Cyclic homology of monogenic extensions in the noncommutative setting

We study the Hochschild and cyclic homology of noncommutative monogenic extensions. As an application we compute the Hochschild and cyclic homology of the rank 1 Hopf algebras introduced in [L. Krop, D. Radford, Finite dimensional Hopf algebras of rank 1 in characteristic 0, Journal of Algebra 302 (1) (2006) 214–230]. © 2008 Elsevier Inc. rights reserved. ,


Introduction
Let k be a commutative ring with 1. A monogenic extension of k is a k-algebra k [x]/ f , where f ∈ k[x] is a monic polynomial. In [F-G-G] this concept was generalized to the noncommutative setting. Examples are the rank 1 Hopf algebras in characteristic zero, recently introduced in [Kr-R]. In the paper [F-G-G], mentioned above, the Hochschild cohomology ring of these extensions was computed. In the present paper we study their Hochschild, cyclic, periodic and negative homology groups, generalizing the results obtained in [B]. We think that the computations of the type cyclic homology groups of these algebra can be a first step in order to calculate other important invariants such as their K -theory groups. We are also interested in these computations because some crossed products can be present as noncommutative monogenic extensions, and we believe that the calculations made out in this paper may help understand the homology of a such crossed product A # f G, where f is a cocycle with values in A, at least when A is a noncommutative smooth algebra. For the problem of computing the type cyclic homology groups of crossed products we refer to [F-T,N,G-J,A-K,K-R]

Preliminaries
In this section we recall some well known definitions and results that we will use in the rest of the paper.

A simple resolution for a noncommutative monogenic extension
In the sequel we recall the definition of noncommutative monogenic extension and we give a brief review of some of its properties (for details and proofs we refer to [F-G-G]). Let k be a commutative ring, K an associative k-algebra and α a k-algebra endomorphism of K . Consider the Ore extension E = K [x, α], that is the algebra generated by K and x subject to the relations xλ = α(λ)x for all λ ∈ K .
Let f = x n + n i=1 λ i x n−i be a monic polynomial of degree n 2, where each coefficient λ i ∈ K satisfies α(λ i ) = λ i and λ i λ = α i (λ)λ i for all λ ∈ K . Sometimes we will write f = n i=0 λ i x n−i , assuming that λ 0 = 1. The monogenic extension of K associated with α and f is the quotient A = E/ f . It is easy to see that {1, x, . . . , x n−1 } is a left K -basis of A. Moreover, given P ∈ E, there exist unique P

P < n.
In this paper, unadorned tensor product ⊗ means ⊗ K , all the maps are k-linear and all the Kbimodule are assumed to be symmetric over k. Given a K -bimodule M, we let M⊗ denote the quotient M/ [M, K ], where [M, K ] is the k-module generated by the commutators mλ − λm, with λ ∈ K and m ∈ M. Let A 2 α r = A α r ⊗ A, where A α r is A endowed with the regular left A-module struc-ture and with the right K -module structure twisted by α r , that is a · λ = aα r (λ), for all a ∈ A α r and λ ∈ K . We recall that Let Υ be the family of all A-bimodule epimorphisms which split as K -bimodule maps.
Theorem 1.1. The complex which are inverse one of each other up to homotopy. These maps are given by Proposition 1.3. ψ * φ * = id and a homotopy ω * +1 from φ * ψ * to id is recursively defined by ω 1 = 0 and Proof. The equality ψ * φ * = id follows immediately from the definitions. For the second assertion see [G-G,

