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Coproper coalgebras
JOURNAL
OF ALGEBRA
54, 203-215 (1978)
Coproper
Coalgebras
H. P. ALLEN* The Ohio State University,
Columbus, Ohio
AND
D. TRUSHIN+ Northern
Illinois
University,
Communicated
De Kalb, Illinois
by I. N. Herstein
Received August 23, 1976; revised October 20, 1977
The theory of finite-dimensional nonsemisimple algebras has been well developed. Using the existence of an identity element one may conclude that such an algebra A contains a set of supplementary orthogonal primitive idempotents {e,} where each Ae, is both indecomposable and projective and that these are precisely the finite-dimensional indecomposable projective A-modules. Beyond this, results on winding numbers, top composition factors, block decomposition among others [cf. 2, Chapter VIII] provide valuable information about A. In this paper we study coalgebras C where C* contains “enough” rational ideals, viz., that the sum of all rational left ideals of C* is dense. We call such a coalgebra left coproper. Much of the indicated material from [2, Chapter VIII] remains valid for coproper coalgebras. Section 1 examines coproperness, Section 2 relates the rational C*-representation theory to the C*rat-representation theory and gives the basic structure theorem for C*rat. This lays the groundwork for a future discussion of Frobenius coalgebras (which are coproper), a parametrization for orthogonality relations for characters on Frobenius coalgebras and a construction of integrals and antipodes for Frobenius bialgebras. Section 3 gives some applications to incidence coalgebras as well as several examples.
1
in
Let C be a coalgebra over a field k. C* has the weak-* topology as described [3]. For any subset X in C*, x denotes closure in this topology, * This research was supported in part by the NSF through grant number + Present address: Bell Telephone Labs, Napenville, Illinois 60540.
4471 -Al.
203 0021-8693/78/0541-0203$02.00/O Copyright 0 1978 by Academic Press, Inc. AU rights of reproduction in any form reserved
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ALLEN AND TRUSHIN
XL={~~Cj(X,cj=O}, for YCC, X’-‘. A C*-module is called rational C-comodule structure.
Y-L={xEC*j(x,Y)=O}and if its module structure
X= arises from a
1.1 PROPOSITION. Let C be a coalgebra over k and let X C C* be a subset.For any rational left (right) C*-module M and subset Y C M, we have X ’ Y = X . Y (Y . X = Y . x). In particular, if X is dense, then X * M = M (M * X = M).
Proof. Let !Pz M ---, M ok with Y(A) = ~~=, hi @ ci . For x E X s.t. (x, ci) = (%, ci) i = then X = C* and so X. M =
C be the comodule structure map and let h E Y any f~ X, x . X = Cy’, (x; ci) Xi . Choose x E X 1,. . ., n. Then clearly 5 . h = x . A. If X is dense,
C* . M = M.
1.2 DEFINITION. Let C be a coalgebra (right) rational ideal of C*.
over k. P(W)
is the maximal
left
Remarks. See [5] for a discussion of P(W’). For the present we note that P(%‘) = 1 {I 1I C C* is a rational left (right) ideal). Furthermore P(V) is a two-sided ideal and in view of 1.1, @(QC) C n {I 1I is a dense right (left) ideal}. 1.3 PROPOSITION. Let C be a coalgebra over k. Then (1)
(CO)l = n{rj
(2)
(K’)l
zc c zs . a cofinite left coideal) E I,(C).
= 0 (I I I C C is a co$nite right coideal) = I,(C).
(3) For any subcoalgebra D C C, I,(C) n D > I,(D) (IT(C) n D 1 &I,(D)). left ideal and so Proof. (1) If A E Co, then C* . A is a finite-dimensional (C* . h)l is a cofinite right submodule of C, i.e. a cofinite left coideal. Thus I,(C) C (Cn)l. Suppose c $1,(C). Then there exists a cofinite left coideal I s.t. c $1, and a h E P- C CO s.t. (A, c) # 0. Thus c $ (CO)i. (2)
Similar
to (1).
(3) Let I be a cofinite coideal in D.
left coideal in C. Then In
D is a cofinite left
1.4 COROLLARY.
C~(K’) is dense in C* -I,(C)
1.5 COROLLARY.
D C C a subcoalgebra. Then It(C) (Ir(C)) = 0 3 It(D)
VrW
(IT(C)) = 0.
= 0.
1.6 COROLLARY. Let C, D be coalgebras, L: D + C a coalgebra injection. Then I *( CO) C Do.
COPROPERCOALGEBRAS
205
1.7 PROPOSITION. Let C be a coalgebra over k. Let 12+(C) = Ii(C) n ker E, and let a: C + C/I,+(C) = D. Then TQ~(C)) 1 IL(D), and hence dim,I,(D) < 1. Furthermore,
3 a monomorphism 0: CD + Do with D* = k + w).
Proof. It(C) = (Cb)l is a subcoalgebra so IL+(C) is a coideal. Let n(c) EI~(D) with c $Il(C), and I a cofinite left coideal s.t. c $1. n(l) is a cofinite left D-coideal, hence V(C) E ~(1) and so c E I + IL+(C) = I. This forces G(C)) 2 4(D). N ow ~-(l~(c)) = IL( C)/lk+(C) has dimension 1, so Ii(D) has dimension at most 1. As IL+(C) C (Cm)‘, Co embeds (as a algebra) in D* via (O(cn), r(c)) = (co, c} and we bote that n*O = 1c0 . rr*(y . B(x)) = n*(y) 71*0(x) = r*(y) . x = Let XECO, YE D*. Then I& X(o) (T”(Y), X(1)) = I&) X(o)(Y, ~*Gw)) = fl*c% 4%)> 3Y, 4%))Y). Since V* is injective, y . O(x) = xtzj 0(x,,,) (y, v(x~,)) and so O(x) E Do. 0 has codemension ~1. The conclusion Since n(ll(C)) = (ecy- we see that OC now follows. 1.8 PROPOSITION. Let T be a coulgebru over k. TFAE: (1)
Co(W)
is dense in C*.
(2) It(C) VT(C)) = 0 (3) For each rational dense in M*. (4)
right
(left)
C*-module
M,
(c*M*)pat
((M&pt)
is
C* contains a dense rational left (right) ideal.
Proof.
(1) e(3)
may be found in [4]. The others are trivial.
1.9 DEFINITION. A coalgebra C over a field k is called left coproper if Co is dense. It is right coproper if qC is dense, and it is coproper if both left and right coproper. Remarks. (2)
(1)
If C is coproper,
In view of 1.5, the property
then clearly Co = qC. of being (left, right) coproper is hereditary.
(3) The duality between left and right coproper is immediate; stated in terms of one side will be assumed for the other.
all results
1.10 PROPOSITION. Let C be a left coproper coalgelmz. Then (1) A vector space M is a left rational and Cm . M = M. (2)
C*-module
+ M is a left Co-module
Every left Co-module contains a unique maximal left rational C*-module.
Proof. It is clear that (1) =S (2), and the “only if” part of 1 is evident from 1.1. If M is a @-module and Co * M = M, th en M is a C*-module with c* . (corn) = (c*co) m. This is well-defined since xim = xzm for xi E Cb *
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ALLEN AND TRUSHIN
(x1 - x2) m = 0 andy (x1 - x2) m = 0 Vy E Co. Now 3e E Co s.t. e(xL - x2) = x1 - x2 . Hence for any c* E C*, (c*xi) m - (c*xs) m = (c*(x, - ~a)) m = (c*e) (x1 - xs) m = 0. Co . M is thus a locally finite C*-module and Vc* E C*, xm E Co M, cc . xm = &j xtO) . m(c*, Ox. EXAMPLES.
