Indecomposable primitively generated abelian Hopf algebrasover a field

ANNALES SOCIETAT1S MATHEMATICAE POLONAE Series I: COMMENTATIONES MATHEMATICAE XXIV (1984) ROCZNIKI POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO

Séria I: PRACE MATEMATYCZNE XXIV (1984)

A

S

(Torurï)

Indecomposable primitively generated abelian Hopf algebras over a field

0. Introduction. Let К be a field of characteristic 2. By abelian Hopf algebra we mean strictly commutative, cocommutative, connected, graded Hopf algebra over К (cf. [12]-[14]). Let H be an abelian Hopf algebra and A the commultiplication of H. An element x of H will be called primitive if d(x) = x ® l-f l® x . If H is different from K, then each of its homogeneous elements of minimal positive degree is primitive. Hence it is useful to think of the primitives as counterparts of the socle of an abelian group. We see that primitively generated abelian Hopf algebras play an important role in the study of all abelian Hopf algebras. Another example of primitively generated abelian Hopf algebras offers the following result proved in [11]: if (R , m) is a local noetherian commutative ring and К its residue field, then the algebra ExtR(K, K) = 0 Extij(AL К ) is a noetherian primitively generated abelian

i = °

Hopf algebra if and only if R is complete intersection such that R = R/ I, where (R , m) is a regular local ring and / is an its ideal generated by an Я-sequence in m3.

Denote by I f the category of all primitively generated abelian Hopf algebras. Recall that I f is a locally noetherian Grothendieck category and L is noetherian object in I f iff Lis finitely generated as К -algebra (cf. [7], [9]).

From [9] it follows that I f — x I f +, where If ~ is the full subcategory of

& consisting of Hopf algebras generated by elements of odd degree and I f + is the full subcategory of I f of all Hopf algebras which are zero in odd degrees. Moreover, every object in I f ~ is a direct sum of objects of type nE

= К [x]/(x2), x primitive of degree 2и + 1, n e N — JO, 1, 2, ...}. The category У'+ is a countably infinite product of the full subcategory I f x of II}

consisting of all Hopf algebras generated, as K-algebra, by elements of degree 2p\ i e N (cf. [9]). Then the full information on the category I f is contained in the category I f x. If К is perfect, then by well-known Borel theorem every

°bject in If \ is a direct sum of objects of type Knr = К [ x ] /( /) , x primitive

°f degree 2p", r, n e N (cf. [7], [10]). If К is non-perfect the structure of

objects of I f \ is not known.

116 A. S k o w r o n s k i

in this paper we investigate the indecomposable primitively generated abelian Hopf algebras over a non-perfect field. Using some trees we give a complete description of so-called 2-indecomposable objects in УД (see Section 3 for definition). Recall that some special trees (see Example 2.5) were used in [12], [14] to the description of indecomposable injective objects in У and in the category of all abelian Hopf algebras. Recall also [6] that the class of 2- idecomposable objects includes the following classes of indecomposable objects: basic, cobasic, with waist, core and cocore. As an application of our result we obtain a description of the above classes of indecomposable objects in module categories over some Artin rings of wild representation type (cf.

[4], [8]).

Section 1 contains fundamental facts on the category УД needed in the paper. We show also some connection between primitively generated abelian Hopf algebras and representations of К -species.

In Section 2 we introduce special trees with ballasts and give some examples.

In Section 3, applying the notion and constructions of Section 2, we prove the theorem on the structure of 2-indecomposable primitively gen­

erated abelian Hopf algebras.

1. Abelian Hopf algebras and representations of species. Let К be a non­

perfect field of characteristic p and m the cardinality of its basis over Kp. For n e N , 1 < r < o c , we denote by "L the polynomial abelian Hopf algebra К [x] with x primitive, deg x = 2pn (degree of x), and put K nr — "ТДх^), nS

= K nl, where x x = 0. We denote by К the full subcategory of У \ consisting of all objects "L. By Mod-R and mod-R we will denote the category of right

R-modules and right finitely generated R-modules over given ring R.

We need the following results on the category УД :

(LI) °S, 2S, ... is a complete list of non-isomorphic simple objects in

^ ([9], p. 136).

(L2) °L, l L, 2L, ... is a family of indecomposable projective generators of УД ([14], Theorem 1.1).

(L3) Every projective object P in УД is isomorphic with a direct sum of objects nL, n e N, aivl any two such decompositions of P are isomorphic ([14], Corollary 1.2).

(L4) Every object in S£ x has a projective cover ([14], Theorem 1.1).

(L5) Global dimension o f УД is 1 and we have an equivalence T : УД -> (Jfop, Mod-Ю given by T(L) = Hom^1 ( —, L), where L is an object in УД ([14], Theorem 1.1).

