Observations on Crossed Products and Invariants of Hopf Algebras

Arch. Math., Vol. 63, 119-127 (1994) 0003-889X/94/6302-0119 $ 3.30/0 1994 Birkh~iuser Verlag, Basel

Observations on crossed products and invariants of Hopf algebras

MARIA E. LORENZ and MARTIN LORENZ

Introduction. Let B = A ~ ~ H denote a crossed product of the associative algebra A with the finite-dimensional Hopf algebra H. By studying the process of induction of modules from A to B in the case where H is pointed we show in Section 1 that the Jacobson radicals of B and A are related by

J(B) dimkH ~ J(A) B.

We then specialize to the situation where A is an H-module algebra, the cocycle a is trivial (so B is a smash product), and the trace map from A to the algebra of H-invariants A n is surjective. Making essential use of the well-known Morita context linking B with the algebra of H-invariants A H we investigate the transfer of properties from A to An. In Section 3 we show, for example, that if A is right Noetherian (right Artinian) then so is A H. In fact, A is Noetherian (Artinian) as right An-module in this case. Furthermore, if Kdim (AA) exists then Kdim (A~H) exists as well and is bounded above by Kdim (AA). The results concerning the Noetherian property have first been obtained by S. Montgomery ([7], Theorem 4.4.2) and have motivated much of our research in this section. Further- more, we extend most of [5], Theorem 3.3 from group algebras to pointed Hopf algebras. For example, we show that if the right A-module W is Noetherian (Artinian) then WA~, the restriction of Wto A H, is likewise. Further, Kdim (W) exists if and only if Kdim (W~) does and in this case both are equal. Finally, we prove the following estimate for the right global dimension of AH:

r.gldim A u < r.gldim B + min {fdim AHA, pdim AA, } < r.gldim A + gldim H + min {fdim AHA, pdim AA~ } .

Here, fdim denotes flat dimension and pdim denotes projective dimension. Most of the results in Section 3 follow in a fairly straightforward manner from some general facts about Morita contexts which are established in Section 2. This is independent of Hopf algebras and may therefore be of interest in its own right. It is a pleasure to thank Susan Montgomery for helpful discussions and for communi- cating Example 1.2 to us. 120 M.E. LORENZ and M. LoRENZ ARCH. MATH.

Notations. Our references for general material about i-topf algebras are the standard texts [1] and [10]. For crossed products in particular we follow the notes [71. Throughout this article, we will keep the following notations.

k denotes a commutative field; H will be a Hopf algebra over k, with comultiplication A and counit ~ ; A denotes an associative k-algebra so that there is a weak H-action on A, denoted (h, a) ~ h' a (h ~ H, a e A); B = A # ~ H will denote a crossed product, with cocycle a : H x H -> A.

Thus B is an associative algebra such that there is an isomorphism of left A-modules A| a| The map a ~ a # 1 identifies A with a subalgebra of B. Defining a k-linear map 7:H~Bby 7(h)=l:~h (hell), we have a), (h) : a # h for a ~ A, h ~ A. It is known (cf. [7], Chap. 7) that 7 is convolution invertible and satisfies the following identity, for h c H and a E A,

(*) v(h) a --- 2(h~" a) ~(h2). Finally, J (-) always denotes the Jacobson radical, e' (.) denotes composition length, and Kdim (.) denotes the Krull dimension in the sense of Gabriel and Rentschler.

1. The Jacobson radical 1.1. Free ring extensions. In this subsection, we collect a few general facts about ring extensions R ~ S (same 1) so that S is free as left R-module. Much of this material is known and we give suitable references to the literature whenever available, Recall that the Loewy length of a module W is the smallest integer t so that WJ t = 0, where J denotes the Jacobson radical, or oo if no such t exists.