(I)
A finite-dimensional
coalgebra
is coproper.
(2) Let C be pointed irreducible. If I is any nonzero right coideal, then I contains a simple C-subcomodule and so contains the unique grouplike of C. Hence by 1.3, C is (left, right) coproper 0 (0) is cofinite 0 C is f.d. Suppose that C = @aEa C, where each C, is a subcoalgebra. C* = nIaoa C* and Vda:~C4!, we define e,E C* by e, Ic = eC , e, jc =O, 01#/3. For g = rgs E C” and c, E C,, , (e,g, c,) = 0 if 01# G, (e,;, c,) =a(g,, , c,,) if 01= y. Thus w = g, and {e&a are the projections associated with the product. Note that {e,),,a are orthogonal idempotents. Let g E Co. Since C*g is f.d., e, *g = 0 for all but finitely many 01and so g E @ C,* C K,X. Let g=g,+ .‘.+g,, g,EC& i= l,...,e. Since gEP, eMig=g,ECC and C*e,,g = C$,gi C Cm. A quick computation shows that C$,g, C Cm: and so Co C @ Can. Thi reverse containment is clear and hence Co ‘= @ C-0. This enables us to prove 1.11 THEOREM. Suppose that C = @C, where each C, is a subcoalgebra. Then C is left coproper c> each C, is left coproper. 1.12. COROLLARY.
Any direct sum of jktite-dimensional
coalgebras is coproper.
1.13 COROLLARY. If C is cosemisimple then C is coproper. A coalgebra is called indecomposable if it cannot be written as a direct sum of two nonzero subcoalgebras. Using the analogous results for cocommutative irreducible coalgebras in [5], one obtains the fact that C is the direct sum of all the maximal indecomposable subcoalgebras of C and any summand of C is a direct sum of a uniquely determined subset of these maximal indecomposable subcoalgebras which will be called indecomposable components (I.C.‘s). I. 14 COROLLARY.
C is left coproper 0 each I.C. is left coproper.
In this section we discuss the structure of the rational duals of coproper coalgebras. The results for left coproper coalgebras have analogous for right coproper coalgebras which we will assume.
COPROPERCOALGEBRAS
207
2.1 THEOREM (Rational Extension Theorem). Suppose that C is a left coproper coalgebra, N is a Jinite dimensional rational right C*-module, and 9: Co ---f N is a morphism of right Co-modules. Then 3! right C* morphism @: C* -+ N which extends y. Moreover Im rp = Im + and qz is a F-morphism. Proof. Letp: C* + End,(N) be the given representation. For z E ker p n C@, (0) = N . z 2 y(Cn) . z = cp(C%). By 1.1, z E C?z, so v(z) = 0. Hence ker p n C@C ker 9. For c* E Cx, choose x E Co with p(c*) = p(x) and define q(c*) = v(x). If y E Co and p(c*) = p(y), then x - y E ker p n Co and so v(x) = v(y). Hence @ is a well-defined map. Evidently Im 9) = Im y and q is linear. Let c*, d* E C* and suppose that x,y E CD are chosen s.t. p(c*) = p(x), p(d*) = p(y). Then p(c*d*) = p(xy) and so q(c*d*) = I = q(x) y = p(y) (y(x)) = p(d*) q(P) = q(c*) d*. Thus g is a C*-morphism and + clearly extends v. As a result, y is also a C*-morphism. Now let Y be any other C*-morphism which extends 9. Then ker p n Cn C ker y C ker Y. Choose x E C” s.t. p(x) = 1, . Y(1) - Y(x) = Y(1) - Y(I) x = 0 and so Y(1) = Y(x) = q(x) = q(l). Now Vc* E C*, Y(c*) = Y(1) c* = F(I) c * = q(c*) and 9, is unique.
2.2 PROPOSITION. Let C be a left coproper coalgebra, M a f.d. rational right F-module and N a right Co-module. If f : M -+ N is a right Co-morphism, then f(M) is a right F-module in which the F-module action extends the Co-module action, f is a F-morphism and (hence) f(M) is rational. Proof. Define p: C* --f End f (M) as follows: choose x E Co with mx = m Vm G M and set p(c*) (f(m)) = f( m) xc* for all c* E C*. Clearly p is a well defined linear map. For c*, d* E C*, dc*d*)
(f(m))
= f (m) . x(c*d*) = f (m . x(c*d*)) = f ((m . xc*) . d*) = f ((m . xc*) . xd*) = f (m * XC*) * xd* = f(d")
(f(c*)
m4N
and so p is a right representation. If co E Cn, then Vm E M, p(G) f (m) = f(m) . xc0 = f(mx) . cn = f (m) . co and so p extends the original action. Clearly f is a C*-module morphism. Note that 2.2 implies that any extension of a finite dimensional rational C*-module which is split as a Co-module is split.
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ALLEN AND TRUSHIN
2.3 PROPOSITION. Let C be a left coproper coalgebra and M afinite-dimensional rational right C*-module. TFAE: (1)
M is C*-projective.
(2) M is Co-projective. (3) M is C@-isomorphicto a Cc summand of Con for some n. Proof.
(1 * 2).
We need the following
2.4 LEMMA. (1) Let M be a right Cm-module. Then 3 a right CGrjection i: M + M @Co C*. In addition, if M is a rational C*-module then M &O C* z M (as C*-modules). (2) Let M be a right Cc-module and let NC M &-o C* be a rational right C*-module. Then NC M &a Ior . Proof. (1) m + m @ 1 clearly works for the first part. Let C* module and define v: M &D C* --f M via C m, @ xi + right C*-morphism and q~/Mocolc* is a Co isomorphism. M&~C*!ZM(&D lc* and apply 2.2. Let x = &, mi @ and choose y E Co with m,y = mi . Then
M be a rational 1 mixi . p7 is a We show that Xi E M @co C*,
X=~m,Z’~Xi=~m~~yxi-~miyX~~lc*. i-l
i=l
i=l
(2) Since N is rational, NC0 = N. Choose x = C mi @ xi and pick y E Co with xy = x. Then x = x m, @ (xiy) = C mi(xiy) @ I,* E M @co lo* . We continue with the proof of 2.3. Let P and Q be right Co-modules, g: P+ Q be a Co-epimorphism and f : M + Q a Co-morphism. By 2.2, f(M) is a rational right C*-module and we may assume that P = g-lf (M) and Q = f (M). NOW the diagram M
induces the diagram
p o&l c* goI+
Q Oco C*
of C*-modules. We will use 0, (resp. 0, , 0,) for the Cm-monomorphism m -+ m @ 1 (resp. p -+p @ 1, q-f q @ 1) of M (resp. P, Q) into M &O C* (resp. P @JonC”,
209
COPROPER COALGEBRAS
Q @co C*). Clearly g @ I is surjective and the C*-projectivity of M produces a C*-morphism h: M -+ P @CmC* s.t. g @I . h = f @ I . OM . By 2.2, h(M) is rational so h(M) _CO,(P). We thus obtain an induced Co morphism h,: M-+ P where 02, . h, = h. We now compute e&f
. w4)
= @1(m) 0 1 = (B 0 4 (h,(m) 0 1) = (g 0 I> Q-4 = (f63
1) . bh)
= eo(f(W>
VrnE M.