Following P. Gabriel [3] a K-species S = (Fh iMj)itjel is a set of division

rings F, containing К in its center together with a set of F, —Fj-bimodules

,Mj such that К operates centrally on ,М,-. A representation X = {Xh ,•</>,) of

Abelian H opf algebras over a field

the К -species S is a set of right F,-modules together with Fy-homomorphisms j(p,\ for all i , j e l ,

where ® = ®Fj. The representations of a given К -species S form a Grothendieck category .^(S) in which a morphism (Xh j(pf) -+(!-, 7> () is given by a set of F(-homomorphisms щ : X {f -> Y{ satisfying

= Uj jCPi-

Now let us consider the following homomorphism of rings vt : К ->Fh where F, = End (*L), given by u;(a)(x) = cf1 x if lL — fC[x] and a e K . Then for /, j e /V, i ^ j, we have a К -module structure on = Horn (•'L, *L) given by the formula

M « )/ = a- f = fuj(a),

where/ eHom (JL, 'L), a e K . It is easy to check that the morphism compos­

ition in К is К -bilinear. Let us observe that F, — K, tDj = K, and the structure of F, — Fj-bimodule on ,Dy is given 'by the formulas a{-d

= a f 'd, d-cij = dcij, where 0 ,-eF,-, ajeFj, de^Dj. Hence for i , j e N , i ^ j , we have dim {iDj)F = 1 and dimf . (,£),•) = We obtain the following К -species

Sn (Fi S (F,-, jM

where

1 I 0 in the opposite case.

Moreover, for each n e N , denote by A„ the matrix ring

^ -^ 1 » 1 ^ 2 » • F2, •

• •? i D

w ‘ • F

where the needed F, — Fy-bimodule homomorphisms

^ijk • i ^ j ® j ^ k i ^ k

are given by сик{хи® х]к) = xfj~J xjk, i < j ^ к, ху е,Т>у, xjkejDk.

Finally, for each n e N , denote by JFln the full subcategory of consisting of all objects L which satisfy the condition: Horn (WL, L) — 0 for m > n + 1 .

We have the following connection between primitively generated abelian Hopf algebras, representations of К -species, and Л„-modules.

T heorem 1.1. There are equivalences of categories (i) sex * -#(S).

(ii) <eXn = R{Sn) = Mod-Ли, n e N .

118 A. S k o w r o n s k i

P ro o f, (i) The required equivalence is a consequence of (L5) and the correspondence between the functors F : J f’op-> Mod-K and the represen­

tations X = (Xh j<Pi) of S given by X-t = F('L), i + x(pi(xi® f i + x) = F (fu + 1 ) (*,-), where x ,e X ) , f i + l eHom (, + 1L, lL).

(ii) Using (L2) it is not hard to check that for each n e N the objects K 0n+X, K Xn, ..., K nl form a family of projective generators of S f\n. Hence we have an equivalence Ж Xn ^ Mod-К), where Ж „ denotes the full sub­

category of Ж* x consisting of objects X,„+1_l, i = 0, n. Then the proof is similar to the proof of (i).

On the other hand it is clear that A„ = End(K0„+1® ... ® K nX). Since K Qn+l® . . . ® K nl is a finitely generated projective generator of JZln then there is the required equivalence SFXn ^ Mod-Л „.

We finish this section by some remarks on the rings A„.

R e m a r k 1. The rings A„, n e N, are artinian if and only if m is finite.

R e ma r k 2. Assume m finite and fix n ^ 3. Then by [8], Theorem 2, and [4] there is a commutative field K' with К a K' c: Fh 0 ^ ^ n, and a full exact embeding

mod-Л -> Mod-Л,,,

where A is any finite dimensional K'-algebra. This shows that a classification of all indecomposable objects in the categories ЖХп, Ж would also furnish classifications of finitely generated modules over all finite dimensional K'- algebras: an impossible task. In Section 3 we will show that some large class of indecomposable objects in Я \ may be still described.

2. Special trees. Throughout this paper we assume that К is a non­

perfect field of characteristic p and we denote by m the cardinality of К over Kp.

Let G = (X, U) be a directed graph with a set of vertices X and a set of edges U (not necessarily finite). For each x e X denote by d£ (x) (resp. d j (x)) the cardinality of the set of all edges with the initial (resp. final) vertex x (cf.

[2], [12]). We say that a vertex x is a node (resp. input, output) iff c/g (x) ^ 2

(resp. dQ (x) = 0, dc (x) = 0). Denote by W{G), Z(G), 1(G) the sets of all

nodes, inputs, outputs of G. If for x, y e X there exists a chain from x to y,

then denote by dG(x, y) the distance from x to y. Whenever no confusion

arises we shall write simply d + (x), d~ (x), d(x, y), W, Z, I instead of dG (x),

dG (x), dG(x, y), W(G), Z(G), 1(G). For each y e X denote by X y the set of all

vertices x in G such that there exists a path from y to x. A path

((xi, x2), ..., (xfc_ l5 xk)) from x x to xk is said to be a branch if the following

condition is satisfied: for each 1 ^ r < k, x, is node iff t = 1 or t = k. A graph

G is said to be normal if for every path ((x1} x2), ..., (хл_!, x j) from x x to xk

and к > 2, (xl5 xk)$U. G is antisymmetric if (y, x)$U whenever (x, y)eU.

