Structure and detection theorems for <span class="mathjax-formula">$k[C_{2}\times C_{4}]$</span>-modules

Structure and Detection Theorems for k[C

C

]-Modules

SEMRAOÈZTUÈ RKKAPTANOGÏLU(*)

ABSTRACT- Letk[G] be the group algebra, whereGis a finite abelianp-group andk is a field of characteristicp. A complete classification of finitely generatedk[G]- modules is available only whenGis cyclic,Cpⁿ, orC2C2. Tackling the first interesting case, namely modules over k[C2C4], some structure theorems revealing the differences between elementary and non-elementary abelian group cases are obtained. The shifted cyclic subgroups ofk[C2C4] are char- acterized. Using the direct sum decompositions of the restrictions of a k[C2C2]-moduleM to shifted cyclic subgroups we define the set of multi- plicities ofM. It is an invariant richer than the rank variety. Certain types of k[C2C4]-modules having the same rank variety ask[C2C2]-modules can be detected by the set of multiplicities, whereC2C2is the unique maximal ele- mentary abelian subgroup ofC2C4.

1. Introduction.

LetMbe a finitely generatedk[G]-module, whereGis a finite group of order divisible bypandkis a field of characteristicp>0. The cyclic group of order n is denoted byC_n. WhenM is considered as a module over a subalgebrak[A] ofk[G] for a subgroupAof the group of units ofk[G], we writeM#_A, and refer to it as therestrictionofMtok[A] (orA) or simply as the restricted module. Some properties of a k[G]-module M, such as complexity, are detected by M#_E as Eruns through elementary abelian subgroups ofG; see [AE], [Kr]. Theorems of this nature are referred to as detection theorems.

The rather rich theory for modules over an elementary abelian p- group Eis not of much use when the group is non-elementary abelian.

(*) Indirizzo dell'A.: Mathematics Department, Middle East Technical University, Ankara, 06531, Turkey.

E-mail: [email protected]

2000 Mathematics Subject Classification. Primary 20C05; Secondary 13C05, 13D02.

An essential motivation for this work was to clearify the reasons for that by presenting a detailed study of modules over the smallest non-ele- mentary abelian 2-group C₂C₄ via their restrictions to various sub- algebras of k[C₂C₄] and k[C₂C₂] where C₂C₂ is the unique maximal elementary abelian subgroup of C₂C₄. Given the fact that k[C2C4] is of wild representation type [Be1, p. 114] what we provide in this article as structure theorems is essential for studying modules over it. It should be noted here that the only abelian non-cyclicp-group whose indecomposable modules are classified is the group C₂C₂; [Ba], [HeRe], [Co]. There are infinitely many non-isomorphic indecomposable k[C₂C₂]-modules.

The work of Carlson defining the rank variety for a module over an elementary abelianp-group is the main source of inspiration for this study [Ca]. However, what we achieve is the result of our new approach; namely, rather than just focusing on whether the restrictions of ak[E]-module are free or not as Carlson does, we further take into account the direct sum decomposition of the restrictions of ak[C2C4]-module to preserve more information so that the module may be characterized by that information.

The direct sum decompositions of the restrictrictions of akG-module atp- points, roughly speaking order p-subgroups, where G is a finite group scheme are studied througly by Friedlander, Suslin and Pevtsova in [FP], [FP1], [SFP]. Examples in an earlier version of this work [Ka1] were mentioned in [SFP]. They consider only cyclic subgroups of order p whereas we consider cyclic subgroups of orderpⁿ,n1. As a result, we can distinguish some modules which are not possible to distinguish by considering only orderpsubgroups.

The restrictions we consider areM#_h1‡xiforxin the Jacobson radical J_G, or simply J, of the group algebra k[G]. The structure theorems, Theorems 4.1, 4.3, 4.5, reveal the structure of the restrictionsM#_h1‡xifor various x. They provide a good insight why modules over elementary abelian groups behave better than modules over non-elementary abelian groups, and indicate what type of changes in the hypotheses lead to similar results when the group is non-elementary abelian. One con- sequence is the characterization of the shifted cyclic subgroups of k[C₂C₄], see (7).

By the formula provided in Corollary 2.2, it is not difficult to compute the multiplicities ofi-dimensional indecomposable summands ofM#_h1‡xi. Hence the direct sum decomposition ofM#_h1‡xican be determined for any xinJ without much difficulty. This leads us to define a new computable invariant,N_G(M), for ak[G]-moduleMcalled theset of multiplicitiesofM

whereGis an abelianp-group see (3). This definition becomes very useful when the group isC2C2 since the domain of definition,J=J², coincides with that of the rank variety, see Definition 1.5. Recovering the rank variety from the set of multiplicities is very easy.

Even though it is not discussed in this paper, we can state that whenG is C2C2C2 or C3C3 the domain of definition of NG(M) becomes J=J³ rather than J=J². That also justifies our choice of groups for this study.

Our detection theorems, Theorems 5.1 and 5.3, are applications of this invariant. They show that certain types of modules can be identified by their set of multiplicities. There is a geometric interpretation of the mul- tiplicities of the indecomposable summands of M#_C whenM is a ``realiz- able''k[G]-module andCis a cyclic 2-subgroup of a groupG[Ka].

In order to state our main results we need to introduce our notation and definitions some of which are similar to the ones in [Ca]. In doing this we also recall some results from the literature to put our results into the proper context. Recall that the notions of projectivity, injectivity and freeness coincide for modules overk[G] whenGis ap-group andkis a field of characteristicp.

The key result concerning the restricted modules M#_h1‡xi is Dade's Lemma [Da, Lemma 11.8] when the group Eis an elementary abelianp- group.