Thus g . h, =f and M is Co-projective. (2 3 3). Since M is f.d. and right rational, 3~: M* CA 0: C for some n. Thus 3 a continuous epimorphism v*: 0% C* + M and so q~* (0: Co) = M. Since M is G-projective the result follows from this. (3 2 1). Let TK COn--+ M be a Co-projection and let ri: Co + Con be the right Co-module injection onto the ith component. The map vi: rr 0 vi: CD -+ M is a right P-module map and so by 2 1 3! qi: C* -+ M extending vi i = l,..., n. Therefore + = @ qi is a well-defined right P-morphism and q(C*n) = M. Let Y: M + (Co)” be a right Co-morphism s.t. v 0 Y = 1,. By 2.2, Y is a right C*-morphism and clearly $j 0 Y = (0 +J 0 Y = VY = I, and so M is a summand of C*n.
2.5 COROLLARY. Let C be a coproper coalgebra and M a f.d. injective rational Then M* is F-projective.
left F-module.
Proof. C*-modules. as before, We now where C is coalgebras,
It is easy to see that M* is projective with respect to rational right Since Co is right rational and 3 a C*-epimorphism r: Co” -P M* M* is C*-projective. turn our attention to questions about the internal structure of Co (left, right) coproper. Again where a result is stated for left coproper the analogous result also holds for right coproper coalgebras.
2.6 PROPOSITION.
Let C be a left coproper coalgebraover k.
3 idempotents in Co. (2) if e E CD is idempotent then Co = Cn(I - e) @ Cue = (1 - e) Co @ eCD. Thus CCe is finite dimensional, and Cue and eCmare C*-projective. (3) e is a promitive idempotent in Co * Cue is indecomposable. (1)
Proof. (1) Since Co is clearly not nil we may find a non-nilpotent element x in Co. Px is a finite-dimensional non-nilpotent subalgebra of Co, hence contains a nonzero idempotent e. (2)
Standard.
(3)
Standard.
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ALLEN AND TRUSHIN
2.7 DEFINITION. Let C be a coalgebra C*-summand of Co is a left rational principal for short). 2.8 PROPOSITION. mands.
over k. An indecomposable left indecomposable F-module (RPIM
Let C be left coproper. Then Cm has no nil right Cm sum-
Proof. Suppose C@ = M @N where M, N are right ideals with N nil. Let y E N and choose x E Co s.t. xy = y. Writing x = m + n with m E M, nENwehavey=xy==my+nyEN.ThusmyEMnN=Oandsoy=ny. Since n is nilpotent, y = 0. 2.9 COROLLARY. Let C be coproper and N a left RPIM. Then X = CCe for some primitive idempotent e E C 0. In particular, N is f.d. and C*-projective. Proof. Using the “right” version of 2.8 we see that Co has no nil left or right summands. Thus N is not nil and being locally finite it contains a nonzero idempotent e. Now N = Ne @ N(l - e), so it follows that N = Ne is finite dimensional. The rest of the argument is standard. At this point it is clear that if C is (left, right) coproper, then CC contains primitive idempotents. By an easy Zorn’s Lemma argument one shows that Co contains maximal sets of primitive orthogonal idempotents. 2.10 THEOREM. Let C be coproper. Then Co = @,Ea Cue, for every maximal set of primitive orthogonal idempotents M = {e, j 01E GE>_CCo. We first prove the following
lemma.
2.11 LEMMA. Let C be right coproper and M = {e, j 01E a} a maximal set of primitive orthogonal idempotents in qC. Set N = .ZK’e, and N’ = {x E CC [ xe, =OQe,EM). ThenK’==N@N’. Proof. Define morphism. Since independent, xe, map rr: DC- N kernel N’. Since
rre: C -+ QZ’e, via VJX) = xe, . Clearly z=, is a left C*-module XX’ is f.d. and for x E DC, the set {xe, 1xe, # 0} is linearly = 0 for all but finitely many e, . We can therefore define a by setting V(X) = C xe, : rr is a C*-module morphism with r IN = I, the result follows.
Proof of 2.10. By 2.8, N’ is not nil if it is nonzero. Thus 3 non-nilpotent x E N’ and so UCx C N’ is f.d. and non-nilpotent. Let e # 0 be an idempotent in N’. Then ee, = 0 for each 01 and in addition, 301, ,..., 01~s.t. ese = 0, B # a1 ,.-., 012. Choosing e with 1 minimal we let f = e - eEIe E N’. A small computation shows thatf is a nonzero idempotent. Now esf = ese - eDeale = 0, P f % t.o.9cyrand e,,f = 0 contradicting the minilality of 1. Thus we may assume that e is orthogonal to each e, . From this we easily obtain a primitive idempotent
COPROPER COALGEBRAS
211
in N’ orthogonal to each e, , thus contradicting the maximality of n/17.Therefore Co = qC = @qCe,= @C”elll. Observe that the same argument may be used to show that Co = Gacd e,Cn and that each e,Cn is a right RPTM. In fact we have the following. 2.12 THEOREM. Let C be a coproper coalgebra over k. Then every right (left) RPIM is$nite dimensional, projective and isomorphic to eCn(C”e) for someprimitive idempotent e E Co. Moreover, CD is a direct sum of right (left) RPIM’s, and every right (left) RPIM is isomorphic to one of these summands. Proof. We need only prove the last assertion. Let N be a right RPIM and suppose that Co = @ olEae,C* according to 2.10. Since N is f.d., N C &, e,iCn = M for some {eel ,..., eUle}.Furthermore, N is a summand of M and M is f.d. so we may apply the Krull-Schmidt Theorem to obtain the result. 2.13 COROLLARY. Let C be copropw and M a f.d. rational projective C*module. Then M s @ Fe, for suitable primitive orthogonal idempotents {e,} C Co. Proof. 2.3, 2.10, and Krull-Schmidt. At this point we observe that one may readily adapt the appropriate results from [2] chapter 8 to coproper coalgebras C and in particular Co. More specifically, if C is coproper, then 3 indecomposable ideals B, c Co s.t. Co = @ B, , {B,} is unique, and if B is an ideal direct summand of Co, then B = @ Be for some subset {B,} _C{B,}. In addition, it is easily seen that each Be = CEO for some 01where (C,) is the complete set of I.C.‘s for C. We also obtain the following. 2.14 THEOREM. Let C be coproper and M a finite-dimensional rational C*module. Then M is injective *ME @ (Cue,)* for suitable primitive orthogonal idempotents {e,> c Co. Proof. Cue, is C*-projective, and so (Cue,)* is Co-injective. Thus @ (Cue,)* is injective for any choice of idempotents {e,} c Co. Conversely, if M is C* injective, M* is C*-projective and thus M* g @ Cue, which gives the result. 2.15 THEOREM. Let C be a coproper coalgebra, {e,} a maximal set of primitive orthogonal idempotents. Then Rad(Cn) = @ Rad(C*) ear. Moreover, v~r, Rad(Coe,) = Rad(Cn) e, = Rad( C*) e, , Rad(Co) is nil, and Rad(Cne,) is ndpotent. We conclude this section with some results on the structures CD0 = {h E Cm* 1ker )I > cofinite ideal in Co}.
of C, C*O and
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ALLEN
AND
TRUSHIN
2.16 THEOREM. Let C be a coproper coalgebra. Then C = @ C e, where (e,} C Co and consists of primitive orthogonal idempotents. Each C . e, is injective, and hence C is injective in the category of C-comodules.
Proof. Co = @ Cue, and so C = C Co = C C . COeu = x C . e, . Clearly this sum is direct. Now C * e, is a f.d. summand of C and so (C . e,)* is isomorphic to a rational summand of C*. Thus (C e,)* is projective and C . e, is injective. 2.17
THEOREM.