Abelian H opf algebras over a field

D e f i n i t i o n

2.1. A special tree is a connected, normal, antisymmetric

graph G = (A, V) without cycles satisfying the following conditions:

(1) for each y e X the set / n X y is not empty;

(2) d~ (x) 1 for each x e X ; (3) if x e W, then d~ (x) = 1.

Denote by 33 the family of all subsets of K.

D e f i n i t i o n

2.2. A quasi-ballast of a special tree G = (X, U) is the

sequence (<p, ф, ^ ), where ^ is a well order in / and (p : W -> /, ф: X -> N are two mappings satisfying the following conditions:

( 1 ) for each i , j e l , i < j whenever ф(Г} < фЦ);

( 2 ) d+(w) is the cardinality of the set tp(w) for each node w;

(3) if there exists a path with an initial vertex x and a final vertex y, then d(x, у) = ф(х)-ф(у).

Remark that from conditions (1) in 2.1 and (3) in 2.2 the mapping ф: X

— ► TV is uniquely determined by its values on the set I.

The special tree G = (X, U) with the quasi-ballast (<p, ф, we will denote by G = (X , U, <p, ф, ^ ).

Now we fix some notation needed in the paper. Let G

= (X, U, q>, ф, be a fixed special tree with a quasi-ballast (<p, ф, ^ ). For each n e N we put

X „= (x eX : ф(х) = n), W„ = W n X „ , I„ = I n X n.

Suppose that W is not empty. Applying induction on n ^ n0 = min \ф{х), w e W

we define for each w e Wn the set

and the element iwe l w. If w e Wn() we put

/ w = \ i e l ; there is a path from w to i]

and take iw to be the minimal element in Iw. If n > n0 and weVF„ put Iw

= Vw u where n — 1

I'w = {tV; w E U such that there exists a branch from w to W}, k =«0 k

1* = J ig /; there exists a branch from w to /},

and take iw to be the minimal element in Iw. It is clear that for each weW, d+ (w) is the cardinality of l w. We will use the following notation

q>(w) = \vWti; i e I w}, weW,

k\v,i vv £ W, i e l w.

1 2 0 A. S k o w r o n s k i

Let us define a function a : X -* I by

a{x) =<

x if x e l , ix if x e W ,

i(x) if хф1 и W and i (x) is an output such that there exists a branch from x to i(x),

iMx) if x $ I v W and w(x) is a node such that there exists a branch from x to w(x).

From weW, m e N , m < \J/(w), we define

Xw,m ^ X m> I\v,m ^ w ^ ^ m,

vm = _{J xeXm} f ® ^{к x} if X m is non-empty, I 0 if X m is empty.

( ® K x if X w<m is non-empty and m X*xw,m

к . if m = i^(vv),

1 0 ^otherwise.

0 K x,

xeXw,m^ w,m V

where K x — К for each x e X .

Furthermore we define the set mappings F • V

л m • w m W+ 1 > m e N ,

as follows: if v — Y ax eVm, then Fm(v) = Y by, where УеХт+1

x e X .

0 tfxkyMx) xeXy,m

if y e W m + 1,

< if (y, x)e U and y $ W m+1,

0 if y e l m+1.

Observe that Fm(Vw m) c Vw>m+1 if weW„, m < n, and denote by

* w,m ' 'w,m * ’'w,m + 1 the restriction of Fm to Vwm.

For weW, put /i(w) = min \n e N ; X Wt„ is non-emptyJ. It is clear that n(w) < «/'(w). Then for weW, r e N , n(w) ^ r < ip{w), we set

Г w,r — l • • • F Wtr) l { 0 ) .

We see that r wr is a К -linear subspace of Vw>r.

Abelian H opf algebras over a field

D

2.3. A quasi-ballast (tp, ф, of a special tree G =( X, U) is a ballast if r w?r cr V^r for each weW, r e N , fi(w) < \(/(w).

Denote by X the class of special trees with ballasts.

E

2.5. m-special trees with ballasts. An m-special tree with a

ballast is a special tree G = (X , U) with a quasi-ballast (tp, ф , <) such that d+ (w) < m and <p(w) is a subset of К consisting of vectors linearly indepen­

dent over Kp, for all weW. Recall that m-special trees with ballast play an important role in the description of indecomposable injective objects in У' and in the category of all abelian Hopf algebras (cf. [12], [14]). It is obvious that m-special trees with ballast are special trees with ballast satisfying the condition: r wr = 0 for we W, r e N , n(w) ^ r < ф(м).