DADE'SLEMMA1.1. Suppose E is an elementary abelian p-group of order pⁿ generated by e₁;. . .;e_n, k is an algebraicaly closed field of ca- haracteristic p, and M is a finitely generated k[E]-module.Then, M is free if and only if M#_h1‡xiis free for all x of the forma1(e1 1)‡. . .an(en 1) where(a1;. . .;an)2kⁿ.

Another result in these lines is the following (Lemma 6.4 in [Ca]).

THEOREM 1.2. Let M be a k[E]-module, and x, y in JnJ² such that xy…modJ²†.Then M#_h1‡xiis free if and only if M#_h1‡yiis free.

DEFINITION1.3. A (cyclic) subgroupSof the units of k[G] is called a shifted (cyclic) subgroupofk[G] wheneverk[G]#_Sis free as ak[S]-module.

Dade's Lemma allowed Carlson to define the shifted cyclic subgroups for an elementary abelian p-group E. In our notation they are the sub- groupsh1‡xiof k[E], wherexis inJnJ². Dade's Lemma together with

Theorem 1.2 made the definition of V_E^r(M), the rank variety of M well defined. It consists of pointsx in J=J² for which M#_h1‡xi is not free to- gether with the point zero.

In order to describe the shifted cyclic subgroups of k[C₂C₄] ex- plicitly, we need to introduce notation for the generators of the group C2C4.

NOTATION1.4. Except in Section 2, in the rest of this article,Gdenotes the groupC₂C₄generated bye,f of orders 2 and 4, respectively;Ede- notes its unique elementary abelian subgroupC₂C₂. Consider the ideals J⁽²⁾J⁽³⁾ofkGcontained inJ²so that

J=J⁽²⁾k(e 1)k(f 1)k(f 1)² k³;

J=J⁽³⁾k(e 1)k(f 1)k(f 1)²k(f 1)³k⁴: …1†

We omit the bars to simplify the notation. WhenM#_h1‡xiandM#_h1‡yihave the same indecomposable summands (up to isomorphism) together with the same multiplicities, that is, whenxandyhave the same Jordan canonical form, we write

M#_h1‡xiM#_h1‡yi

forx;yinJprovided thatkh1‡xiandkh1‡yiare isomorphic subalgebras.

Our main structure theorems are as follows.

THEOREM4.1. - (i)If x2J, then k[G]#_h1‡xiis free if and only if x62J⁽²⁾. (ii) If M is a k[G]-module, then M is free if and only if the re-

striction M#_h1‡xi is free for all x in JnJ⁽²⁾.

The first part of Theorem 4.1 is the analogue of [Ca, Lemma 6.1]. By that the shifted cyclic subgroups ofk[G] can be written in the formh1‡xi for anyxinJ=J⁽²⁾, see Remark 4.2. The second part of Theorem 4.1 is the counterpart of Dade's Lemma.

THEOREM 4.3. - Let M be a k[G]-module, x, y in JnJ⁽²⁾, and xy…modJ⁽²⁾†.

(i) If x²is not zero, then M#_h1‡xiM#_h1‡yi. (ii) M#_h1‡xiis free if and only if M#_h1‡yi is free.

The second part of the above theorem implies that the freeness of M#_h1‡xi is well defined modulo J⁽²⁾. In other words, J=J⁽²⁾ for G is the analogue of theJ=J²for the elementary abelian groups. In the terminology of [FP] this amounts to showing thath1‡xiandh1‡yiare in the same equivalence class asjh1‡xij-points ofk[G]. Moreover, by Theorem 4.5 the Jordan form ofxis well defined moduloJ⁽³⁾.

THEOREM 4.5. - Let M be a k[G]-module, x, y be in JnJ², and xy…modJ⁽³⁾†.Then

M#_h1‡xiM#_h1‡yi:

The analogue of Theorem 4.5 for the groupEis given below.

THEOREM4.7. -Let M be a k[E]-module and let x, y be in JnJ²such that xy…modJ²†.Then

M#_h1‡xiM#_h1‡yi:

Due to the fact that all indecomposable k[C2]-modules, up to iso- morphism, arekandk[C2],M#_h1‡xi(k)^h1(x)(k[h1‡xi])^h2(x)whenh1‡xi is isomorphic toC2.

Thus the pairh(x)ˆ(h₁(x);h₂(x)) consisting of the multiplicities of the indecomposable summands describes the restricted moduleM#_h1‡xi up to isomorphism completely and h_i(x)'s are easy to compute by the formula given in Corollary 2.2(i).

DEFINITION1.5. For ak[E]-moduleM, theset of multiplicitiesis de- fined as

N_E(M)ˆn

x;h(x) x2J=J²n0o

; it is well defined by Theorem 4.7.

The rank varietyV_E^r(M) can be recovered from the set of multiplicities N_E(M) easily, see Remark 4.4.

DEFINITION1.6. Ak[H]-moduleMis called anisotypical k[H]-module (of type N) wheneverMis isomorphic tomcopies of an indecomposable k[H]-moduleNfor somem1.

THEOREM 5.1. - If M is a finitely generated isotypical k[E]-module, then NE(M) determines M completely (up to isomorphism) except that

isotypical modules of type Vⁿ(k) and of type V ⁿ(k) can not be dis- tinguished, whereVⁿ(k)denotes the n-th Heller shift (or n-th syzygy) of the trivial module k.

Following the definition and notation given in [Be2, p. 190], for z in H^l(H;k),Lzdenotes the kernel of thek[H]-homomorphism that represents the image ofz under the isomorphismH^l(H;k)Hom_k[H](V^l(k);k) ; see (4). We writeL_z instead ofL_zwhen the group isG.