Let C be a coproper coalgebra over k. Then
(1) croci = coc*o = L(C) (6: c--t c*y. (2) C”OCO = pip = coo. (3)
coo is counitary.
(4) Coo g C as coalgebras and the dual of Co C-t C* is an epimorphism of c*O to coo. Proof. (1) Note first that L(C) is the largest rational right C*-module of C*O, since f(x) E L(C) @ C*O implies x E I(C). Thus C*OP, being rational, is contained in l(C). But L(C) CD = L(C) and so C*sCu = c(C). Similarly COC*O = 40 (2) Let x E Coo and let I be a cofinite ideal (of Co) contained in leer x. Cu = Q&r e,Cn and so all but finitely many e, are in I. Let e = zem4, e, . Then clearly xe = x and so xe E CW’O. Thus CnOCo = PO. Similarly CnCcO = COO.
(3) By (2), Coo = G&a Coo . err. Define x E (Coo)* by x /cam, = e, Ic~O.,a (e, considered in (COO)*). For y E (PO)*, (xy, CDs . e,) = (x, y@O . e,) = = (h, co> Let c E C. If rr o L(C) = 0, for co E Co .7r(C*O) C Coo, and T is comultiplicative. then 0 = (V 0 L(C), c”) = (co, c) and so c E (Cn)l = (0). Thus rr o I is injective. Let 0 # e, E Co. Then 3c E C s.t. 0 # (e, , c) = (e,e, , c) = (e, , ce,) = Thus x # 0. (e, , I~To L(ce,)) = (e, , x o L(C)e,> (since r is comultiplicative). we need only show that E ocno v = l cti to conclude the remainder. Let ecuor = (x, m(h)) = (x, C n(h) e,) = ((e,), x r(h) es) = h E C*O. Then <(eJ, C +bD
=
Let “0” denote this (4) By 1.10 and (2), C q o is a rational left C*-module. module structure and #: Coo -+ Co @ L(C) be the associated comodule structure. ForxECnandX=y.yECOo=CO.COo,xoh=(IOx)(IOrr)ICl(h)onthe
213
COPROPERCOALGEBRAS
one hand and xoh=xo(y.y)=xy.~=x.(y.r)=x.X=(IOx)f(X) on the other. Now C* 0 X is finite dimensional, so by 2.1 c* I+ c* 0 X is the unique extension of x -+ x o h, i.e. (I @ 7~)# = d. Therefore d(Cns) _CPO @ r 0 L(C) and by (3), Coo 2 n 0 c(C). 2.18 THEOREM. Let C be coproper. Then (1)
C*O = c(C) @ (Cn)l as coalgebras.
(2) If M is a locally finite F-module, then ME {m GM j Cum = (0)).
Mrat @ M’ where M’ =
(1) r: C*O c-+ Coo is an epimorphism and r /‘cc) is an isomorphism. (2) Since M is locally finite, 3~: M c--+ 0, C*O = @, t(C) @ Or (Co)+ CnM = Mrat C M. For m E M, write vrn = x, + xi with x, E @JrL(C) and xi E 0, (C 0>I. Now 3e E Cm such that ex, = x, and exi = 0. Thus em = ex, E M and so xi E M also. Then M z M n 0, c(C) @ M n @I (CD)*. We conclude this section by noting that the torsion preradical for Lt. C*-modules M--f Mrat is split for coproper coalgebras. Proof.
3. EXAMPLES We have already seen that coproper coalgebras are hereditary. In this section we present an example of a coalgebra C with Co = (0) = qC’ yet having subcoalgebras D and E with D 1 E, qD = (0) Do dense (in D*) and E coproper. We begin with some preliminary results on incidence coalgebras. Throughout C = C,(X, S) will be a coassociative weighted incidence coalgebra for the relation set S on X (cf. [I] and [5]). (x, t) d enotes an element of S and e(x, y) is the element of C* taking the value 1 on (x, y) and 0 on every other segment. 3.1 THEOREM. It(C) has a basis {(a, 6) 1(--co, u) n (-co,
b) is infinite>.
Proof. I,(C) = (P)l (1.3) is a subcoalgebra therefore by [I] is spanned by the segments which it contains. We will show that a segment (a, 6) is excluded by some cofinite left coideal I if and only if (- CO,a) n (- CO,b) is finite. One direction is trivial for if the latter set is finite then {(x, y) ] y # b} u {(x, b) 1(x, a) # S} is cofinite in S and clearly spans a left coideal excluding (a, 4. In the other direction let I be a cofinite left coideal with (a, b) $1. For x E (-co, u) n (-co, b) a quick computation with (x, b) . e(x, a) shows that (x, a) $1. If (-m, u) n (-00, b) is infinite then there must be a dependence relation of the form C &(x6 ,6) E I for Xi # 0, xi E (- co, u) n (-co, b). Acting
214
ALLEN AND TRUSHIN
on this relation on the right by e(x, , x1) forces (x1 , b) E 1, a contradiction. (-00, u) n (-co, b) is finite.
Thus
3.2 COROLLARY. Let C,(X, S) be a coassociative weighted incidence coalgebra satisfying the condition : (-co, u) n (-a~, b) is injkite V(a, b) E S. Then co = (0). 3.3 COROLLARY. Let C,(X, S) be a coassociative incidence coalgebra satisfying the condition: 3(a, b) E S s.t.(-q a) n (-00, b) is finite. Then Co # (0). Moreover if this condition holds V(a, b) E S then Co is dense. One can modify the proof of 3.1 to display a finite-dimensional ideal in C*; viz., J = h{e(z, b) 1z E (-co, u) n (-co, b)}.
rational
left
3.4 PROPOSITION. Let C,(X, S) be a weighted coassociative incidence coalgebra. Assume that Vx E X 3 cofinite left coideal excluding (x, x). Then Co is dense. Proof. As observed in the proof of Theorem 3.1 I,(C) is a subcoalgebra. If subcoalgebra of It(C). By ([I]) IL(C) # (0) let D # (0) b e a minimal D = k{(y, x) 1y, z E (x, x)} for some fixed x E X. But then (x, x) E D lies in every cofinite left coideal, a contradiction. The examples now follow directly. Let X = 77. with the usual ordering, S = {(n, m) 1n < m> and trivial weighting h = 1, C = C,(X, S). Since every ray is infinite (3.2) implies that K’ = (0) = Co. Let Y = N with the usual ordering T = S n (N x N). The inclusions N + Z, T + S induce a monomorphism of D = C,(Y, T) into C. Since every right ray in Y is infinite and every left ray finite (3.2) and (3.3) sh ow that qD = (0) and that Do is dense in D*. Let Z = N, U = {(n, m) 1 1 < n < m < 2n). A small computation shows that the inclusions N - N, U + T induce a monomorphism of E = C,(Z, U) into D. Since every right and left ray in Z is finite we see that E is coproper. We finally note that since the corresponding graphs are connected, each of C, D, E is indecomposable. Finally note that one can easily obtain a coproper subcoalgebra F of C which is an incidence coalgebra of a partially ordered set having neither an upper nor a lower bound. Indeed let X = Z and let I’ = U u {(n, m) 1m < - 1,2m < n ,( m} u ((0, l), (0, O), (--1,O)). Th e inclusion of X in X induces a monomorphism of F = C,(X, V) into C.
REFERENCES 1. H. P. ALLEN AND C. FERRAR, Weighted incidence coalgebras, J. Algebra, to appear. Theory of Finite Groups and Associative 2. C. CURTIS AND I. REINER, “Representation Algebras,” Interscience, New York, 1966.