E xample 2.6. Let m ^ 3 . Take elements a , b , c , d , e e K such that a, d, ефКр and 1, b, c are linearly independent over Kp. Let us consider the special tree G

We have / = [x0, x i, * 2 > *з> xi}-> W — xs) and Z = Define the well order in 1 putting

X 0 < X J < X 2 < X 3 < x 7 and the mapping ф: X —> N by

»M*o) = Ф(х 1 ) = Ф(х 2 ) = ф{х з) = О,

ФЫ = Ф(х5) = Ф{хь) = Ф(х

) = 1, ф{х8) = 2 ,

Ф(Х9) = 3.

Then we obtain / = (хь x 2 j, iXs = x l5 Finally, define the mapping cp: W -*■ ® putting

<?(*s) = \Vx5.Xl, Vx5.x2], <P(X8) = lVx8,x0, VX8,Xi, Vx8’xl ’ Vx8tX3 ’ Vx8’xl i ’

1 2 2 A. S k o w r o r i s k i

where vx$tXl = 1, vX5<X2 = a, vXgtX0 = L, vXgtXl = b, vXgtX3 = vX8<xl = c , f =

— dp — epb. Obviously (q>, ф, <) is a quasi-ballast of G. We shall show that it is also a ballast of G. In our notation the mappings

F0- К = K XQ® K Xl® K X2@KX3 -*■ Vx = К Х4ф К Х5@КХьф К Х1, F i : V i = К Х4Ф К Х5Ф КХ6® К Х1 - V2 = K Xs,

are defined by formulas

F0(u0, u1, u 2, м3) = « , «1 +up2a, up3, 0), F^Ut, u5, и6, m 7) = (up4- d pup6) + (up5- e pup6)b + upiC, where ще К, i — 0, 7. Hence

f x5,o = о c f ; 5>0,

ГХ8Л = K(d, e, 1, 0) cz К Х4Ф К Х5Ф КХ6 = к;8>1,

^ 8,„ = 0 <= к;8,0,

since а, ефКр, and the equalities up0 = et, и^ + up2a = ft, иръ — t, t e K , give u0

= Mj = u2 — u3 = 0. Consequently, ((p, ф, is a ballast of G. On the other hand, r xg l Ф 0, so the tree G = (X, U, (p, ф, is not m-special.

3. Indecomposable primitively generated abelian Hopf algebras. Following [1], [5], [6] we introduce a hierarchy of classes of indecomposable objects in the category L i .

Let L be an object in L l . For two subobjects M, N of L, by M + N we denote the minimal subobject of L which contains M and N. A subobject N of L is called non-superfiuous if there exists a proper subobject M of L such that M + N = L. The concept of non-superfiuous subobject is dual to the notion of a non-essential subobject of an object. Let n be a positive integer.

We say that L is n-indecomposable if the intersection of any its n + l non-

n + 1

superfluous subobjects iVl5 ..., N n+i such that £ jVf = L is different from

i = 1

К (К is zero object in L x). We say that L is oc-indecomposable if L is n- indecomposable for all n ^ 1. The object L has a core if the intersection of all its non-superfiuous subobjects is not K. This intersection, whether К or not, will be called the core of L and will be denoted C(L). Dually, the cocore of L, C°(L), is the sum of all non-essential subobjects of L. We say that L has a cocore if C°{L) Ф L. The object L is basic (resp. cobasic) if C(L) = L (resp.

C°(L) = K). Finally, we say that L has a waist if there exists a proper subobject W of L such that every subobject of L contains W or is contained in W.

The following proposition gives a complete description of all basic

objects in !£ x.

Abelian H opf algebras over a field

P

3.1. An object L in x is basic if and only if L ^ K nrfor a certain n s N , 1 ^ r ^ oc.

P ro o f. From [7], Proposition 7.8, we know that each subobject of "L

= К [x] is generated by for certain 1 ^ r ^ oo. Thus, the objects K„r are basic. Conversely, suppose L is basic and v : 0 is the projective

ie l

cover of L. Since v('jL)+ £ vÇlL) = L, we see that L = C(L) a v(njL) for

j * i

each j e / . Hence L is generated by one element and our hypothesis follows from (L2).

For the description of the remaining classes of indecomposable objects in S f x we define a representation of % in !£ x. Let G = (X, U, cp, ф, be a special tree with a ballast. In the notation of Section 2 we define the set of admissible relations o f G as the disjoint union

Ip ( w ) - 1

П G )= и и Г»,

weW (G) r=t i( w)

if W(G) Ф 0 and r(G) = 0 if W(G) = 0 . Let us consider also the set mapping

t : F ( G ) - 0 *<f>L

iel(G)

which assigns to each v — £ av e r w r the element

* 6 * w , r

*(*>) = Z ахУ£(х)Ф<Т(Х) if ^ L = K[_yif iel(G).