We set the cohomology algebras as follows:

H(G;k)ˆk[t1;t]L(v); and z_aˆa2t²₁‡a1t;

H(E;k)ˆk[t1;t2]; and j_aˆa2t1‡a1t2; …2†

whereaˆ(a₁;a₂) is ink²nf(0;0)g, andt₁,t₂,vare of degree 1, andtis of degree 2. It can be shown that the rank varieties of the modulesLzaand of the induced modulesLja"^G_Eare the same; namely, the line throughaand the origin. However, restrictions of these modules to the shifted cyclic subgrouphu_aiofk[E] corresponding to the pointaare have different de- compositions. Thus these modules can be distinguished by their sets of multiplicities when considered as modules overE, see (9) for the compu- tation ofNE(L_jⁿ_a).

THEOREM5.3. -If M is a finitely generated isotypical k[G]-module of typeL_zⁿ_a or induced from an isotypical k[E]-module of type L_jⁿ_a, then M is completely determined (up to isomorphism) by its set of multiplicities NE(M#_E)when considered as a k[E]-module.

As the last theorem of the introduction we state a well known theorem due to Chouinard [Ch] used several times in the article.

THEOREM 1.7 (Chouinard). Let H be a finite group, M be a k[H]- module.Then M is a free k[H]-module if and only if M#_A is a free k[A]- module for all elementary abelian p-subgroups of H.

The proofs of our theorems are self-contained and use linear algebra methods and basic homological algebra techniques. Section 2 is devoted to results that are valid for any abelian p-group. ThereafterGis the group C₂C₄. Section 3 consists of preliminary lemmas fork[C₂C₄]-modules.

Section 4 contains the structure theorems. Section 5 is devoted to examples and applications of our multiplicities set which are referred as detection theorems.

2. General Results on Restrictions of Modules.

In this section,Gdenotes an abelianp-group. WhenGis the cyclicp- group of order pⁿ, the list of indecomposable k[G]-modules (up to iso- morphism) is given by thei-dimensional vector spaces which are ideals of k[G], namely, theJ^{pn i}, foriinf1;. . .;pⁿg. Hence for any abelianp-group G, a finitely generated k[G]-module M, and x in J_GnJ²_G with kh1‡xi k[Cpⁿ] we have the direct sum decomposition

M#_h1‡xi(k)^h1(x)(J^{pn 2})^h2(x) (J)^hpn1(x)(kh1‡xi)^hpn(x) whereh_i(x), or simplyh_i, denotes the multiplicity of thei-dimensional in- decomposable summands of M#_h1‡xi, and JˆJ_h1‡xi. Thus the module M#_h1‡xi can be represented by h(x)ˆ(h₁(x);. . .;h_pn(x)). Theseh_i(x)'s are easy to compute by our formula given in Corollary 2.2, however it is not clear how to find the suitable ideal, sayIG, such that, congruence moduloIG

implies the equality h(x)ˆh(y) forx and y in J. If such an ideal is de- termined, then the set of multiplicities N_G(M) will be well-defined over J=I_G, where

N_G(M)ˆn

x;h(x) x2J=I_Gn0o : …3†

ForN_C2C4(M) see Remark 4.6(2).

LEMMA 2.1. Let X be a dd nilpotent matrix over a fieldF andh_t denote the number of tt Jordan blocks in the Jordan form of X.Then

(i) h_t ˆrank (X^{t 1}) 2rank (X^t)‡rank (X^t‡1)for t1;

(ii) the number of Jordan blocks in X of dimension less than or equal to t is

rank (X⁰) rank (X) rank (X^t)‡rank (X^t‡1);

the number of those of dimension greater than t is rank (X^t) rank (X^t‡1).

The proof of Lemma 2.1 depends on the fact that rank ([j_t]^r)ˆ t r; if r<t;

0; if rt;

(

where [jt] denotes thettJordan matrix belonging to the eigenvalue zero.

The following are immediate corollaries of Lemma 2.1.

COROLLARY2.2. Let G be an abelian group whose order is divisible by p, and M be a k[G]-module.If x is in J andh1‡xiis a cyclic subgroup of k[G]such that kh1‡xiis isomorphic to k[Cp^s], then

(i) M#_h1‡xi(k)^h1(J^{ps 2})^h2(J^{ps 3})^h3 (J²)^hps2(J)^hps1 (k[Cp^s])^hps, where

h_iˆdim (x^{i 1}M) 2dim (xⁱM)‡dim (x^i‡1M);

(ii) the number of indecomposable summands of M#_h1‡xiof dimen- sion less than or equal to t isdim (M) dim (xM) dim (x^tM)‡

‡dim (x^t‡1M), the number of those of dimension greater than t is dim (x^tM) dim (x^t‡1M).In particular, the number of non-free indecomposable summands of M#_h1‡xi is equal to dim (M)

dim (xM) dim (x^{ps 1}M)ˆdim (M^h1‡xi) dim (x^{ps 1}M).

(iii) M#_h1‡xi is free if and only if dim (xM)ˆp^s 1

p^s dim (M) if and only if dim (x^{ps 1}M)ˆdim (M)

p^s if and only ifdim (xM)‡

‡dim (x^{ps 1}M)ˆdim (M).

COROLLARY2.3. If G is an abelian group whose order is divisible by p, M is a k[G]-module, and x;y are in J, then the following are equivalent.

(i) The restrictions M#_h1‡xi and M#_h1‡yi have the same decom- position.

(ii) For all i,h_i(x)andh_i(y)are the same.

(iii) For all i,dim (xⁱM)anddim (yⁱM)are the same.

REMARK 2.4. Note that the number of non-free indecomposable summands of M#_h1‡xi is equal to the dimension of the Tate cohomology group

H^⁰(h1‡xi;M#_h1‡xi):

Lethuibe a shifted cyclic subgroup ofGand letzbe inHⁿ(G;k). Denote the image ofzunder the restriction map res^G_huibyz#_hui. Whenzis zero,L_zis defined as Vⁿ(k)V¹(k); otherwise L_z is the kernel of a k[G]-homo- morphism^zrepresentingz, i.e. it fits in the following short exact sequence

0 !Lz !Vⁿ(k) !^{^z} k !0:

…4†

For more information on Vⁿ,V ⁿ, andLz, we refer to [Be2, p. 190]. We writeVⁿ_G(k) if the group needs to be indicated.