Mixed complexes
In this subsection we recall briefly the notion of mixed complex. For more details about this concept we refer to [K] and [Bu].
A mixed complex (X, b, B) is a graded C -module (X r ) r 0 , endowed with morphisms b : X r → X r−1 and B : X r → X r+1 , such that bb = 0, B B = 0 and Bb + bB = 0.
A morphism of mixed complexes f : By deleting the positively numbered columns we obtain a subcomplex BN(X ) of BP(X ). Let BN (X ) be the kernel of the canonical surjection from BN(X ) to (X, b). The quotient double complex BP(X )/BN (X ) is denoted by BC(X ). The homology groups HC * (X ), HN * (X ) and HP * (X ), of the total complexes of BC(X ), BN(X ) and BP(X ) respectively, are called the cyclic, negative and periodic homology of X (the nth module of the total complex is the product of all the modules which are in the nth diagonal). The homology HH * (X ), of (X, b), is called the Hochschild homology of X . If we truncate BP(X ) to the left of the pth column we obtain a complex BC(X ) [2p]. Note that and that there is a natural epimorphism It is immediate that Tot(BP(X )) = lim p Tot BC(X ) [2p] and that there is a diagram of short exact se- Taking homology in the above diagram we obtain the following commutative diagram with exact rows The rows in this diagram are name the SBI Connes periodicity exact sequences of X . Finally, it is clear that a morphism f : X → Y of mixed complexes induces a morphism from the double complex BP(X ) to the double complex BP(Y). Let A be a noncommutative monogenic extension of K . The normalized B), where b is the canonical Hochschild boundary map and in which [a 0 ⊗ · · · ⊗ a r ] denotes the class of a 0 ⊗ · · · ⊗ a r in A ⊗ A ⊗ r ⊗. The cyclic, negative, periodic and Hochschild homology groups HC K * (A), HN K * (A), HP K * (A) and HH K * (A) of A, are the respective homology groups of (A ⊗ A ⊗ * ⊗, b, B).

The perturbation lemma
Next we recall the perturbation lemma. We give the version introduced in [C]. A homotopy equivalence data consists of the following: (1) Chain complexes (Y , ∂), (X, d) and quasi-isomorphisms i and p between them.
(2) A homotopy h from ip to id.
A perturbation δ of (1) is a map δ : X * → X * −1 such that (d + δ) 2 = 0. We call it small if id − δh is invertible. In this case we write Δ = (id − δh) −1 δ and we consider A deformation retract is a homotopy equivalence data such that pi = id. A deformation retract is called special if hi = 0, ph = 0 and hh = 0.
In the case considered in this paper the map δh is locally nilpotent, and so (id − δh) −1 = ∞ j=0 (δh) j .
Theorem 1.4. (See [C].) If δ is a small perturbation of the homotopy equivalence data (1), then the perturbed data (2) is a homotopy equivalence. Moreover, if (1) is a special deformation retract, then (2) is also.

Hochschild homology of A
Let k, K , α, f = X n + λ 1 X n−1 + · · · + λ n and A be as in Subsection 1.1. Given an A-bimodule M, we let [M, K ] α j denote the k-submodule of M generated by the twisted commutators [m, λ] α j = mα j (λ) − λm. As usual, we let A e and H K * (A, M) denote the enveloping algebra A ⊗ k A op of A and the Hochschild homology of A relative to K , with coefficients in M, respectively. When M = A we will Theorem 2.1. Let M be an A-bimodule. With the notations introduced in Theorem 1.2, we have: where the boundary maps d * are defined by (2) The maps The composition ψ * φ * is the identity map, and the family of maps is a homotopy from φ * ψ * to the identity map.
Proof. For the first item, apply the functor M ⊗ A e − to the resolution C S (A), and use the identification Let ψ * and φ * be the morphisms induced by the comparison maps ψ * and φ * . The second and third items follow easily from Theorem 1.2 and Proposition 1.3 in a similar way. 2 When M = A we will write C S (A) and C S * (A) instead of C S (A, A) and C S * (A, A), respectively. The following result will be used in the proof of Theorem 3.6.