Lemma. Let R c= S be an extension of rings so that S is j'ree as left R-module. Then: (a) R is a direct summand of S as left R-module. (b) J (S) c~ R ~ J (R). (c) For any right R-module V, the annihilator ann s (V | is the largest ideal orS that is contained in annR(V)S. (d) If there exists afinite upper bound d for the Loewy lengths ofall S-modules V| a S, where Visa simple right R-module, then J (S) a ~= J(R)S. P r o o f. (a) Let {s~} be a left R-module basis of S and write 1 = ~ risi (ri ~ R). Putting I = ~ riR we must have I = R, for otherwise I S # S, by freeness of S over R, contradict- ing the fact that 1 ~ IS. Thus we can write 1 = 52rit i with h ~ R and defining ~ :RS~RR by ~(sl) = tl we obtain the required projection which is the identitiy on R. (b) With U(. ) denoting unit groups, part (a) implies that U(S) c~ R = U(R) which in turn yields (b) (cf. [8], hemma 7.1.3). (c) This is proved in [3], 10.4. Vol. 63, 1994 Crossed products of Hopf algebras 121

(d) By assumption and (c), J(X)d~= anns(V@RS ) ~= annR(V)S holds for all simple R-modules V. Consequently,

J(S) a ~= ~ (annR(V)S) = (0 annR(V))S = J(R)S, V V where Vruns over the simple R-modules and where the first equality uses freeness of S over R. This completes the proof. []

1.2. Induced modules. We now return to the ring extension A ~ B = A 4~oH. Part (a) of the lemma below implies that if H is finite-dimensional and pointed then the hypothesis of Lemma 1 (d) is satisfied, with d = dimk H. (Note that the composition length clearly is an upper bound for the Loewy length.)

Lemma. Assume that H is finite-dimensional and pointed. Let V be a right A-module. Then: (a) vz (V@A BA) = ~ (V) " dim k H. Consequently, ~ (V| a Be) < ~e(V) dimk H. (b) V is Noetherian (Artinian) if and only if V| BA is. In this case, V| Be has the same property. (c) The Loewy length of V| BA is bounded above by the product of the Loewy length of V with the length of the eoradical series of H. P r o o f. Let {H,} denote the coradical filtration of H and let G = G(H) denote the set of group-like elements of H. So

H_I=O~Ho=kG~= =...=~HnCH.+I =...=~Ht=H n for some t. Using formula (.) and the fact that AH, ~= ZHi | H,-i we see that each 0 A ~r'(H,) is an A-A-subbimodule of B which is a direct summand of B as left A-modules (since A B "~ A| Thus we can define A-submodules W~ __c V| A by VV, = V| A), (H,). Put W = W,/VV,_ 1 and H, = H,/H,_ 1 (n > 0). The lemma is a consequence of the following more precise assertion: Each W is isomorphic to a direct sum of dimk H~, many G-conjugates of V. Since each G-conjugate of V has the same (composition and Loewy) length as V, and is Noetherian (Artinian) if and only if V is, all claims in the lemma about V| AB A follow from this assertion. The statements about V| are immediate. For the proof we use the Taft-Wilson theorem (cf. [7], p. 64) which implies that every h ~ H, can be written in the form h = ~ hx with x~G (**) hx~H,,Ahx=x| x rood H,| H,_I . Fix such elements hx ~ H, (for various x e G) so that their images in H, form a k-basis of H, and let V ~ the image of V| A A 7 (hx) in W. Then, as k-space, W is the direct sum of the various V ~. Moreover, for v ~ V, a e A one computes using (*) and (**) 122 M.E. LORENZ and M. LORENZ ARCH. MATH.

(v|174 modW,_ 1 =v(x.a)| modW,_ 1. Thus, letting vx denote the image of v | 7(h~) in W, we have a k-linear isomorphism V--~+ V x, v ~ vx, so that

v~ a = (v(x' a)) ~ . This shows that V x is the conjugate of Vcorresponding to (the inverse of) the automor- phism a ~ x - a of A, and our assertion is proved. [] The following example shows that part (a) of the above Lemma does not extend to the non-pointed case, even in the special case of smash products. The example is due to Susan Montgomery and Don Passman. E x a m p 1 e. Let G be a finite group and put A = k [G], the group algebra of G, and H = (K[G])*. Then A is an H-module algebra via h. g = h(g)g (h e H, g E G) and so we can form the smash product B = A # H, Now let V be any right A-module and put V= V| A. Then