Now if Г is a subset of T(G), then by L(G, I) we denote the quotient primitively generated abelian Hopf algebra

(g) *(f)L/S{G, Г),

iel(G)

where S(G, Z) is the ideal of 0 Ф(1)Ь generated in the case Z ( G ) # 0 ie/(G)

by elements

(1) Ф ), veZ,

(

2

yfH'"-"n - k WJÿ C * ~ * W'>’ t e l „ w e W { G ) ,

(3) z e Z{ G)

and in the case Z(G) = 0 by elements of type (1), (2) only, where *(i)L

= К [у,] and 0 is the minimal element of 1(G). The image of y, by the natural epimorphism 0 ф{,)L -+L(G, Z) we will denote by .

ie l(G )

Notice that if W(G) = 0 , then L(G, 0 ) — K m t, where т — ф(0), t

124 A. S k o w r o n s k i

= ф(г) — ф(0), and conversely for n e N , 1 ^ r < со, K w = L(G„r, 0 ), where Gnr : 0 *- 1 r

and ф(0) = n.

For each G in X, put <5(G) = max [r e N ; # 0} if W Ф 0 and 0(G) = ф(0) if fF = 0 . Moreover, for each L in we denote by /(L) the ideal of L generated by all elements of positive degrees.

We are now able to prove the main result of this paper.

T

3.2. Let L be an object in T £ x . The following statements are equivalent :

(i) L is 2-indecomposable.

(ii) L is o c -indecomposable.

(iii) L % L(G, Z) for certain G in X and I <= T(G).

For the proof of the theorem we need two lemmas.

L

3.3. Let G e î , I c T(G), n e N, and a be a non-zero homogeneous primitive element of degree 2pn in L(G, Z). Then

(i) a has a unique expression in the form

Z

^х^а(х) ’ ^ К •

x e X „

(ii) I f a£/(L(G , l) ) 2, then o f ф 0 for all 0 ^ r < ф(г) — n.

P ro o f, (i) First observe that monomials is, ..., ixeI(G), j s, . . . , j xe N , s ^ l , which satisfy the conditions:

(a) is > .... > ix,

S

/ t v

V '

*<*r>

(b) 2] Jrp = P ’

r = 1

(c) for each 1 < t ^ s, the monomial is not linearly dependent (over К ) upon the monomials with m, < j t,

form a basis of L(G, Z)2p„ over K. Let

be an expression of a in this basis. Then the element a has the expression

Z

a j(r) < + b,

r= 1

, . п-фиг)

where aj(r) = a0...J(....0, j r = p

b = I “Jr-Jitf■■■*!!>

( j s , . . . J x )eNsa

and Nsa is the set of all tuples (js, . . . , j x) e N s such that j t Ф 0 Ф j r for certain

Abelian H opf algebras over a field

t Ф r, Y JrPФЬг) — Pn and ajs..Ji ^ 0. We will.show that Nsa is empty. It is

r — 1

clear if s = 1. Assume s ^ 2 and Nsa Ф 0 . Then if (ms, mj) is its maximal element in the lexicographical order, it is easy to check that the summand d

2 pn- 1

of A (a) which belongs to ® L(G, Z)i®L(G, I ) has the expression u p ®

- “ii + c>

where c contains no monomials of the form d u p ® u p f ...u p , de K.

On the other hand, A(a) = a®l + l ®a and we have a contradiction since am „ ^ 0- Consequently Nsa is empty. Then, by relations (2) in the definition of L(G, I), the element a has the required expression.

(ii) Let a = Y ax м£<*),M*) and аф1 (L(G, I))2. Since the set A

x e X n

= [xeX„; ax Ф 0} is finite, so from some node w in G there exists a path to each x in A. Thus a = Y ахиаП (х)'1'а{Х)• From our assumption A n l w n # 0 and consequently v — Y ахФ^,п- Hence ифГ^„ and

x e X w „

/ ( W )-Ww)

Q aP*'») - » = _ t . . . FwJ (v) uf

Therefore, ( f Ф 0 for 0 ^ r < i]/(z) — n, and the proof is complete.

L

3.4. Let L be an object in x. Then L is 2-indecomposable if and only if the intersection o f any two non-superfiuous subobjects of L is different from K.

P ro o f. First observe that if L is 2-indecomposable and M, N are non-superfiuous subobjects of L, then L + M + N = L and M n N

= L n M n N Ф K. Now suppose that any two non-superfiuous subobjects of Lhave non-zero intersection. Let M, N, P be non-superfiuous subobjects of L and M + N + P = L. Consider the projective cover / : (x) "*L-> Lof Land fix

n

ief

i e l . The object P ==/("‘L) is non-superfiuous in L, so the intersections M n R, N n R, P n R, are non-zero. But the subobjects of R are linearly ordered and we conclude that the intersection M n N n P is non-zero in . Thus L is 2-indecomposable and we are done.