Corollary 2.2. There is a special deformation retract
The aim of this subsection is to compute the Hochschild homology of A relative to K , with coefficients in A, under suitable hypothesis. We let Z(K ) denote the center of K .
Hence, it will be sufficient to check that if i is not congruent to 0 module n, then [K , K ] α mn+i = K . But this follows immediately from the facts that [λ ,λ] Theorem 2.4. Under the hypothesis of Theorem 2.3, the boundary maps of C S (A) are given by for all m 0. Consequently, if λ n = 0, then the odd boundary maps d 2 * +1 are zero.
where the last equality follows from Theorem 2.3. Again by item (1) of Theorem 2.1 and Theorem 2.3, Theorem 2.4 implies that λλ n − α n (λ)λ n ∈ [K , K ] α mn for all λ ∈ K and m 0. Indeed, this can be proved directly from the hypothesis at the beginning of this paper and then it is true with full generality. In fact, Corollary 2.5. Under the hypothesis of Theorem 2.3, Assume now that k is a field, the hypothesis of Theorem 2.3 are fulfilled, K is finite dimensional over k and α is diagonalizable. Let ω 1 = 1, ω 2 , . . . , ω s be the eigenvalues of α and let K ω h be the eigenspace of eigenvalue ω h . Write Note that 1, λ n ∈ K 1 . We assert that there is a primitive nth root of 1 in k (which implies that the characteristic of k does not divide n), and that all the nth roots of 1 in k are eigenvalues of α. In fact, since α is diagonalizable, we can writeλ = x 1 + · · · + x s , where x i is an eigenvector of eigenvalue w i . Since . . , w s are nth roots of 1 and the least common multiple of their orders is n. Hence, there exist i 1 , . . . , i s ∈ N such that w := w i 1 1 · · · w i s s is a primitive nth root of 1, and so (x i 1 1 · · · x i s s ) i is an eigenvector of eigenvalue w i of α, because α is an algebra morphism.
Theorem 2.6. The chain complex C S (A) decomposes as a direct sum C S

Moreover the boundary maps d
Proof. It follows easily from Theorems 2.3 and 2.4, since the fact that λ n ∈ K 1 implies that if λ ∈ K ω h , then λλ n ∈ K ω h (and so C S,ω h (A) is a subcomplex of C S (A)).
Note that if α n has finite order v (that is α nv = id and α nj = id for 0 < j < v), then for all m 0.
Example 2.8. Let k be a field, K = k [G] the group k-algebra of a finite group G and χ : G → k × a character, where k × is the group of unities of k. Let α : K → K be the automorphism defined by α(g) = χ (g)g and let f = x n + λ n ∈ K [x] be a monic polynomial whose coefficients satisfy the hypothesis required in the introduction. Let Z(G) be the center of G. Assume that there exists g 1 ∈ Z (G) such that χ (g 1 ) is a primitive nth root of 1. Here we apply the results obtained in Section 2 to compute the Hochschild homology of A = K [x, α]/ f relative to K , with coefficients in A (if the characteristic of k is relative prime to the order of G, then k [G] is a separable k-algebra and so, by [G-S, Theorem 1.2], HH K * (A) coincides with the absolute Hochschild homology HH * (A) of A). Note that the hypothesis of Theorem 2.3 are fulfilled, takingλ = g 1 . Since α is diagonalizable Theorem 2.6 and Corollary 2.7 apply. In this case {ω 1 , . . . , ω s } = χ (G), Next we consider another situation in which the cohomology of A can be computed. The following results are very close to the ones valid in the commutative setting.
Theorem 2.9. If α is the identity map, then

Hochschild homology of rank 1 Hopf algebras
Let k be a characteristic zero field and let n 2 be a natural number. Recall that k × denotes the group of unities of k. Let G be a finite group and χ : G → k × a character. Assume there exists g 1 ∈ Z(G) such that χ (g 1 ) is a primitive nth root of 1. In this section we compute the Hochschild homology of the k-algebra A = k [G][x, α]/ x n − ξ(g n 1 − 1) , where ξ ∈ k and α ∈ Aut(k [G]) is defined by α(g) = χ (g)g. We divide the problem in three cases. The first and second ones give the Hochschild homology of rank 1 Hopf algebras. For the sake of completeness we recall from [Kr-R] that A is the underlying algebra of a rank 1 Hopf algebra if ξ(g n 1 − 1) = 0 or χ n = 1. In both cases the comultiplication Δ is determined by Δ(x) = x ⊗ g 1 + 1 ⊗ x and Δ(g) = g ⊗ g for all g ∈ G, the counit by (x) = 0 and (g) = 1 for all g ∈ G, and the antipode S by S(x) = −g −1 1 x and S(g) = g −1 for all g ∈ G.
Let C n ⊆ k be the set of all nth roots of 1. [G]. Since K is separable over k, we know that HH * (A) = HH K * (A). So, by Corollary 2.7, ξ = 0 and χ n = 1. In this case f = x n − ξ(g n 1 − 1) satisfies the hypothesis required in the preliminaries.
In fact α ξ g n 1 − 1 = ξ g n 1 − 1 since α(g n 1 ) = χ (g n 1 )g n 1 = χ (g 1 ) n g n 1 = g n 1 , and ξ g n 1 − 1 λ = α n (λ)ξ g n 1 − 1 for all λ ∈ k [G], since ξ(g n 1 − 1) ∈ Z(G) and α n (λ) = λ, because χ n = 1. Note also that C n is the set of eigenvalues of α, since G is a multiplicative basis of eigenvectors of α, the eigenvalue χ (g 1 ) of g 1 is a primitive nth root of 1 and the eigenvalue χ (g) of every g ∈ G is an nth root of 1 (again because χ n = 1). Moreover, the algebra K = k [G] is separable over k and so, HH * (A) = HH K * (A). By Corollary 2.7, . ξ = 0 and χ n = 1. Let g ∈ G such that χ n (g) = 1. Since we conclude that the ideal x n − ξ(g n 1 − 1) coincides with the ideal x n , g n 1 − 1 . So, A = k[G/ g n 1 ][x, α]/ x n , where α is the automorphism induced by α. We consider now K = k[G/ g n 1 ] and f = x n . These data satisfy the hypothesis of Theorem 2.6 withλ the class of g 1 in G/ g n 1 . Moreover the algebra K = k[G/ g n 1 ] is separable over k and so, HH * (A) = HH K * (A). Thus, by Corollary 2.7,