~- V| with the diagonal action of G on the right. To see this, note that V= ~ V| p~, where pooH is the projection, po(x)= 6o, ~ for g, xeG. The G-action is given by (v | Po)x = v x | Px-1 o" Thus the required isomorphism is given by v | pg ~-, v | 9-t. As is well-known, V| a ~- A~ im~vJ, and hence ~ A~ im~v) , the free right A-module of rank dimk V. Consequently, r (V) = dim,, V. r (A). Now assume that G is non-abelian and let k be a splitting field for G so that char k does not divide the order of G. Assume further that V is irreducible of maximal dimension. Then dim k V. e(A) = dimk V'~ dim k W> ~ (dimk W) 2 -= dimk H, W W where W runs over a full set of irreducible A-modules. Thus (a) of the Lemma does not hold here. - We also note that, by choosing k so that char k divides the order of G in the foregoing, one sees that V| A need not be completely reducible, even though V is irreducible and H is semisimple (cf. [7], Example 7.4.4).

1.3. Jacobson radicals. The following proposition is the main result of this section. Part (a) is immediate from Lemma 1 (b) while part (b) is a consequence of Lemma 1 (d) and Lemma 2.

Proposition. (a) J(B) c~ A ~ J(A). (b) If H is finite-dimensional and pointed then J (B) dim~ ~ J (A) B. Vol. 63, 1994 Crossed products of Hopf algebras 123

2. Some results on Morita contexts.

2.1. Morita eontexts. In this section, let (R, S, RPs,sQ~t,f, g) be a fixed Morita context. Here, R and S are arbitrary rings (with 1) and

f:P| and g:Q| are bimodule maps which satisfy the associativity conditions

f(p|174 and g(q|174 for p, p' E P and q, q' ~ Q. Throughout this section we will assume that the map f is surjective. We note that this assumption implies that RP and QR are generators, Ps and sQ are finitely generated projective, and f is an isomorphism (cf. [2], Theorem II.3.4).

2.2. Chain conditions. In the following lemma, we use ~(. ) to denote the lattice of submodules of the module in question.

Lemma. For each S-module Vs, there exist order preserving maps

~(V| ~ ~ ' ~(Vs) with 7c o # = id. Moreover, 7r respects direct sums.

P r o o f. For any submodule U of V | QR define #(U) = Im(U| in--cl| V@sQ| idv| ' V| = r).

Then #(U) is a submodule of Vs, being the image of an S-module homomorphism, and p is clearly order preserving. Similarly, for any submodule W of Vs, define (W) =Im (W | QR- incl| ida, V| s QR) .

Again, ~(W) is a submodule of V| R and ~ is order preserving. Moreover, the map incl | id e is injective, since sQ is projective. Thus, if W1 and Wz are submodules of Vs with zero intersection, then ~(wl | w~) = ~(wl) | ~(w2) Finally, u(#(U)) = U "f (P | . To see this, note that rc(#(U)) is the image of the map

U @RP| in~cl| | V| |174 idv|174 V|174 = V|

Consider u = ~ vi | qi e U. Then the image of u | p | q under this map is

2 v~g (qi | P) | q = ~2 vi | g (q~ | P) q = ~2 v~ | ql f (P | q) = u f (p | q), which proves the above equality. Since f(P| R, we get u(#(U))= U, as required. [] 124 M.E. LORENZ and M. LORENZ ARCH. MATH.

Proposition. (a) Let Vs be an S-module. Then: [1) If Vs is Noetherian (Artinian) then V| R is likewise. (2) ~*(V| <= e(Vs). (3) IfKdim (Vs) exists then so does Kdim (V | and Kdim [V | <= Kdim (Vs). (4) If Vs is completely reducible then so is V| R. ib) If S is right Noetherian (right Artinian, semisimplei then so is R. Furthermore, if Kdim (Ss) exists then Kdim (RR) exists too and is bounded above by Kdim (Ss). P r o o f. (a 1)-(a 3) are clear from the lemma. For (a4), assume that Vs is completely reducible. Then for any submodule U of V| QR there exists a submodule Wof V with tl(U) e W= V. Applying ~ to this, we obtain U | 7~(W) = 7z(V) = V| Therefore, V| R is completely reducible. For (b), take V= S s in ta) to deduce that QR is Noetherian ~Artinian, completely reducible) if Ss has these properties. Moreover, if Kdim (Ss) exists then Kdim (Qe) exists too and is bounded above by Kdim (Ss). Since Q~ is a generator. RR inherits all these properties, with Kdim (RR) _---