LEMMA 2.5. Let G be an abelian p-group, hui be a shifted cyclic subgroup of k[G]andz be in Hⁿ(G;k)n0.Then V^m_huiVⁱ_hui(k)where i is 0, or 1 if m is even, or odd respectively.The following isomorphisms hold:

(i) Vⁿ_G(k)#_huiVⁿ_hui(k)(khui)^sfor some s inN.The same is true if n is replaced by n.

(ii) For some s and l inN,

(L_z)#_hui L_z#hui(khui)^s; if z#_hui6ˆ 0;

Vⁿ_hui(k)V¹_hui(k)(khui)^l; if z#_huiˆ0:

8<

:

PROOF. (i) follows from the definitions of Vⁿ(k),V ⁿ(k), and shifted cyclic subgroups, and the property that a short exact sequence remains short exact when restricted to khui. (ii) follows from the definition ofL_z

and part (i). p

3. Restrictions ofk[C2C4]-Modules.

Here we return to the groupGˆC₂C₄. This section is devoted to answers to various questions that naturally arise in studying ak[G]-module through its restricitions to shifted cyclic subgroups. In the rest of this section,Bis the following basis

…5† B ˆf1;e 1;f 1;f² 1;(f 1)³;(e 1)(f 1);(e 1)(f 1)²;(e 1)(f 1)³g:

Some properties of the nilpotent elements of the group algebrak[G] are given in Lemma 3.1, Theorem 3.2, and Lemma 3.3.

Thenorm elementnHof a group algebrak[H] is the sum of all elements ofH. WhenHis a cyclic group generated byhof orderpⁿ the norm ele- mentn_Hˆ P

g2Hgcan also be expressed as (h 1)^{pn 1}.

LEMMA 3.1. Let x, y be in Jn0and xy…modJ²†.Then the following hold.

(i) x⁴ˆ0, and x²ˆa(f² 1)for some a in k.

(ii) If x is in J², then x²ˆ0.

(iii) If x is in J³, then xJ²ˆ0.

(iv) x³ˆ0if and only if x²ˆ0.

(v) x²ˆ0if and only ifh1‡xi C2(k[G]).

(vi) x²6ˆ0if and only ifh1‡xi C₄(k[G]).

(vii) x²6ˆ0if and only if k[h1‡xi]k[C4].

(viii) kh1‡xiis a direct summand of k[G]#_h1‡xi.

(ix) x²ˆy², x³ˆ0 iff y³ˆ0, h1‡xi  h1‡yi, and kh1‡xi 

kh1‡yi.

PROOF. Writexandyusing the basisBgiven in (5) as

xˆa(e 1)‡b(f 1)‡c(f 1)²‡r₁n_hfi‡r₂(e 1)(f 1)‡r₃n_E‡r₄n_G yˆa(e 1)‡b(f 1)‡c(f 1)²‡s1n_hfi‡s2(e 1)(f 1)‡s3n_E‡s4n_G …6†

for somea;b;c;ri;siink. Then we have

x²ˆb²(f² 1) and x³ˆab²nE‡b³n5f>‡r2b²nG:

It is clear that the coefficientbplays a significant role. Proofs of (i)-(ix) are

now easy verifications. p

The restricted module k[G]#_hf²_i is free as hf²i is a subgroup of G.

However, (f 1)²is zero inJ=J². Thus it is clear that we need a substitute forJ=J²to extend Dade's Lemma. By Lemma 3.1 the non-zero elements of Jare of two types depending on whether their square is zero or not. The set of elements of J whose square is zero is denoted by JE, that is, JEˆJEJ⁽²⁾ˆk(e 1)k(f² 1)J⁽²⁾, whereJ⁽²⁾is as given in (1).

THEOREM3.2. Let x, y be inJ_EnJ⁽²⁾such that xy…modJ⁽²⁾†.There is a unit uˆ1‡n with n in JEnJ⁽²⁾ such that if Exˆ hu;1‡xi and Eyˆ hu;1‡yi, then

(i) the groups Ex, Eyare shifted subgroups of k[G]which are iso- morphic to E,

(ii) the group algebras k[E], k[E_x], k[E_y] are isomorphic sub- algebras of k[G].

PROOF. (i) By hypothesis,xˆx‡J⁽²⁾is non-zero. Then there is annin J_Esuch thatfx;ngis a basis forJ_E=J⁽²⁾. We can writexandyas in (6) with bˆ0. Let

nˆne(e 1)‡n_f²(f 1)²‡n1(f 1)³‡n2(e 1)(f 1)‡n3nE‡n4nG; wheren_e;n_f2;n_iare ink. By hypothesis, (a;c) and (n_e;n_f2) are not equal to (0;0). Moreover,xandnarek-linearly independent moduloJ⁽²⁾. This is true if and only ifan_f² cneis non-zero. Since the field is of characteristic 2,

an_f²‡cneˆan_f² cneis non-zero. Hence

nxˆ(an_f²‡cn_e) 1‡an₁‡cn₂‡r₁ne‡r₂n_f² an_f2‡cn_e (f 1)

n_E

is not zero. Set

v_xˆ1‡an₁‡cn₂‡r₁n_e‡r₂n_f²

an_f2‡cn_e (f 1):

It is clear thatv_xis a unit ink[G]. Sincenis inJ⁽²⁾, we haven²ˆ0 by Lemma 3.1 (ii). Therefore uˆ1‡n is a unit of order 2 in k[G]. Note also that u(1‡x)ˆ1‡x‡n‡xnis a unit ink[G] different from 1‡xand 1‡n.