Cyclic homology of A
Let k, K , α, f = X n + λ 1 X n−1 + · · · + λ n and A be as in Subsection 1.1. In this section we get a mixed complex, simpler than the canonical of Tsygan, computing the cyclic homology of A relative to K .
A simple tensor a 0 ⊗ · · · ⊗ a r ∈ A ⊗ A ⊗ r will be called monomial if there exist λ ∈ K \ {0}, 0 i 0 < n and 1 i 1 , . . . , i r < n such that a 0 = λx i 0 and a j = x i j for j > 0. We define the degree of a monomial . , x n−1 is a basis of A as a left K -module, each element a ∈ A ⊗ A ⊗ r can be written in a unique way as a sum of monomial tensors. The degree deg(a), of a, is the maximum of the degrees of its monomial tensors.

Proof. Let x
By the definition of ω r+1 it suffices to show that ω r+1 (x 1 ) is a sum of tensors of the form with j 0 + · · · + j r+2 i 1 + · · · + i r . Using the formulas for φ r and ψ r establish in Theorem 1.2 it is easy to see that The fact that w r+1 (x 1 ) can be expressed as a sum of simple tensors satisfying the mentioned above property follows now by induction on r, since Proof. By Theorem 2.1 we already know that the Hochschild homology of (C S * (A), d * , D * ) is the Hochschild homology of A relative to K . Let By the perturbation lemma, in order to prove the assertion for the cyclic homology it suffices to check that there is a special deformation retract Finally, in order to prove the assertion for the periodic and negative homology it suffices to show that the maps Φ, Ψ and W commute with the canonical surjections In fact, from this, the fact that and (3), it follows that there is a special deformation retract which immediately implies the assertion for the periodic homology, and also for the negative homology, because from the existence of a commutative diagram with exact rows 0 with Φ and Φ quasi-isomorphisms, it follows that there is a quasi-isomorphism Tot BN(X ) → Tot BN(X ) making the diagram commutative.
Next we prove there is a special deformation retract (3) satisfying the above required conditions. Let be the special deformation retract obtained in Corollary 2.2. Consider the perturbation induced by B.
Applying the perturbation lemma we obtain a special deformation retract and d n = j 0 ψ n−2l+2 j+1 (Bω) j Bφ n−2l on C S n−2l (A). In order to finish the proof it suffices to show that ψ r+2 j+1 (Bω) j Bφ r = 0 for all j > 0. Assume first that r = 2m. By the definition of φ 2m and Proposition 3.1, On the other hand ψ 2m+2 j+1 vanishes on elements of degree less than (m + j)n. The fact that ψ r+2 j+1 (Bω) j Bφ r = 0 for all j > 0 follows by combining theses facts. The case r = 2m + 1 is similar. 2 Recall from Subsection 1.1, that given P ∈ E, there exist unique P and