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COALGEBRAS
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3. R. G. HEYNEMAN AND D. E. F~ADFORD, Reflexicity and Coalgebras of Finite Type, J. Algebra 28 (1974), 215-246. 4. BERTRAVEL I-PENG LIN, “On Homological Properties of Coalgebras,” Ph. D. Dissertation, University of Illinois, Circle Campus, 1973. 5. M. E. SWEEDLER, “Hopf Algebras,” Benjamin, New York, 1969.
481/5411-15
OF ALGEBRA
54, 203-215 (1978)
Coproper
Coalgebras
H. P. ALLEN* The Ohio State University,
Columbus, Ohio
AND
D. TRUSHIN+ Northern
Illinois
University,
Communicated
De Kalb, Illinois
by I. N. Herstein
Received August 23, 1976; revised October 20, 1977
The theory of finite-dimensional nonsemisimple algebras has been well developed. Using the existence of an identity element one may conclude that such an algebra A contains a set of supplementary orthogonal primitive idempotents {e,} where each Ae, is both indecomposable and projective and that these are precisely the finite-dimensional indecomposable projective A-modules. Beyond this, results on winding numbers, top composition factors, block decomposition among others [cf. 2, Chapter VIII] provide valuable information about A. In this paper we study coalgebras C where C* contains “enough” rational ideals, viz., that the sum of all rational left ideals of C* is dense. We call such a coalgebra left coproper. Much of the indicated material from [2, Chapter VIII] remains valid for coproper coalgebras. Section 1 examines coproperness, Section 2 relates the rational C*-representation theory to the C*rat-representation theory and gives the basic structure theorem for C*rat. This lays the groundwork for a future discussion of Frobenius coalgebras (which are coproper), a parametrization for orthogonality relations for characters on Frobenius coalgebras and a construction of integrals and antipodes for Frobenius bialgebras. Section 3 gives some applications to incidence coalgebras as well as several examples.
1
in
Let C be a coalgebra over a field k. C* has the weak-* topology as described [3]. For any subset X in C*, x denotes closure in this topology, * This research was supported in part by the NSF through grant number + Present address: Bell Telephone Labs, Napenville, Illinois 60540.
4471 -Al.
203 0021-8693/78/0541-0203$02.00/O Copyright 0 1978 by Academic Press, Inc. AU rights of reproduction in any form reserved
204
ALLEN AND TRUSHIN
XL={~~Cj(X,cj=O}, for YCC, X’-‘. A C*-module is called rational C-comodule structure.
Y-L={xEC*j(x,Y)=O}and if its module structure
X= arises from a
1.1 PROPOSITION. Let C be a coalgebra over k and let X C C* be a subset.For any rational left (right) C*-module M and subset Y C M, we have X ’ Y = X . Y (Y . X = Y . x). In particular, if X is dense, then X * M = M (M * X = M).
Proof. Let !Pz M ---, M ok with Y(A) = ~~=, hi @ ci . For x E X s.t. (x, ci) = (%, ci) i = then X = C* and so X. M =
C be the comodule structure map and let h E Y any f~ X, x . X = Cy’, (x; ci) Xi . Choose x E X 1,. . ., n. Then clearly 5 . h = x . A. If X is dense,
C* . M = M.
1.2 DEFINITION. Let C be a coalgebra (right) rational ideal of C*.
over k. P(W)
is the maximal
left
Remarks. See [5] for a discussion of P(W’). For the present we note that P(%‘) = 1 {I 1I C C* is a rational left (right) ideal). Furthermore P(V) is a two-sided ideal and in view of 1.1, @(QC) C n {I 1I is a dense right (left) ideal}. 1.3 PROPOSITION. Let C be a coalgebra over k. Then (1)
(CO)l = n{rj
(2)
(K’)l
zc c zs . a cofinite left coideal) E I,(C).
= 0 (I I I C C is a co$nite right coideal) = I,(C).
(3) For any subcoalgebra D C C, I,(C) n D > I,(D) (IT(C) n D 1 &I,(D)). left ideal and so Proof. (1) If A E Co, then C* . A is a finite-dimensional (C* . h)l is a cofinite right submodule of C, i.e. a cofinite left coideal. Thus I,(C) C (Cn)l. Suppose c $1,(C). Then there exists a cofinite left coideal I s.t. c $1, and a h E P- C CO s.t. (A, c) # 0. Thus c $ (CO)i. (2)
Similar
to (1).
(3) Let I be a cofinite coideal in D.
left coideal in C. Then In
D is a cofinite left
1.4 COROLLARY.
C~(K’) is dense in C* -I,(C)
1.5 COROLLARY.
D C C a subcoalgebra. Then It(C) (Ir(C)) = 0 3 It(D)
VrW
(IT(C)) = 0.
= 0.
1.6 COROLLARY. Let C, D be coalgebras, L: D + C a coalgebra injection. Then I *( CO) C Do.
COPROPERCOALGEBRAS
205
1.7 PROPOSITION. Let C be a coalgebra over k. Let 12+(C) = Ii(C) n ker E, and let a: C + C/I,+(C) = D. Then TQ~(C)) 1 IL(D), and hence dim,I,(D) < 1. Furthermore,
3 a monomorphism 0: CD + Do with D* = k + w).
Proof. It(C) = (Cb)l is a subcoalgebra so IL+(C) is a coideal. Let n(c) EI~(D) with c $Il(C), and I a cofinite left coideal s.t. c $1. n(l) is a cofinite left D-coideal, hence V(C) E ~(1) and so c E I + IL+(C) = I. This forces G(C)) 2 4(D). N ow ~-(l~(c)) = IL( C)/lk+(C) has dimension 1, so Ii(D) has dimension at most 1. As IL+(C) C (Cm)‘, Co embeds (as a algebra) in D* via (O(cn), r(c)) = (co, c} and we bote that n*O = 1c0 . rr*(y . B(x)) = n*(y) 71*0(x) = r*(y) . x = Let XECO, YE D*. Then I& X(o) (T”(Y), X(1)) = I&) X(o)(Y, ~*Gw)) = fl*c% 4%)> 3Y, 4%))Y). Since V* is injective, y . O(x) = xtzj 0(x,,,) (y, v(x~,)) and so O(x) E Do. 0 has codemension ~1. The conclusion Since n(ll(C)) = (ecy- we see that OC now follows. 1.8 PROPOSITION. Let T be a coulgebru over k. TFAE: (1)
Co(W)
is dense in C*.
(2) It(C) VT(C)) = 0 (3) For each rational dense in M*. (4)
right
(left)
C*-module
M,
(c*M*)pat
((M&pt)
is
C* contains a dense rational left (right) ideal.
Proof.
(1) e(3)
may be found in [4]. The others are trivial.
1.9 DEFINITION. A coalgebra C over a field k is called left coproper if Co is dense. It is right coproper if qC is dense, and it is coproper if both left and right coproper. Remarks. (2)
(1)
If C is coproper,
In view of 1.5, the property
then clearly Co = qC. of being (left, right) coproper is hereditary.
(3) The duality between left and right coproper is immediate; stated in terms of one side will be assumed for the other.
all results
1.10 PROPOSITION. Let C be a left coproper coalgelmz. Then (1) A vector space M is a left rational and Cm . M = M. (2)
C*-module
+ M is a left Co-module
Every left Co-module contains a unique maximal left rational C*-module.