P r o o f o f T h e o re m 3.2. Implication (ii) ->(i) is obvious. Now we show

(iii) ->(ii). Let G belongs to T and Г is a subset of T(G). Since the subobjects

of each "Lare linearly ordered, for our aim, it suffices to show that for any

non-zero homogeneous primitive elements a, b of L(G, I ) the intersection

(a)n(b) is non-zero in £ f lt where (a), (b) denote the abelian Hopf algebras

generated, as X-algebras by a and b respectively. Since L(G, Г) is an object

of deg a = 2pn, deg b — 2pm for suitable m, n e N . Thanks to Lemma 3.3

(И) we can assume m = n. If n ^ <5(G), then a = lb, I e K, and (я)п Ф ) * К .

126 A. S k o w r o n s k i

Assume n < <5(G). From Lemma 3.3 (i) the elements a and b have expressions

X

axupa{x) , b = 2 , bxupa(x)

x e X n x e X n

Put A — {xeX„; ax Ф 0} and B = |x eX „; ЬХ Ф 0}. The sets A and B are finite, n < <5(G), so from some node w in G there exist paths to all elements of A u B . Using Lemma 3.3 (ii) again, we conclude that

и w ) - " _ £WP * / ' ( W ) ^ ( w ) - n _ I u p i j / ( w ) - t ( i w)

‘w ’ lw ’

where к, / are non-zero elements of K. Then (a)n(b) ф К and (iii)->(ii) is proved. For the proof of the theorem it remains to show that (i) implies (iii).

Let L be a 2-indecomposable object in and q : (x)"‘L->L, where

i e l

"lL — К [x,], its projective cover. Denote by the image of x, by q. From the minimality of q we have у{ф1(Ь)2. Denote by pht(y^, p-height of yh the minimal element from the set JVu {ooj such that yf" = 0 .

Let us consider the set

Y = {yf"; i e l , p/it(y,)}

and introduce in Y the relation ~ given by

yf" ~ j f iff yf” = k y f for some non-zero к e К .

We see that ~ is an equivalence in Y By [yf"] we will be denote the class of equivalence of yf". Now put

X = У / - ,

U = {(x, y ) e l x l ; there exist i e l , n e N,

such that x = [yf"] and y = [yf" 1]}.

We will write also x -»y instead of (x, y) eU. We will prove that the pair G

= (X, U) satisfies the conditions of Definition 2.1. It is clear that G is normal, antisymmetric and without cycles. Observe that the abelian Hopf algebras (yf), i e l , generated, as К -algebras, by y, are non-superfluous in L and, using Lemma 3.4, we conclude that (у,) п(у7) Ф К for all i , j e l . Hence if x = [yf”], у = [yf”] e X , then yf* = ly*f for some t ^ n, r ^ m, 0 Ф l e K, we have the chain

x = b r t o f 1] = b f w ”] = y,

and G is connected. It is obvious that the set /(G) of outputs of G is equal to [y,-; i e l ) . We will identify 1(G) with I and [y,] with i. If x = [yf"]eX , we have the path

x = [y f " ] -[y f" _1] M = i

and condition (1) in 2.1 is satisfied. For the proof of 2.1 (2) suppose that x t

Abelian H opf algebras over a field

->x, x 2 ->x, and x = Qf"], xx = [yjm],

= [yf]. Then, according the definition of edge in G, we have the equalities [yjm ] = [yf”] = [y f ] and consequently xx = x 2. This shows that d~ (x) ^ 1, and (2) holds. Now assume that*d~ (w) = 0 and d+ (w) ^ 2 for some w = [ÿf~\ in X. This means that y?” = 0 and there exist edges w -> x, w ->■ y with x Ф y . But if x = [yfm], У = Ьт]> we have [ y f ' *] = [y f] = [ y f +1], so y f +1 = y f +1 = 0. Since x ф y, the elements y f and y f are linearly independent, and so (y,) n (У»)

= К. We have a contradiction, so (3) in 2.1 holds. Therefore G = (X, U) is a special tree.

Now we will define a quasi-ballast (<р, ф, <) of the tree G. If x = [y f]

we put ф(х) = п1 + п. Let ^ be an arbitrary well order in / satisfying condition (1) of 2.2. Finally, if w is a node, then as <p(w) we take the set {vWti;

i e l w], where vw>i are determined by equalities

М - щ P^w)~niw

Уг v w,i s i w

Evidently (q>, ф, satisfies conditions (1)— (3) of 2.2.

Assume that there exist weW, fi(w) < r < (w), and an element v

= Y, ax *n r w>r which does not belong to V^r. Then there exists an

xeXw r

element i in I w<r which belongs to {xeX w>r; ax Ф 0}. Take y = aiyi + X

jc 6 XM

■ tyo(x)

<r(x)

The element у is non-zero because q is minimal. Further v e r wr, so P^(w)-

= w,ll/{w) — 1 • • • F w,r) ( ^ ) У

1 — U.

