both [ ] V1 and V[ ]2 are subrings of [ V1ÅV2], and we identify

[ Å ] = [ ¼ ¼ ¼ ¼ ]

V1 V2  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 ter-minal variable inV₁*ÅV₂*.

LEMMA3.1. Assume the notation of the previous paragraph. Lety_{i j,} ÎV_j* be a terminal variable. Then the degrees of y_{i j,} in Vj* andV1*ÅV2* are equal.

Proof. Since[ ] Í [ ÅVjG  V1 V2]G, we have that the degree of yi j, in Vj* is bigger than its degree inV₁*ÅV₂*. On the other hand, the restriction map [ ÅV V]  [ ]G  V

jG

1 2 given f f∣Vjpreserves

any power of the form y_{i j}d_,. This gives the reverse inequality. □

We denote the Hilbert Ideals [ ÅV1 V2]+G[ ÅV1 V2], [ ]V1G+[ ]V1 and[ ]V2+G[ ]V2 with , 1 and 2, respectively.

THEOREM 3.2. Assume that ₁ and ₂ are generated by the powers of the variables in V₁* and V₂*, respectively, and that the variablesy1,1,¼,ym1,1,y1,2,¼,ym2,2are terminal variables. Then  is generated by the union of the generating sets for1and 2.

Proof. Assume that 1 is generated by x ,¼,xn ,y ,¼,y d m d 1,1 1,1 1,1 m,1 1,1 1 ,1 _{and } 2 is generated by ¼ ¼ x , ,xn ,y , ,y d m d 1,2 2,2 1,2 m,2 1,2 2 ,2_{. We show that d}

i j, is equal to the degree of the variable yi j, for £ £i m

1 j and1£ £j 2. For simplicity, we set =i j=1 and denote the degree of y1,1with d . Since 1is generated by monomials, each monomial in a polynomial in 1is divisible by one of its monomial generators. So we getd1,1£d. On the other hand, since y1,1d1,1is a member of 1there is a positive degree invariant with a monomial that divides y_1,1d1,1_{. So by the minimality of d , we get}

£

d d1,1 as well. By Lemma3.1, di j, is also equal to the degree of yi j, in V[ Å1 V2]G. We claim that the union of the generating sets for 1 and 2 generate . Otherwise, there exists a poly-nomial f in  that contains a non-constant mopoly-nomial_{1£ £i m,1£ £j 2}y_{i j}e_,

j

i j,

with0£ei j, <di j,. Let Di j, denote the derivation with respect to the terminal variable yi j,. Then applying Di j

e ,

i j,

successively to f for1£ £i mj, 1£ £j 2yields an invariant with a non-zero constant. This is a contradiction

by Lemma2.4sinceei 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:

EXAMPLE3.3. LetH= ás t, ñbe the Klein 4-group,  a field of characteristic 2 andm³2. Let W ( )-m _{ be a vector space of dimension m=2n+1 over . Choose a basis {x x, ,¼,x ,y,}

m

1 2 1

¼ ₊}

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 s( ) =yj yj +xj,s( ) =xj xj,t( ) =yj yj +xj-1,t( ) =xj xj using the convention that x0= =

+ xm 1 0.

The variablesy y1, 2,¼,ym+1are terminal. One can readily check that

+ + + + +

y₂2 x y_{2 2} x y_{1 2} x y_{2 1} x y_{1 3} y x_{1 3}

is invariant under H (note the last term is zero ifm=2), so y₂ has degree 2. On the other hand, y₂ is not fixed by either s or t, which means NH( )y

2 has degree 4. It is interesting to note that

+ + + Î 

x y_{1 2} x y_{2 1} x y_{1 3} y x_{1 3} , so we still havey₂2 Î .

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. LetG=Zprdenote a cyclic group of order pr. Fix a generator s of G. There are pr indecom-posable G-modulesV1,¼,Vprover , and each indecomposable module Vi is afforded by s-1acting via a Jordan block of dimension i with ones on the diagonal. For an arbitrary G-module V , we write

= ( £ £ )

=

V V with 1 n p for alli ,

i k n i r 0 i

⨁

where each Vni is spanned as a vector space by e1,i,¼,en ii,. Then the action of s

-1 _{is given by} s ( ) =- +

+ ej i ej i ej i 1

, , 1, for1£ <j niand s-1(en ii,) =en ii,. Note that thefixed point space V G _is -linearly spanned byen1,0,¼,en kk, . The dual Vn* is isomorphic to Vi ni. Let x1,i,¼,xn ii, denote the

corresponding dual basis, then we have

[ ] = [ £ £ £ £ ]