2.3. Homologieal dimension. Lemma. Let VR and WR be right R-modules. Then there are third quadrant spectral sequences Ef 'q = Ext~(Tor~(V, P), HomR( Q, W)) T Ext](V, W) and E f'q = ExtP(V| Ext,(Q, W)) ~ Ext](V, W).

P r o o f. Let U s be a given right S-module. Letting 93ls denote the category of right S-modules and similarly for other rings we define functors G : ~J~R--~ ~i~S, G(X) = X | s and F : ~s ~ ?O~g, F(Y) --- Horn s (Y, U). Then FG(X) = Horn s (X | U) ~ Homg (X, Horn s (P, U)) via the adj oint isomorphism ([9], Theorem 2.11). Also, since Ps is projecti~e, G (X) is right F-acyclic if Xg is projective, that is, Extis (G(X), U) = 0 holds for i > 0. Grothendieck's theorem ([9], Theorem 11.40) now gives a third quadrant spectral sequence

Ef 'q = Ext~ (Torff (V, P), U) T Ext,(V, Horn s (P, U)).

Now, for a given WR, take U = Horn R (Q, W), viewed as right S-module via (fs)(q) = f(s q), as usual. Then the adjoint isomorphism and the fact that P @s Q -~ R via f together imply that Homs (P, U) ~ Homa(P| W) ~- HomR (R, W) ~ W. VoL 63, 1994 Crossed products of Hopf algebras 125

Thus the above spectral sequence yields the first spectral sequence of the lemma. The second spectral sequence is established similarly using the functors G : !~J~R ~ 9Jrs, G(X) = Horn R (Q, XJ and F : 9J~s -~ .~i~z, F(Y) = Horn s (V| Y). []

Corollary. r.gldim R < r.gldim S + rain {fdimg P, pdim QR} - P r o o f. Ifn = p + q > r.gldim S + fdimeP then p > r.gldim S or q > fdimeP and so Ext] (Tor~ (., P),.) = 0. The first spectral sequence in the lemma now gives Ext~ (.,.) = 0 which proves that r.gldim R < r.gldim S + fdimg P. Similarly, the second spectral sequence implies that r.gldim R < r.gldim S + pdim QR- [] The above spectral sequences also yield estimates for the homological dimensions of modules. For example, if VR is a right R-module then arguing as above using the second spectral sequence of the lemma one obtains pdim VR < pdim V| s + pdim QR. For further results on homological dimensions in Morita contexts we refer to [6].

3. Applications to smash products and invariants 3.1. Preliminaries. Throughout this section, A will be a left H-module algebra and H will be finite-dimensional. Fix a left integral 0 # t ~ H. Then there is a Morita context (R, S, R Ps,sQR, f, 9) with S = A 44=H, the smash product of A and H, R = A n, the algebra of H-invariants, and P = Q = A, with suitable bimodule actions. For details we refer to [7], Sect. 4.5. Letting t : A -~ A n, t(a) = t . a denote the trace map afforded by t, the maps fand g in the Morita context are given by f = (',") : A @A#H A ---->A H, (al, a2) = t(a 1 a2) and 9=[',']:A| ~ H, [al,a2]=alta2. We will assume throughout this section that f is surjective or, equivalently, that the trace map is sutjective. Note that this assumption is independent of the choice of t, since t is determined up to a scalar. Furthermore, the assumption holds, for example, if H is semisimple. For any right A~H-module Vand any linear form ~b ~ H*, we put V~={v~Vlvh=(9(h)v for allh~H}. Thus V~ = V H, the H-invariants in V. It is easily checked that each V~ is in fact a module over A n. Below, the case where ~b = cr is the so-called distinguished group-like element will be important (cf. [7], 2.2.3). The element e satisfies th=c~(h)t (h~H). 126 M.E. LoRENZ and M. LORENZ ARCH. MAT~.