Then we obtain that E_xˆ hu;1‡xi is isomorphic to E, and that k[E_x]ˆkknkxknx is contained in k[G], where nxˆn_Ex is non- zero. We also have

nExk[G]ˆknEx‡k(f 1)nEx ˆkvxnE‡kvxnG;

which has dimension 2 as a vector space; hence the restrictionk[G]#_Ex is free. Similarly fory, the element

v_yˆ1‡an₁‡cn₂‡s₁n_e‡s₂n_f² an_f²‡cne (f 1)

is ink[G]. Thenn_Ey ˆ(an_f2‡cn_e)v_yn_Eis non-zero andk[G]#_Ey is free.

(ii) Definef_x:k[Ex] !k[E] by

f_x(u)ˆ1‡n_e(e 1)‡n_f²(f² 1)‡n₃n_E; f_x(1‡x)ˆ1‡a(e 1)‡c(f² 1)‡r₃n_E:

Thenf_x(n_Ex)ˆ(an_f²‡cne)n_Eis non-zero. Hencef_xis an isomorphism of group algebras. For similarly defined f_y, we have the equivalence f_x(x)f_y(y)…modJ_E²†. Thus we have the following equalities and iso- morphisms

JE=J²_Eˆkf_x(n)kf_x(x)ˆkf_y(n)kf_y(y)JEx=J_E²_x JEy=J²_Ey: p LEMMA3.3. Let x be in J.

(i) If x²is non-zero, thenJ⁽²⁾is contained in xJ.

(ii) If x is not in J², thenJ⁽³⁾is contained in xJ.

PROOF. Writexas in (6). Under the hypothesies of (i) or (ii)xJcontains (e 1)Jand (f 1)J.

(i) The hypothesis thatx²is non-zero implies thatbis non-zero; thenJ⁽²⁾ is a subset of (f 1)J.

(ii) By hypothesis, we have (a;b) different than (0;0). Ifb is not zero, then the first part implies the result, becauseJ⁽³⁾is contained inJ⁽²⁾. Ifais

not zero, thenJ⁽³⁾is a subset of (e 1)J. p

The following corollary is a special case of Lemma 2.5 and is essential for proving Theorems 5.1 and 5.3. The cohomology algebras are as given in (2) and

H(C₂;k)ˆk[t]; H(C₄;k)ˆk[t]L(v):

We also writekhgiinstead ofk[hgi] for simplicity.

COROLLARY3.4. (i)The Heller shiftVⁿ_C2(k)and L0are isomorphic to k for all n while Lt is zero.

(ii) The dimensions ofVⁿ_E(k) and L_j^m are 2n‡1and2m, respec- tively, where j is in H¹(E;k).Moreover, the following iso- morphisms hold:

Vⁿ_E(k)#_hubi(k)Vⁿ_hubi(k)(khubi)ⁿk(khubi)ⁿ and

(Lj^m_a)#_hubi (khubi)^m; if kfbg 6ˆkfag iff j^m_a#hubi6ˆ0;

(k)²(khubi)^{m 1}; if kfbg ˆkfag iff j^m_a#hubiˆ0:

(

(iii) When the group is C₄, L_t is the zero module, and the Heller shifts are

Vⁿ_C4(k) k; if n is even;

V¹_C4(k)J_C4; if n is odd:

(

(iv) When the group is G, the dimensions of the Heller shifts and Lz's forzHⁿ(G;k)are as follows:

dim (Vⁿ_G(k))ˆ 4n‡1; if n is even;

4n‡3; if n is odd;

and

dim (Lz)ˆdim (Vⁿ_G(k)) 1:

WhenL_zis restricted to E, it takes the form

L_z#_E L_z#E (k[E])ⁿ⁼²; if n is even;

L_z#E (k[E])^(n‡1)=2; if n is odd:

(

For z_a in degree2, the restrictions of Lza to the shifted cyclic subgroupshCgiof k[G]take the forms

Lz^ma#hCgi (khCgi)^4m; if z^m_a#hCgi6ˆ0;

(k)²(khCgi)^{4m 1}; if z^m_a#_hCgiˆ0; wherehCgi C2; (

and

Lz^m_a#_hCgi (khCgi)^2m; if z^m_a#_hCgi6ˆ0;

kV¹_hCgi(k)(khCgi)^{2m 1}; if z^m_a#_hCgiˆ0; wherehCgiC4: 8<

:

PROOF. (i) and (iii) follow from the fact that the minimalk[Cpⁿ]-free resolution ofkis of the form

!k[Cpⁿ] !k[Cpⁿ] !k !0 for anypandn.

(ii) follows from the classification of k[E]-modules (see [Ca]) and the definitions ofVⁿ(k) andL_z.

(iv) follows from the fact that the minimalk[G]-free resolution ofkis of the form

!(k[G])³ !(k[G])² !(k[G])¹ !k !0;

and the definitions ofVⁿ(k) andL_z. p

4. Structure Theorems.

In this section we prove our theorems for modules overk[G] andk[E]

for our fixed group GˆC₂C₄ and its unique elementary subgroup EˆC₂C₂.

Theorems 4.1 and 4.3 contain generalizations of Dade's Lemma and Carlson's analoguous theorems for modules over elementary abelian p- groups. Theorems 4.5 and 4.7 guarantee the well-definedness of the set of multiplicities of a module overk[G] andk[E], respectively. In the proofs we use Corollary 2.2 or 2.3 to determine whether a module is free or not, and Lemmas 3.1, 3.3 for the properties of the elements ofJ. We use the no- tation of (6) forxinJ.

THEOREM4.1. (i) If x2J, then k[G]#_h1‡xiis free if and only if x62J⁽²⁾. (ii) If M is a k[G]-module, then M is free if and only if the re- striction M#_h1‡xi is free for all x in JnJ⁽²⁾.

PROOF. Note thatxis not inJ⁽²⁾if and only if (a;b;c) in (6) is not equal to (0;0;0).