Proof. It is clear that (1) =S (2), and the “only if” part of 1 is evident from 1.1. If M is a @-module and Co * M = M, th en M is a C*-module with c* . (corn) = (c*co) m. This is well-defined since xim = xzm for xi E Cb *
206
ALLEN AND TRUSHIN
(x1 - x2) m = 0 andy (x1 - x2) m = 0 Vy E Co. Now 3e E Co s.t. e(xL - x2) = x1 - x2 . Hence for any c* E C*, (c*xi) m - (c*xs) m = (c*(x, - ~a)) m = (c*e) (x1 - xs) m = 0. Co . M is thus a locally finite C*-module and Vc* E C*, xm E Co M, cc . xm = &j xtO) . m(c*, Ox. EXAMPLES.
(I)
A finite-dimensional
coalgebra
is coproper.
(2) Let C be pointed irreducible. If I is any nonzero right coideal, then I contains a simple C-subcomodule and so contains the unique grouplike of C. Hence by 1.3, C is (left, right) coproper 0 (0) is cofinite 0 C is f.d. Suppose that C = @aEa C, where each C, is a subcoalgebra. C* = nIaoa C* and Vda:~C4!, we define e,E C* by e, Ic = eC , e, jc =O, 01#/3. For g = rgs E C” and c, E C,, , (e,g, c,) = 0 if 01# G, (e,;, c,) =a(g,, , c,,) if 01= y. Thus w = g, and {e&a are the projections associated with the product. Note that {e,),,a are orthogonal idempotents. Let g E Co. Since C*g is f.d., e, *g = 0 for all but finitely many 01and so g E @ C,* C K,X. Let g=g,+ .‘.+g,, g,EC& i= l,...,e. Since gEP, eMig=g,ECC and C*e,,g = C$,gi C Cm. A quick computation shows that C$,g, C Cm: and so Co C @ Can. Thi reverse containment is clear and hence Co ‘= @ C-0. This enables us to prove 1.11 THEOREM. Suppose that C = @C, where each C, is a subcoalgebra. Then C is left coproper c> each C, is left coproper. 1.12. COROLLARY.
Any direct sum of jktite-dimensional
coalgebras is coproper.
1.13 COROLLARY. If C is cosemisimple then C is coproper. A coalgebra is called indecomposable if it cannot be written as a direct sum of two nonzero subcoalgebras. Using the analogous results for cocommutative irreducible coalgebras in [5], one obtains the fact that C is the direct sum of all the maximal indecomposable subcoalgebras of C and any summand of C is a direct sum of a uniquely determined subset of these maximal indecomposable subcoalgebras which will be called indecomposable components (I.C.‘s). I. 14 COROLLARY.
C is left coproper 0 each I.C. is left coproper.
In this section we discuss the structure of the rational duals of coproper coalgebras. The results for left coproper coalgebras have analogous for right coproper coalgebras which we will assume.
COPROPERCOALGEBRAS
207
2.1 THEOREM (Rational Extension Theorem). Suppose that C is a left coproper coalgebra, N is a Jinite dimensional rational right C*-module, and 9: Co ---f N is a morphism of right Co-modules. Then 3! right C* morphism @: C* -+ N which extends y. Moreover Im rp = Im + and qz is a F-morphism. Proof. Letp: C* + End,(N) be the given representation. For z E ker p n C@, (0) = N . z 2 y(Cn) . z = cp(C%). By 1.1, z E C?z, so v(z) = 0. Hence ker p n C@C ker 9. For c* E Cx, choose x E Co with p(c*) = p(x) and define q(c*) = v(x). If y E Co and p(c*) = p(y), then x - y E ker p n Co and so v(x) = v(y). Hence @ is a well-defined map. Evidently Im 9) = Im y and q is linear. Let c*, d* E C* and suppose that x,y E CD are chosen s.t. p(c*) = p(x), p(d*) = p(y). Then p(c*d*) = p(xy) and so q(c*d*) = I = q(x) y = p(y) (y(x)) = p(d*) q(P) = q(c*) d*. Thus g is a C*-morphism and + clearly extends v. As a result, y is also a C*-morphism. Now let Y be any other C*-morphism which extends 9. Then ker p n Cn C ker y C ker Y. Choose x E C” s.t. p(x) = 1, . Y(1) - Y(x) = Y(1) - Y(I) x = 0 and so Y(1) = Y(x) = q(x) = q(l). Now Vc* E C*, Y(c*) = Y(1) c* = F(I) c * = q(c*) and 9, is unique.
2.2 PROPOSITION. Let C be a left coproper coalgebra, M a f.d. rational right F-module and N a right Co-module. If f : M -+ N is a right Co-morphism, then f(M) is a right F-module in which the F-module action extends the Co-module action, f is a F-morphism and (hence) f(M) is rational. Proof. Define p: C* --f End f (M) as follows: choose x E Co with mx = m Vm G M and set p(c*) (f(m)) = f( m) xc* for all c* E C*. Clearly p is a well defined linear map. For c*, d* E C*, dc*d*)
(f(m))
= f (m) . x(c*d*) = f (m . x(c*d*)) = f ((m . xc*) . d*) = f ((m . xc*) . xd*) = f (m * XC*) * xd* = f(d")
(f(c*)
m4N
and so p is a right representation. If co E Cn, then Vm E M, p(G) f (m) = f(m) . xc0 = f(mx) . cn = f (m) . co and so p extends the original action. Clearly f is a C*-module morphism. Note that 2.2 implies that any extension of a finite dimensional rational C*-module which is split as a Co-module is split.
208
ALLEN AND TRUSHIN
2.3 PROPOSITION. Let C be a left coproper coalgebra and M afinite-dimensional rational right C*-module. TFAE: (1)
M is C*-projective.
(2) M is Co-projective. (3) M is C@-isomorphicto a Cc summand of Con for some n. Proof.
(1 * 2).
We need the following
2.4 LEMMA. (1) Let M be a right Cm-module. Then 3 a right CGrjection i: M + M @Co C*. In addition, if M is a rational C*-module then M &O C* z M (as C*-modules). (2) Let M be a right Cc-module and let NC M &-o C* be a rational right C*-module. Then NC M &a Ior . Proof. (1) m + m @ 1 clearly works for the first part. Let C* module and define v: M &D C* --f M via C m, @ xi + right C*-morphism and q~/Mocolc* is a Co isomorphism. M&~C*!ZM(&D lc* and apply 2.2. Let x = &, mi @ and choose y E Co with m,y = mi . Then
M be a rational 1 mixi . p7 is a We show that Xi E M @co C*,
X=~m,Z’~Xi=~m~~yxi-~miyX~~lc*. i-l
i=l
i=l
(2) Since N is rational, NC0 = N. Choose x = C mi @ xi and pick y E Co with xy = x. Then x = x m, @ (xiy) = C mi(xiy) @ I,* E M @co lo* . We continue with the proof of 2.3. Let P and Q be right Co-modules, g: P+ Q be a Co-epimorphism and f : M + Q a Co-morphism. By 2.2, f(M) is a rational right C*-module and we may assume that P = g-lf (M) and Q = f (M). NOW the diagram M
induces the diagram
p o&l c* goI+
Q Oco C*
of C*-modules. We will use 0, (resp. 0, , 0,) for the Cm-monomorphism m -+ m @ 1 (resp. p -+p @ 1, q-f q @ 1) of M (resp. P, Q) into M &O C* (resp. P @JonC”,
209
COPROPER COALGEBRAS
Q @co C*). Clearly g @ I is surjective and the C*-projectivity of M produces a C*-morphism h: M -+ P @CmC* s.t. g @I . h = f @ I . OM . By 2.2, h(M) is rational so h(M) _CO,(P). We thus obtain an induced Co morphism h,: M-+ P where 02, . h, = h. We now compute e&f
. w4)
= @1(m) 0 1 = (B 0 4 (h,(m) 0 1) = (g 0 I> Q-4 = (f63
1) . bh)
= eo(f(W>
VrnE M.