P < n.
Theorem 3.3. The Connes operator D * is given by Proof. Besides the notations introduced in Theorem 1.2 we use the following ones.
Lastly, ψ 2m+1 (Υ i ) = 0 except if 1 = · · · = m = n − 1. In this last case i 1 = · · · = i m = n. So The expression for D 2m follows immediately from all these facts. 2 Remark 3.4. Another formula for D 2m useful for some computations is the following This follows from Theorem 3.3 and the fact that

Explicit computations
Let k, K , α, f = X n +λ 1 X n−1 +· · ·+λ n and A be as above. In this subsection we compute the cyclic homology of A relative to K , under suitable hypothesis. We will freely use the notations introduced at the beginning of Section 2 and below Corollary 2.5. Recall that by Theorem 2.3, if there exists then λ 1 = · · · = λ n−1 = 0 and Moreover, by Theorem 2.4, the Hochschild boundary maps of the mixed complex (C S * (A), d * , D * ) are given by We now compute the Connes operator D * .
Proof. It follows immediately from Theorem 3.3. 2 Theorem 3.6. Assume the hypothesis of Theorem 2.6 are fulfillled. Then the mixed complex ( decomposes as a direct sum Proof. It follows immediately from Theorem 3.5. 2 In the rest of this section we assume that k is a characteristic zero field and that hypothesis of Theorem 2.6 are fulfilled. We let HC Theorem 3.7. The cyclic, negative and periodic homology of A relative to K decompose as Moreover, we have: We first compute the homology in degree 2m. Let ι : X 0 → X 2m ⊕ X 2m−2 ⊕ · · · ⊕ X 0 be the canonical inclusion. By using that each D ω h 2i+1 map is an isomorphism it is easy to see that ι induces an epimorphism A direct computation shows now that the boundary of [ζ 2m+1 ]x n−1 , . . . , [ζ 1 ]x n−1 ∈ X 2m+1 ⊕ · · · ⊕ X 1 equals ι ([λ]) if and only if and . The assertion about HC K ,ω h 2m (A) follows easily from these facts. We now are going to compute the homology in degree 2m + 1. It is immediate that [ζ 2m+1 ]x n−1 , . . . , [ζ 1 ]x n−1 ∈ X 2m+1 ⊕ · · · ⊕ X 1 is a cycle of degree 2m + 1 if and only if it satisfies (4) and ζ 2m+1 λ m+1 Remark 3.8. Theorem 3.7 applies in particular to the monogenic extensions of finite group algebras K = k [G] considered in Example 2.8. Note that since K is a separable k-algebra, this computes the absolute cyclic homology, as follows easily from [G-S, Theorem 1.2] using the SBI-sequence.

Cyclic homology of rank 1 Hopf algebras
Let k, G, χ , g 1 , α and A be as in Subsection 2.2. Here we compute the cyclic homology of A. Let C n ⊆ k be the set of all nth roots of 1. As in the above mentioned subsection we consider three cases. ξ = 0. That is A = K [x, α]/ x n , where K = k [G]. Since K is separable over k, from Theorem 3.7 it follows that HC 2m (A) = K [K , K ] , x n−1 . ξ = 0 and χ n = 1. In this case A = K [x, α]/ x n − ξ(g n 1 − 1) , where K = k [G]. Arguing as in Subsection 2.2, but using Theorem 3.7 instead of Corollary 2.7, we obtain {λ ∈ K ω : λ(g n 1 − 1) m+1 ∈ [K , K ] ω } [K , K ] ω x n−1 . ξ = 0 and χ n = 1. In this case A = K [x, α]/ x n , where the algebra K = k[G/ g n 1 ] and α is the automorphism induced by α. Since K is separable over k, from Theorem 3.7 it follows that HC 2m (A) = K [K , K ] , x n−1 .

The periodic and negative homology
Assume that k is a characteristic zero field and that the hypothesis of Theorem 2.6 are satisfied.
The aim of this section is to compute the periodic and negative homology of A when α has finite order.
In the following remark we compute the maps of the SBI exact sequence of the mixed complex ω h * ) of Theorem 3.6. We will use the notations introduced above Theorem 3.7.
Remark 4.1. From the computations of Theorem 3.7 it follows that: (1) If h = 1 or ω n h = 1, then the map is the identity map.