(i) By Corollary 2.2 (iii)k[G]#_h1‡xiis free if and only if dim (xk[G]) is 4 or 6 depending onbis zero or not, respectively. Computing the rank of the matrix representingxshows that this dimension requirement is satisfied as (a;b;c)6ˆ(0;0;0).

(ii) SupposeMis a freek[G]-module andx2J. By (i)k[G]#_h1‡xiis free if and only ifx62J⁽²⁾, henceM#_h1‡xiis free for allx62J⁽²⁾. Conversely assume that M#_h1‡xiis free for allx2JnJ⁽²⁾. As usual, letEdenote the maximal elementary abelian subgroup ofG. SinceJEJ andJ²_EJ⁽²⁾,M#_h1‡ziis free for allz2J_EnJ²_E. Then by Dade's LemmaM#_Eis a free module. Hence Mis a freek[G]-module by Chouinard's Theorem, Theorem 1.7, asEis the only maximal elementary abelian subgroup ofG. p REMARK 4.2. The first part of the above theorem implies that the shifted cyclic subgroups ofk[G] are of the formh1‡xifor anyxinJ=J⁽²⁾. If the natural one-to-one correspondance betweenk³andJ=J⁽²⁾is used (see (1)), then shifted cyclic subgroups ofGcan also be defined ashC_gifor each gˆ(g₁;g₂;g₃) ink³, where

C_gˆ1‡g₁(e 1)‡g₂(f 1)‡g₃(f² 1):

…7†

Obviously, hC_gi is isomorphic to C₄ if and only if g₂ is not zero;

otherwise, it is isomorphic to C₂. When g₂ is non-zero, then hC²_gi is h1‡g²₂(f² 1)i. It is clear that the lines through the origin in k³, parametrize the shifted cyclic subgroups of k[G]. However note that the points (0;1;1;) and (0;1;0) give the same group algebra, k[h1‡f 1‡f² 1i]ˆkh1‡f 1i]ˆkhfi]. Thus we can use points of k³ andJ=J⁽²⁾ interchangebly to write the shifted cyclic subgroups of k[G]. The latter definition is consistent with Carlson's definition of shifted cyclic subgroups of k[E], namely, huai for aˆ(a1;a2) in k², where u_a ˆ1‡a₁(e 1)‡a₂(f² 1). The muliplicities set in this no- tation can be written as N_E(M)ˆ

a;h(u_a 1) a2k²n0 .

THEOREM 4.3. Let M be a k[G]-module, x, y be in JnJ⁽²⁾, and xy…modJ⁽²⁾†.

(i) If x²is not zero, then M#_h1‡xiM#_h1‡yi. (ii) M#_h1‡xi is free if and only if M#_h1‡yiis free.

PROOF. Sincexis not inJ⁽²⁾(a;b;c) of (6) is not equal to (0;0;0), hence xy…modJ²†.

(i) Sincex² is not zero, Lemma 3.3 implies thatJ⁽²⁾is contained inxJ.

Thenyˆx(1 r) for somerinJasx yis inJ⁽²⁾. This proves the claim because 1 ris a unit.

(ii) The assumptionxy…modJ⁽²⁾†implies thatx²ˆy²(see Lemma 3.1 (ix)). In the case x² is non-zero, Theorem 4.3 (i) implies that M#_h1‡xiM#_h1‡yi. This, of course, implies thatM#_h1‡xiis free if and only if M#_h1‡yiis free.

In the case x² is zero, without loss of generality we can write xˆa(e 1)‡c(f² 1) and yˆx‡w where w2J⁽²⁾. Note that y²ˆ0 and we can writewˆqrfor someq,rinJwithq²ˆ0,r⁴ ˆ0.

Letz_Mdenote the map formMtoMgiven byz_M(m)ˆzmfor allmin M,zinJ.

SupposeM#_h1‡xi is free. Sincex²ˆ0, we have ker (x_M)ˆxM.

Claim: M#_h1‡yi is free. Let Nˆker (yM)=yM, yˆqr with q²ˆ0, r⁴ˆ0 as written above. It remains to show thatNˆ0. Consider the map r_N:N !N, multiplication byr. Letm be in ker (r_N). Then we havemin ker (y_M) andrminyM. Thus,ymˆ(x‡qr)mˆ0 andrmˆynfor somen in M. Since q²ˆ0, we obtain xmˆ qrmˆqrmˆq(x‡qr)nˆxqn.

Hence m‡qn is in ker (xM). Thusm‡qnˆxs for somes in M. Multi- plying the last equation by r we obtain that rmˆrqn‡rxs. We had rmˆynˆxn‡qrn above, hence rxsˆxrsˆxn. Thus n rs is in ker (x_M)ˆxM. Hence we can writenˆrs‡xtfor sometinM. We have mˆqn‡xsˆq(rs‡xt)‡xsˆqrs‡qxt‡xs‡q²rtˆ(x‡qr)(s qt)ˆ

ˆy(s qt). Thusm ˆ0, showing thatrNis injective. On the other handr is nilpotent, this forcesNto be zero which proves the claim.

Similarly,M#_h1‡yiis free implies thatM#_h1‡xi is free. p REMARK4.4. SupposeMis ak[G]-module. Then

V_E^r(M#_E)ˆ f0g [

x2J_E=J²_Eh₂(x)6ˆdim (M)=2 :

The restrictionM#_hCgiis free if and only if the restrictionM#_hC²_giis free by Chouinard's Theorem, Theorem 1.7, which holds if and only if the restric- tionM#_hf²_iis free. Note thath1‡f²iis the shifted cyclic subgroup ofk[E]

corresponding to the point (0;1).