For unimodular H (e.g., for H semisimple), one has 7 = e.

3.2. Chain conditions. The following result is an application of Proposition 2.2. Similar conclusions could be drawn in the case where g is surjective (which holds, e.g., when A ~ H is a simple ring). The Noetherianness statements in part (b) below are due to S, Mont- gomery ([7], Theorem 4.4.2).

Theorem. (a) Let V be a right A #H-module Then: (1) If V is Noetherian (Artinian) then V~ is likewise (as module over An), (2) e(V~) < e(V). (3) If Kdim (V) exists then so does Kdim (V~) and Kdim (V~) < Kdim (V). (4) If V is completely reducible then so is V~. (b) If A is right Noetherian (right Artinian) then so is An. In fact, A is Noetherian (Artinian) as right An-module in this case. Furthermore, if Kdim (An)exists then Kdim (A~H) exists too and is bounded above by Kdim (AA). Finally, if A and H are semisimple Then so is An. P r o o f. (a) In view of Proposition 2.2(a), it suffices to show that

V~ ~ V| as right An-modules. For this, fix a c A with {(a) = ] and put e = at ~ A 4~ H. Then tat = t holds in A~H, and hence e 2 = e, and AenA _~ A~Ht = A#He as A~H-A H- bimodules. Therefore, V| A_~ Ve as An-modules. Finally, one checks that Ve = V~ which completes the proof of (a). (b) This is an application of Proposition 2.2 (h). We only have to make sure that the above properties for A entail the corresponding properties for A~H. Inasmuch as A ~: H is a finitely generated right A-module, this is clear for right Noetherianness, right Artinianness, and existence of Krull dimension, with Kdim (A~HAe~)< Kdim (A #k HA) _-< Kdim (AA). Finally, semisimplicity of A :~ H :follows from semisimplicity of A and H by [7], Theorem 7.4.2: [] We now consider restriction of A-modules to A n. The following results extends most of [5], Theorem 3.3 from group algebras to pointed Hopf algebras.

Proposition. Assume that H is pointed. Let Wbe a right A-module. Then: (1) If W is Noetherian (Artinian) then WAn is likewise. (2) ((WAH) < r dimkH. (3) Kdim (W) exists if and only if Kdim (WAH) does and in this case both are equal. P r o o f. The result is an application of part (a) of Proposition 2.2, with Vthe induced A~H-module V= W@AA#H. Note that V@A4~ItAAH ~ WAH, Thus we have to keep track of the transfer of properties from W to V for which we use Lemma 1.2. This result immediately takes care of the Noetherian, Artinian, and length statements, thereby proving parts (1) and (2). For Kdim, note that if Kdim (WAH)exists then certainly Kdim (W) does and is bounded above by Kdim (WAH). SO assume that Kdim (W) exists. As we have shown in the proof of Lemma 1.2, VA has a finite series of submodules with factors isomorphic to Conjugates Vol. 63, 1994 Crossed products of Hopf algebras 127 of IV.. This implies that Kdim (VA) exists and is equal to Kdim (W). Consequently, Kdim (VA~H) also exists and is bounded above by Kdim (W). Proposition 2.2 (a3) now implies that Kdim (WA~) exists and is bounded above by Kdim (W), which completes the proof of (3). []

3.3. Homological dimension. Proposition. r.gldim A n < r.gldim A #H + rain {fdimA,A, pdim AA~} < r.gldim A + gldim H + min {fdim AnA, pdim AA~ } . P r o o f. The first inequality is an application of Corollary 2.3 and the second inequality follows from [4], Corollary 4. (Note that r.gldim H = 1.gldim H, because the antipode of H is an antiautomorphism of H.) [7

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Eingegangen am 9.9. 1993 Anschriften der Autoren: M. E. Lorenz M. Lorenz Department of Mathematics Department of Mathematics University of Pittsburgh Temple University Pittsburgh, PA 15260 Philadelphia, PA 19122 USA USA