Moreover, (y)+ £ (ys) = L, so (y) is non-superfluous in L. On the other jel\{i}

hand, w is a node, so there exists an element j in I w different to /. But (y)^{yj) = K and this is an absurd since L is 2-indecomposable. Then r wr cz V^ r for all w eW, fi(w) < r < and (q>, ф, is a ballast of G.

Now we define the subset X of L(G) as the disjoint union ф(у»)~ 1

£ = и U

\veW r = /i(w)

where

£„., = { I ^{ax e T wy} £ «,ÆM,| = 0Î.

x e X w,r x e X w,r

It is easy to check that there exists a commutative diagram L(G, X) --- 5 ---- ► L

where s is the canonical epimorphism. Then r(u,) = ts(xi) = q(x,) = у,- and t is

128 A. S k o w r o n s k i

an epimorphism. We claim that t is also a monomorphism. Let a be a homogeneous primitive element of L(G, Z) such that t(a) = 0. If deg a = 2p", then by Lemma 3.3 (i) a has an expression in the form £ . Put

x e X n

A = {xeX„; ax Ф 0}. Assume А Ф 0 , if n ^ <5(G), Л has one element, say x, and 0 -(x) = O. Then the equality 0 = t (a) = ax у£(х)Ф<т(х) = ax y%" ф(0) implies y^Q ф(0) = 0 and consequently n^\f/{z), a = 0. If n< S(G ), then for some node w in G there exist paths from w to all elements of the finite set A.

Hence A is contained in X w „ and

o = ( X

ахл;;а(х)

x e A

pil*(w) ~ n

« J /

x e A

since f ( a) = £ ах/" (х)фа(х). Therefore £ ax e Z WiH and a = £ ахиЦх)фв{^

x e A x e A x e A

= 0. This shows that f is a monomorphism. The proof of the theorem is complete.

T

3.5. Let L b e an object o f ££x. The following statements are equivalent :

(i) L has a core.

(ii) L ^ L(G, Z) for certain G eX, X c T(G), with 0(G) < oo.

Pr oof. From Theorem 3.2 it remains to show that L(G, Z) has a core iff 0(G) < oo. Suppose that L(G, Z) has a core. If L(G, Z) is basic, then W(G) Ф 0 and ô(G) = ф(0) < oo. In the other case, the core of L(G, Z) is contained in f) (щ) and is equal to (utf") for some m e N . Hence <5(G) = m

ieI(G)

+ Ф( 0) < oo. Assume now ô (G) = r < oo. We claim that C(L(G, Z))

= ( m ^ ^(0)). It is clear in the case г = ф(0) because then IF(G) = 0 and L(G, Z) is basic. Let r >i ^( 0) and H be a non-superfluous subobject of L(G, Z). Then Я ф /(L(G, Z))2 and there exists a homogeneous primitive element a in H, say of degree 2p”, which does not belong to / (L(G, Z))2. By Lemma 3.3 a = ]Г ах ь^"х)ф<т(х), where w is a node in G such that ф ^ ) = r.

x e X w , n

Using Lemma 3.3 again, we have

0 ^ a p' _ ” = (F„,r. 1...F w,„)( £ а ,)и Г 'И°>-

xeXw,n

But <r(w) = 0 and C(L(G, Z)) contains (i$ ф{0)). On the other hand, it is easy to see that (и0) n (ц) = (u£ ^<0)) for each element i e Iw different to 0.

Therefore C(L(G, Z)) = (w^ *(0)) and the proof is finished.

T

3.6. Let L b e an object in S fx. The following statements are equivalent :

(i) L has a cocore.

Abelian Hopf algebras over a field

(ii) L is cobasic.

(iii) L ^ L(G, r(G )) for certain G from H.

P ro o f. The implication (ii) —>(i) is obvious. Conversely, suppose that L has a cocore and let q: 0 "‘L ->Lbe its projective cover. Since C°(L) Ф L,

i e l

there exists i e l such that qÇlL) глН Ф К for each non-zero subobject H of L.

Consequently L is cobasic, since the subobjects of qÇlL) are linearly ordered, and we have (i)-*(H). Now observe that if L is cobasic, then every two non-superfluous subobjects of L have a non-zero intersection and from Lemma 3.4 Lis 2-indecomposable. Thus, for (ii) <-► (iii), it remains to show that L(G, I) is cobasic iff I = L(G). It is obvious that we can consider G in T with W{G) Ф 0 . Let G be such tree and let I c T(G). Then if v — ]T ax

x e X w >r

is an element of Twr which does not belong to I , a = £ ах и£(х)фа{х)

_ , * 6 * w . r

Ф 0 and

( f

■ ФЧш)

0 .