 V  xj i, ∣1 j ni, 0 i k ,

and the action of s is given by s (xj i,) =xj i, +xj-1,i for 1< £j ni and s (x1,i) =x1,i for £ £i k

0 . Notice that the variables xn ii, for0£ £i k are terminal variables. We follow the nota-tion of Secnota-tion2and denote xn ii, with xi. We show that Theorem1.2applies completely to G by computingdeg( )xi explicitly for0£ £i k. For each0£ £i k, let ai denote the largest integer such thatni>pai-1.

LEMMA4.1. We havedeg( ) =xi pai. In particular, we may takeN x( ) =i NG( )xi.

Proof. From [7, Lemma 3], we get that deg( )xi is at least pai. On the other hand, since

³ >

-pa n p

i a 1

i i _{, a Jordan block of size ni has order p}ai_{. That is, this block affords a faithful}

module of the subgroup of G of size pai_{. It follows that the orbit of xi has p}ai _{elements and so}

that the orbit productNG( )x

i is a monic polynomial that is of degree paiin xi. □ Applying Theorem1.2, we obtain the following.

PROPOSITION 4.2. Assume the notation of Theorem 1.2with specializationG=Zpr. We have an isomorphism [ ¼ ] @ [ ¼ ]/( ¼ )  X, ,Xk  t, ,tk tp , ,tk . p 0 0 0 a0 _ak

Moreover, [ ] VGis free as a module over X[ 0,¼,Xk]. □

Now let H denote the Klein 4-group andp=2. For each indecomposable H -module V , there exists a basis of V * with one of the terminal variables xi satisfyingdeg( ) = [xi H:Hxi], see [10].

In this source, it is also proven that with the exception of the regular module, each basis consists offixed points and the terminal variables, and the Hilbert Ideal of every such module is generated byfixed points and the powers of the terminal variables. So we have by Theorems1.2and3.2: PROPOSITION4.3. Let V be a H -module containingk+1 indecomposable summands. There is a basis{x0,x1,¼,xn}of V * in which x0,x1,¼,xk are terminal variables, each coming from one summand, such that V[ ]H is free as a module over its subalgebra  generated by the images

¼

X0,X1, ,Xk of the terminal variables. Moreover,@ [ ¼ ]/( t , ,tk ta,¼,tk ) a

0 00 k , wheret0,¼,tkare independent variables, and for each i, we haveai=2or 4.

PROPOSITION4.4. Let V be a H -module such that V does not contain the regular module H as a summand. Then there exists a basis of V * such that[ ]V₊H[ ]V is generated by powers of basis elements. In particular, [ ] V+H[ ]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 offinite groups generated by reflections, Amer. J. Math. 77 (1955), 778–782.

3. E. Dufresne and A. Maurischat, On thefinite generation of additive group invariants in posi-tive characteristic, J. Algebra 324 (2010), 1952–1963.

4. J. Elmer and M. Kohls, On separating a fixed point from zero by invariants, Comm. Algebra 45(2017), 371–376.

5. N. J. Fine, Binomial coefficients modulo a prime, Amer. Math. Monthly 54 (1947), 589–592. 6. R. Kane, Poincaré duality and the ring of coinvariants, Canad. Math. Bull. 37 (1994), 82–88. 7. M. Kohls and M. Sezer, Degree of reductivity of a modular representation, Commun.

Contemp. Math. 19 (2017), 1650023, 12.

8. M. Sezer, Decomposing modular coinvariants, J. Algebra 423 (2015), 87–92.

9. M. Sezer and R. J. Shank, On the coinvariants of modular representations of cyclic groups of prime order, J. Pure Appl. Algebra 205 (2006), 210–225.

10. 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.

11. G. C. Shephard and J. A. Todd, Finite unitary reflection groups, Canadian J. Math. 6 (1954), 274–304.

12. L. Smith, A modular analog of a theorem of R. Steinberg on coinvariants of complex pseudo-reflection groups, Glasg. Math. J. 45 (2003), 69–71.

13. R. Steinberg, Differential equations invariant under finite reflection groups, Trans. Amer. Math. Soc. 112 (1964), 392–400.

14. R. Tanimoto, An algorithm for computing the kernel of a locallyfinite iterative higher deriv-ation, J. Pure Appl. Algebra 212 (2008), 2284–2297.