THEOREM 4.5. Let M be a k[G]-module, x, y be in JnJ², and xy…modJ⁽³⁾†.Then

M#_h1‡xiM#_h1‡yi:

PROOF. By Lemma 3.3, we know that J⁽³⁾ is contained in xJ. Then yˆx(1 r) for somerinJasx ybelongs toJ⁽³⁾. This proves the claim as

1 ris a unit. p

REMARK 4.6. (1) The hypothesis x²6ˆ0 in Theorem 4.3 (i) can be replaced by any one of the (a) - (e) : (a) xyˆ0 and x²ˆ0, (b) xy…modkn_Ekn_G†,(c)M#_heiorM#_hfiorM#_hf2iis the trivial module,(d) M#_hfi has no free summands,(e)(e 1)(f 1)Mˆ0.

(2)Note that Theorem 4.5 makes it clear that the set of multiplicities is well-defined for ak[G]-module when we takeI_GˆJ⁽³⁾in equation (3). That is, for ak[G]-moduleM, itsset of multiplicitiesis defined as

N_G(M)ˆn

x;h₁(x);h₂(x);h₃(x);h₄(x) x2J=J⁽³⁾n0o

;

note thath₃(x)ˆh₄(x)ˆ0 forxinJwithx²ˆ0. Further, we define afil- trationof it by the subsets

Nⁱ_G(M)ˆn

x;h₁(x);h₂(x);h₃(x);h₄(x) 06ˆx2J=J⁽³⁾; h_j(x)ˆ0 forjio : It is obvious thatNⁱ_G(M) form a nested sequence

N¹_G(M) N²_G(M) N³_G(M) N⁴_G(M);

…8†

andNⁱ_G(M)nN^i‡1_G (M) gives thosex's for whichM#_h1‡xihas only i-dimen- sional indecomposable summands. Recall that the Loewy length of a non- free indecomposablek[G]-module is at most 4, and the Loewy length ofMis iif and only ifh_i(x) is non-zero for somexinJnJ², which in turn holds if and only if N_G(M)ˆ Nⁱ_G(M). For a non-free k[G]-module M, we can write N_G(M)ˆ [_iNⁱ_G(M).

The analogue of Theorem 4.5 for the group E is given below which shows thatIEˆJ², see (3) for the definition ofIE.

THEOREM4.7. Let M be a k[E]-module and let x, y be in JnJ²such that xy…modJ²†.Then

M#_h1‡xiM#_h1‡yi:

PROOF. Recall thatEˆ he;f²i. Without loss of generality, assume that Mis non-free and indecomposable. Then the number ofk[E]-free summands inMis zero; equivalently (e 1)(f² 1)MˆJ²Mˆ0. Thereforexmˆym for allminMas (x y)mis inJ²M; thusM#_h1‡xiM#_h1‡yi. p

5. Detection Theorems and Examples.

This section is devoted to the applications of our definitions and theo- rems. These theorems are based on the observation that, when restricted toh1‡xi, the indecomposable k[E]-moduleLz have either two copies of the trivial modulekor none, see (9) below. The proofs uses only Corollary 3.4, and Theorem 4.7.

THEOREM5.1. If M is a finitely generated isotypical k[E]-module, then N_E(M)determines M completely (up to isomorphism) except that isotypical modules of type Vⁿ(k) and of type V ⁿ(k) can not be dis- tinguished, where Vⁿ(k) denotes the n-th Heller shift of the trivial module k.

PROOF. By the classification ofk[E]-modules given in [Ca], a finitely generated indecomposable k[E]-module is isomorphic to one of the fol- lowing:k,k[E],Vⁿ(k),V ⁿ(k), orL_zⁿ_a for each [a] inP¹_k, whereaˆ(a₁;a₂) is ink²n0, andnis a positive integer. The modules given have dimensions 1, 4, 2n‡1, 2n‡1, and 2n, respectively, as vector spaces overk. Thus an isotypicalk[E]-moduleMis isomorphic tomcopies of one of the modules listed for some positive integer m. First we compute the set of multi- plicities of each of the modules listed, then multiplying each multiplicity by m gives the set of multiplicities of M. We have the obvious iso- morphisms

k#_hubik and k[E]#_hubi(khubi)²;

whereu_bis a shifted cyclic subgroup ofk[E] andbis ink²nf0g. Therefore we have

N_E(k)ˆ f ‰b;1;0Šjb2k²n0g and N_E(k[E])ˆ f ‰b;0;2Šjb2k²n0g:

By Corollary 3.4 (ii), we know that the restrictionsVⁿ(k)#_hubiandV ⁿ(k)#_hubi are both isomorphic tok(k[hubi])ⁿ. Hence the set of multiplicities for any

two of them is f(b;1;n)jb2k²n0g. Therefore they cannot be dis- tinguished by the set of multiplicities. By Corollary 3.4, we have the iso- morphisms

L_jⁿ_a#_huai(k)²(k[C2])^{(n 1)} and L_jⁿ_a#_hubi(k[C2])ⁿ forbnot inkfagwhich is the rank varietyV_E^r(L_(za)ⁿ) ofL_(za)ⁿ. Hence

N_E(L_(ja)ⁿ)ˆ f‰a;2;n 1Šg [

‰(s;l);0;nŠj(s;l)2k²nf0;ag : …9†

Therefore the possibilities forNE(M) are (i) f[a;m;0]ja2k²n0g,

(ii) f[a;0;2m]ja2k²n0g, (iii) f[a;m;mn]ja2k²n0g,

(iv) f[a;2m;mn m]g [ f[(s;l);0;mn]j(s;l)2k²nf0;ag g.

It is clear what the module should be in the first two cases. The third one impliesMis eithermcopies ofVⁿ_E(k) orV_Eⁿ(k). In the fourth case,h₁ can be zero or non-zero. In this case,mˆh₁=2 and nˆ(2h₂‡h₁)=h₁ for the non-zero h₁ and the corresponding h₂. In each case m can be de- termined easily. ThusNE(M) determinesM(up to isomorphism) in each

case. p

EXAMPLE5.2. LetMbe ak[E]-module. Item 2) below shows that if the hypothesis isotypical is removed from Theorem 5.1, then its conclusion is no longer true.