Thus g . h, =f and M is Co-projective. (2 3 3). Since M is f.d. and right rational, 3~: M* CA 0: C for some n. Thus 3 a continuous epimorphism v*: 0% C* + M and so q~* (0: Co) = M. Since M is G-projective the result follows from this. (3 2 1). Let TK COn--+ M be a Co-projection and let ri: Co + Con be the right Co-module injection onto the ith component. The map vi: rr 0 vi: CD -+ M is a right P-module map and so by 2 1 3! qi: C* -+ M extending vi i = l,..., n. Therefore + = @ qi is a well-defined right P-morphism and q(C*n) = M. Let Y: M + (Co)” be a right Co-morphism s.t. v 0 Y = 1,. By 2.2, Y is a right C*-morphism and clearly $j 0 Y = (0 +J 0 Y = VY = I, and so M is a summand of C*n.
2.5 COROLLARY. Let C be a coproper coalgebra and M a f.d. injective rational Then M* is F-projective.
left F-module.
Proof. C*-modules. as before, We now where C is coalgebras,
It is easy to see that M* is projective with respect to rational right Since Co is right rational and 3 a C*-epimorphism r: Co” -P M* M* is C*-projective. turn our attention to questions about the internal structure of Co (left, right) coproper. Again where a result is stated for left coproper the analogous result also holds for right coproper coalgebras.
2.6 PROPOSITION.
Let C be a left coproper coalgebraover k.
3 idempotents in Co. (2) if e E CD is idempotent then Co = Cn(I - e) @ Cue = (1 - e) Co @ eCD. Thus CCe is finite dimensional, and Cue and eCmare C*-projective. (3) e is a promitive idempotent in Co * Cue is indecomposable. (1)
Proof. (1) Since Co is clearly not nil we may find a non-nilpotent element x in Co. Px is a finite-dimensional non-nilpotent subalgebra of Co, hence contains a nonzero idempotent e. (2)
Standard.
(3)
Standard.
210
ALLEN AND TRUSHIN
2.7 DEFINITION. Let C be a coalgebra C*-summand of Co is a left rational principal for short). 2.8 PROPOSITION. mands.
over k. An indecomposable left indecomposable F-module (RPIM
Let C be left coproper. Then Cm has no nil right Cm sum-
Proof. Suppose C@ = M @N where M, N are right ideals with N nil. Let y E N and choose x E Co s.t. xy = y. Writing x = m + n with m E M, nENwehavey=xy==my+nyEN.ThusmyEMnN=Oandsoy=ny. Since n is nilpotent, y = 0. 2.9 COROLLARY. Let C be coproper and N a left RPIM. Then X = CCe for some primitive idempotent e E C 0. In particular, N is f.d. and C*-projective. Proof. Using the “right” version of 2.8 we see that Co has no nil left or right summands. Thus N is not nil and being locally finite it contains a nonzero idempotent e. Now N = Ne @ N(l - e), so it follows that N = Ne is finite dimensional. The rest of the argument is standard. At this point it is clear that if C is (left, right) coproper, then CC contains primitive idempotents. By an easy Zorn’s Lemma argument one shows that Co contains maximal sets of primitive orthogonal idempotents. 2.10 THEOREM. Let C be coproper. Then Co = @,Ea Cue, for every maximal set of primitive orthogonal idempotents M = {e, j 01E GE>_CCo. We first prove the following
lemma.
2.11 LEMMA. Let C be right coproper and M = {e, j 01E a} a maximal set of primitive orthogonal idempotents in qC. Set N = .ZK’e, and N’ = {x E CC [ xe, =OQe,EM). ThenK’==N@N’. Proof. Define morphism. Since independent, xe, map rr: DC- N kernel N’. Since
rre: C -+ QZ’e, via VJX) = xe, . Clearly z=, is a left C*-module XX’ is f.d. and for x E DC, the set {xe, 1xe, # 0} is linearly = 0 for all but finitely many e, . We can therefore define a by setting V(X) = C xe, : rr is a C*-module morphism with r IN = I, the result follows.
Proof of 2.10. By 2.8, N’ is not nil if it is nonzero. Thus 3 non-nilpotent x E N’ and so UCx C N’ is f.d. and non-nilpotent. Let e # 0 be an idempotent in N’. Then ee, = 0 for each 01 and in addition, 301, ,..., 01~s.t. ese = 0, B # a1 ,.-., 012. Choosing e with 1 minimal we let f = e - eEIe E N’. A small computation shows thatf is a nonzero idempotent. Now esf = ese - eDeale = 0, P f % t.o.9cyrand e,,f = 0 contradicting the minilality of 1. Thus we may assume that e is orthogonal to each e, . From this we easily obtain a primitive idempotent
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211
in N’ orthogonal to each e, , thus contradicting the maximality of n/17.Therefore Co = qC = @qCe,= @C”elll. Observe that the same argument may be used to show that Co = Gacd e,Cn and that each e,Cn is a right RPTM. In fact we have the following. 2.12 THEOREM. Let C be a coproper coalgebra over k. Then every right (left) RPIM is$nite dimensional, projective and isomorphic to eCn(C”e) for someprimitive idempotent e E Co. Moreover, CD is a direct sum of right (left) RPIM’s, and every right (left) RPIM is isomorphic to one of these summands. Proof. We need only prove the last assertion. Let N be a right RPIM and suppose that Co = @ olEae,C* according to 2.10. Since N is f.d., N C &, e,iCn = M for some {eel ,..., eUle}.Furthermore, N is a summand of M and M is f.d. so we may apply the Krull-Schmidt Theorem to obtain the result. 2.13 COROLLARY. Let C be copropw and M a f.d. rational projective C*module. Then M s @ Fe, for suitable primitive orthogonal idempotents {e,} C Co. Proof. 2.3, 2.10, and Krull-Schmidt. At this point we observe that one may readily adapt the appropriate results from [2] chapter 8 to coproper coalgebras C and in particular Co. More specifically, if C is coproper, then 3 indecomposable ideals B, c Co s.t. Co = @ B, , {B,} is unique, and if B is an ideal direct summand of Co, then B = @ Be for some subset {B,} _C{B,}. In addition, it is easily seen that each Be = CEO for some 01where (C,) is the complete set of I.C.‘s for C. We also obtain the following. 2.14 THEOREM. Let C be coproper and M a finite-dimensional rational C*module. Then M is injective *ME @ (Cue,)* for suitable primitive orthogonal idempotents {e,> c Co. Proof. Cue, is C*-projective, and so (Cue,)* is Co-injective. Thus @ (Cue,)* is injective for any choice of idempotents {e,} c Co. Conversely, if M is C* injective, M* is C*-projective and thus M* g @ Cue, which gives the result. 2.15 THEOREM. Let C be a coproper coalgebra, {e,} a maximal set of primitive orthogonal idempotents. Then Rad(Cn) = @ Rad(C*) ear. Moreover, v~r, Rad(Coe,) = Rad(Cn) e, = Rad( C*) e, , Rad(Co) is nil, and Rad(Cne,) is ndpotent. We conclude this section with some results on the structures CD0 = {h E Cm* 1ker )I > cofinite ideal in Co}.
of C, C*O and
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2.16 THEOREM. Let C be a coproper coalgebra. Then C = @ C e, where (e,} C Co and consists of primitive orthogonal idempotents. Each C . e, is injective, and hence C is injective in the category of C-comodules.
