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Observations on Crossed Products and Invariants of Hopf Algebras

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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 By 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.
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