1)LetMˆ(L_t3

1)²andM⁰ˆ(L_t2

1)³bek[E]-modules. Then by Corollary 3.4, M and M⁰ are both of dimension 12 and they have the same rank variety, namely, the linekf(0;1)g. However, [(0;1);4;4] is inNE(M) and [(0;c);6;3] is inN_E(M⁰). Thus their set of multiplicities are not the same.

2) Let MˆL_t²

1‡t²₂L_t⁵

1 and M⁰ˆL_t⁴

1‡t⁴₂L_t³

1 be two periodic k[E]- modules that are not isotypical. Then by Corollary 3.4,MandM⁰are both of dimension 14, they have the same rank variety, and they have the same set of multiplicities, namely,

‰(1;1);2;6Š;‰(0;1);2;6Š [

‰(s;l);0;7Šj(s;l)2k²nkf(1;1)g [kf(0;1)g : THEOREM5.3. If M is a finitely generated isotypical k[G]-module of typeL_zⁿ_a or induced from an isotypical k[E]-module of type L_jⁿ_a, then M is completely determined (up to isomorphism) by its set of multiplicities N_E(M#_E)when considered as a k[E]-module.

PROOF. Let aˆ(a1;a2) be in k² and dˆdim (M). Then the shifted cyclic subgroup huaiis isomorphic to C2. Suppose thatM ismcopies of L_(a2t²

1‡a1t)ⁿ for some m and a in k²n0. By Corollary 3.4 (iv), we know dim (M)ˆm(4(2n))ˆ8mn, and for zˆ(a2t²₁‡a1t²₂)ⁿ, we have the iso- morphisms

M#_E(Lz)^m(k[E])^mn; M#_huai(k)^2m(k[C2])^{4mn m}; M#_hubi(k[C2])^4mn: Hence

NE(M#E)ˆ

‰a;2m;4mn mŠ [

‰b;0;4mnŠjb2k²nfa;0g : Using the information on the non-free part, we obtain mˆh₁=2, nˆ2h2‡h1=4h1, and dim (M)ˆ8mn. Note thatjG=Ej ˆ2; then for

M(L_(a2t1‡a1t2)ⁿ)^m"^G_E; we have M#_k[E](L_(a2t2

1‡a1t²₂)ⁿ)^2m; where m and a are in k²n0. By Corollary 3.4 (iv), we know dim (M#_E)ˆ4mnˆdim (M), and forzˆ(a₂t²₁‡a₁t²₂)ⁿ, we have the iso- morphisms

M#_E(Lz)^2m; M#_huai((k)²(k[C2])^{n 1})^2m; M#_hubi(k[C2])^2mn: Thus

N_E(M#_E)ˆ

‰a;4m;2mn 2mŠ [

‰b;0;2mnŠjb2k²nfa;0g : Using the information on the non-free part, we obtain mˆh₁=4, nˆ2h₂‡h₁=h₁, and dˆ4mn. In each case, n andm, and hence M are determined (up to isomorphism) byNE(M#_E). p EXAMPLE5.4. If we drop the hypothesis isotypical, then Theorem 5.3 fails, see the items 1) and 2) below.

1) Let MˆL_t⁴

1‡t²L_t⁴

1 and M⁰ˆL_t²

1‡tL_t⁶

1. By Corollary 3.4, we know that they are both of dimension 32, and we have the isomorphisms M#ELt⁴₁‡t⁴₂(k[E])²Lt⁴₁(k[E])²; M⁰#ELt²₁‡t²₂k[E]Lt⁶₁(k[E])³: Their rank variety as a module overk[E] is a unionkf(1;1)g [kf(0;1)gof two lines. In addition, their sets of multiplicities are both equal to

‰…0;c†;2;15Š;‰…c;c†;2;15Šjc2kn0 [

‰…s;l†;0;16Šj…s;l† 2 k²nkf…1;1†g[kf…0;1†g :

2)LetMˆL_t⁴

1‡t⁴₂"^G_EL_t³

1"^G_EandM⁰ˆL_t²

1‡t²₂"^G_EL_t⁵

1"^G_E. SincejG=Ej ˆ2, we have the isomorphisms

M#_E(L_t⁴

1‡t⁴₂)²(L_t³

1)² and M⁰#_E(L_t²

1‡t²₂)²(L_t⁵

1)²: By Corollary 3.4 (ii), we know thatMandM⁰are of dimension 28, and their rank variety as a module over k[E] is the union of two lines, namely, kf(1;1)g [kf(0;1)g. In addition, their sets of multiplicities are both equal to the set

f ‰ …0;c†;4;12Š;……c;c†;4;12†jc2kn0g

[ f ‰ …s;l†;0;14Šj…s;l† 2k²nkf…1;1†g [kf…0;1†g g:

3)There are non-isomorphic indecomposablek[G]-modulesMandM⁰ such that their restriction tok[E] are isomorphic, henceNE(M)ˆ NE(M⁰).

LetMˆV¹(L_t²

1) andM⁰ˆL_t²

1. We know thatM⁰is of period 2, henceMis not isomorphic to M⁰. They both are isomorphic to L_t²

1k[E] when re- stricted tok[E].

Acknowledgments. I thank Amir Assadi for introducing this subject to me. It all started with his question whetherM#_h1‡xiandM#_h1‡yi had the same number of non-free indecomposable summands, whereMis a finitely generated module over an elementary abelianp-group, andx,yinJwith xy…modJ²†. I thank Peter May for reading this article, and for his suggestions making this article more appealing regardless of the techni- calities involved.

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Manoscritto pervenuto in redazione il 4 aprile 2009.