Proof. Co = @ Cue, and so C = C Co = C C . COeu = x C . e, . Clearly this sum is direct. Now C * e, is a f.d. summand of C and so (C . e,)* is isomorphic to a rational summand of C*. Thus (C e,)* is projective and C . e, is injective. 2.17
THEOREM.
Let C be a coproper coalgebra over k. Then
(1) croci = coc*o = L(C) (6: c--t c*y. (2) C”OCO = pip = coo. (3)
coo is counitary.
(4) Coo g C as coalgebras and the dual of Co C-t C* is an epimorphism of c*O to coo. Proof. (1) Note first that L(C) is the largest rational right C*-module of C*O, since f(x) E L(C) @ C*O implies x E I(C). Thus C*OP, being rational, is contained in l(C). But L(C) CD = L(C) and so C*sCu = c(C). Similarly COC*O = 40 (2) Let x E Coo and let I be a cofinite ideal (of Co) contained in leer x. Cu = Q&r e,Cn and so all but finitely many e, are in I. Let e = zem4, e, . Then clearly xe = x and so xe E CW’O. Thus CnOCo = PO. Similarly CnCcO = COO.
(3) By (2), Coo = G&a Coo . err. Define x E (Coo)* by x /cam, = e, Ic~O.,a (e, considered in (COO)*). For y E (PO)*, (xy, CDs . e,) = (x, y@O . e,) =
=
Let “0” denote this (4) By 1.10 and (2), C q o is a rational left C*-module. module structure and #: Coo -+ Co @ L(C) be the associated comodule structure. ForxECnandX=y.yECOo=CO.COo,xoh=(IOx)(IOrr)ICl(h)onthe
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one hand and xoh=xo(y.y)=xy.~=x.(y.r)=x.X=(IOx)f(X) on the other. Now C* 0 X is finite dimensional, so by 2.1 c* I+ c* 0 X is the unique extension of x -+ x o h, i.e. (I @ 7~)# = d. Therefore d(Cns) _CPO @ r 0 L(C) and by (3), Coo 2 n 0 c(C). 2.18 THEOREM. Let C be coproper. Then (1)
C*O = c(C) @ (Cn)l as coalgebras.
(2) If M is a locally finite F-module, then ME {m GM j Cum = (0)).
Mrat @ M’ where M’ =
(1) r: C*O c-+ Coo is an epimorphism and r /‘cc) is an isomorphism. (2) Since M is locally finite, 3~: M c--+ 0, C*O = @, t(C) @ Or (Co)+ CnM = Mrat C M. For m E M, write vrn = x, + xi with x, E @JrL(C) and xi E 0, (C 0>I. Now 3e E Cm such that ex, = x, and exi = 0. Thus em = ex, E M and so xi E M also. Then M z M n 0, c(C) @ M n @I (CD)*. We conclude this section by noting that the torsion preradical for Lt. C*-modules M--f Mrat is split for coproper coalgebras. Proof.
3. EXAMPLES We have already seen that coproper coalgebras are hereditary. In this section we present an example of a coalgebra C with Co = (0) = qC’ yet having subcoalgebras D and E with D 1 E, qD = (0) Do dense (in D*) and E coproper. We begin with some preliminary results on incidence coalgebras. Throughout C = C,(X, S) will be a coassociative weighted incidence coalgebra for the relation set S on X (cf. [I] and [5]). (x, t) d enotes an element of S and e(x, y) is the element of C* taking the value 1 on (x, y) and 0 on every other segment. 3.1 THEOREM. It(C) has a basis {(a, 6) 1(--co, u) n (-co,
b) is infinite>.
Proof. I,(C) = (P)l (1.3) is a subcoalgebra therefore by [I] is spanned by the segments which it contains. We will show that a segment (a, 6) is excluded by some cofinite left coideal I if and only if (- CO,a) n (- CO,b) is finite. One direction is trivial for if the latter set is finite then {(x, y) ] y # b} u {(x, b) 1(x, a) # S} is cofinite in S and clearly spans a left coideal excluding (a, 4. In the other direction let I be a cofinite left coideal with (a, b) $1. For x E (-co, u) n (-co, b) a quick computation with (x, b) . e(x, a) shows that (x, a) $1. If (-m, u) n (-00, b) is infinite then there must be a dependence relation of the form C &(x6 ,6) E I for Xi # 0, xi E (- co, u) n (-co, b). Acting
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on this relation on the right by e(x, , x1) forces (x1 , b) E 1, a contradiction. (-00, u) n (-co, b) is finite.
Thus
3.2 COROLLARY. Let C,(X, S) be a coassociative weighted incidence coalgebra satisfying the condition : (-co, u) n (-a~, b) is injkite V(a, b) E S. Then co = (0). 3.3 COROLLARY. Let C,(X, S) be a coassociative incidence coalgebra satisfying the condition: 3(a, b) E S s.t.(-q a) n (-00, b) is finite. Then Co # (0). Moreover if this condition holds V(a, b) E S then Co is dense. One can modify the proof of 3.1 to display a finite-dimensional ideal in C*; viz., J = h{e(z, b) 1z E (-co, u) n (-co, b)}.
rational
left
3.4 PROPOSITION. Let C,(X, S) be a weighted coassociative incidence coalgebra. Assume that Vx E X 3 cofinite left coideal excluding (x, x). Then Co is dense. Proof. As observed in the proof of Theorem 3.1 I,(C) is a subcoalgebra. If subcoalgebra of It(C). By ([I]) IL(C) # (0) let D # (0) b e a minimal D = k{(y, x) 1y, z E (x, x)} for some fixed x E X. But then (x, x) E D lies in every cofinite left coideal, a contradiction. The examples now follow directly. Let X = 77. with the usual ordering, S = {(n, m) 1n < m> and trivial weighting h = 1, C = C,(X, S). Since every ray is infinite (3.2) implies that K’ = (0) = Co. Let Y = N with the usual ordering T = S n (N x N). The inclusions N + Z, T + S induce a monomorphism of D = C,(Y, T) into C. Since every right ray in Y is infinite and every left ray finite (3.2) and (3.3) sh ow that qD = (0) and that Do is dense in D*. Let Z = N, U = {(n, m) 1 1 < n < m < 2n). A small computation shows that the inclusions N - N, U + T induce a monomorphism of E = C,(Z, U) into D. Since every right and left ray in Z is finite we see that E is coproper. We finally note that since the corresponding graphs are connected, each of C, D, E is indecomposable. Finally note that one can easily obtain a coproper subcoalgebra F of C which is an incidence coalgebra of a partially ordered set having neither an upper nor a lower bound. Indeed let X = Z and let I’ = U u {(n, m) 1m < - 1,2m < n ,( m} u ((0, l), (0, O), (--1,O)). Th e inclusion of X in X induces a monomorphism of F = C,(X, V) into C.
REFERENCES 1. H. P. ALLEN AND C. FERRAR, Weighted incidence coalgebras, J. Algebra, to appear. Theory of Finite Groups and Associative 2. C. CURTIS AND I. REINER, “Representation Algebras,” Interscience, New York, 1966.
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3. R. G. HEYNEMAN AND D. E. F~ADFORD, Reflexicity and Coalgebras of Finite Type, J. Algebra 28 (1974), 215-246. 4. BERTRAVEL I-PENG LIN, “On Homological Properties of Coalgebras,” Ph. D. Dissertation, University of Illinois, Circle Campus, 1973. 5. M. E. SWEEDLER, “Hopf Algebras,” Benjamin, New York, 1969.
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