We see that in this case (а) п(и,- ) = K. Hence if L(G, I ) is cobasic, then T = L(G). As in the proof of Theorem 3.2, for the proof that L(G, L(G)) is cobasic we need only to show that if a is a non-zero homogeneous primi­

tive element, say of degree 2pm, in L(G, L(G)), then </ Ф 0 whenever 0 < r < ф(г) — m. If m ^ 6(G) it is clear. If m < 5 (G) there exist a node w and an expression of a in the form

« - I

x e X w

axit

Hence from the equality ярГ = 0 , 0 ^ r < ij/(z) — m, it follows that

a x e r w m c :

L(G), so a

0, and the proof of the theorem is complete.

x e X w.m

A special tree of the form -♦ we call simple. Observe that in our representation of special trees with ballasts in the category the simple tree with the ballast ф(0) = n corresponds to the simple object "S.

C o r o l l a r y

3.7. Let L be a non-simple object in The following

statements are equivalent:

(i) L has a waist.

(ii) L has a core and a cocore.

(iii) L ^ L(G, T{G)) for certain non-simple G in Î with <5(G) < x . P ro o f. From Theorems 3.5 and 3.6, we have (i) ->(ii) -*• (iii). For (iii) -»(i), suppose that G is a non-simple special tree with a ballast and 3(G)

< x . If W(G) = 0 , then L(G, L(G)) = L(G, 0 ) = K nr, where n = ф(0), r

= ф( 2 ) — ф(0), by Proposition 3.1, has a waist. Assume W(G) Ф 0 . Let a be a non-zero homogeneous primitive element, say of degree 2p", in L(G, T{G)).

9 — Prace Matematyczne 24.1

130 A. S k o w r o n s k i

From our assumption it follows that there exists only one node w in G with

I

l/(w) = <5(G). As in the proof of Theorem 3.6, we prove that ( f Ф 0 for r

< ф(z) — n. Put M = (uo"), where m = <5(G) —1//(0). We see that (a) contains M if n ^ d(G) and is contained in M if n > <5(G). Hence M is contained in every subobject of L(G, L(G)) which has a non-zero homogeneous element of degree ^ 2p^G) and contains every subobject of L(G, T(G)) without such elements. Consequently, M is a waist in L(G, T(G)) and the proof is complete.

We end the paper by showing an example of indecomposable object in which is not 2-indecomposable.

E

3.8. Let m ^ 3 and 1, a, b be elements of К linearly indepen­

dent over Kp. Let us consider the quotient abelian Hopf algebra

where xl9 x 2, x 3 are primitives of degree 2. Since (x1) n ( x 2) = K , L is not 2- indecomposable. We claim that Lis indecomposable. It is easy to verify that L has no nilpotent elements and the space P(L)2p of its primitive elements of degree 2p, has dimension 2 (over K). Suppose that L = H ® M , where H and M are proper subobjects of L. Then dimK H 2 = 1, dimK M 2 — 2, or conver­

sely. In this case, choose a non-zero vector dx = cl l x l + cl2 x 2 + cl3 x 3 in H 2 and two vectors d2 = c2l x l + c22x 2+c23 x 3, d3 = c3l x l +c32x 2 + c33 x 3 in M 2 linearly independent over K. Since L2 = Я 2© М 2, the determinant of the matrix

is non-zero. On the other hand, H r \M = K, dimK P(L)2p = 2, L has no nilpotents, so the primitive elements

are linearly dependent over K. Then the determinant of the matrix L = K [ x u x 2, x3] /( x ^ - axp2- b x p3),

d P2 = (fix a + cp22xp2 + cp2l b + cp23xp3, dp3 = cp3 la + cp32xp2 + cp3 lb + cp33xp3,

cp2 ia + cp22 cp21b + (f2 3 cp31a + cp32 cp31b + (f3 3

is zero. But 1, a, b are linearly independent over Kp, and we obtain that

с 22^зз ~ С2ЪСЪ2’ с 2 х с ъъ = с 2з c 3i> c 2 i c 32 — c 2 2 c 3i • This is a contradiction

since det С Ф 0. Therefore L is indecomposable.

Abelian H opf algebras over a field

References

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[6] E. L. G r e e n , On the structure o f indecomposable modules, in: Representation theory o f algebras (ed. by R. G o r d o n ), Lecture Notes in Pure and Appl. Math. 37 (Marcel Dekker) (1976), 369-380.

[7] J. M iln o r , J. C. M o o r e , On the structure o f H opf algebras, Ann. Math. 81 (1965), 211- 264.

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[9] C. S c h o e lle r , Etudé de la catégorie des algèbres de H opf commutatives connexes sur un corps, Man. Math. 3 (1970), 133-155.

[10] D. S im s o n , A. S k o w r o n s k i, On the category o f commutative connected graded Hopf algebras over a perfect field, Fund. Math. 101 (1978), 137-149.

[11] G. S jô d in , A characterization o f local complete intersections in terms o f the Ext -algebra, Stockholms Univ. Math. Inst., Preprint Series No. 2 (1976).

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INSTITUTE OF MATHEMATICS, NICHOLAS COPERNICUS UNIVERSITY TORUN, POLAND