Module Structures on the Hochschild and Cyclic Homology of Graded Rings

B.H. Dayton Department of Mathematics Northeastern Illinois University Chicago, IL 60625, U.S.A.

C.A. Weibel Mathematics Department Rutgers University New Brunswick, NJ 08903, U.S.A.

1 Introduction

Module structures on various cohomology and K-theory groups have proved useful in

performing calculations (see [Bloch, D, DW, GRW, LR, S-TP, S-OE, SvdK, W-MVS,

W-MGR, WG]). In this paper we will show how to define module structures on

Hochschild homology, Cyclic homology and Kähler differentials for certain graded

rings.

In this paper k will be a commutative ring (usually a field or Z), R will be a

commutative k–algebra and Λ a unitary associative R–algebra. We will write HH_{∗}(Λ) for the Hochschild homology group HH_{∗}^{k}(Λ; Λ) and HC_{∗}(Λ) = HC_{∗}^{k}(Λ) for the Cyclic homology group of the k–algebra Λ. Ω_{Λ/k}^{n} is the module of Kähler differentials, H_{dR}^{n}(Λ) is the classical deRahm cohomology and HH_{dR}^{n}(Λ) is the non-commutative deRahm cohomology. Finally, W (R) is the ring of Witt vectors over R, whose underlying abelian group is 1 + tR[[t]].

Theorem 1.1 Let k be a commutative ring, R a commutative k–algebra and Λ = R ⊕ Λ_{1} ⊕ Λ_{2} ⊕ ⋯ a unitary associative graded R–algebra. Then the following have natural W (R)–module stuctures.

• Ω_{Λ/k}^{n}/Ω_{R/k}^{n} if Λ is commutative.

• H_{dR}^{n}(Λ)/H_{dR}^{n}(R) if Λ is commutative.

• HH_{n}(Λ)/HH_{n}(R)

• HH_{dR}^{n}(Λ)/HH_{dR}^{n}(R)

• HC_{1}(Λ)/HC_{1}(R)

• HC_{n}(Λ)/HC_{n}(R) if Q ⊆ k or R = k.

Our technique, following [W-MGR], is to deduce the general graded case from the special case Λ[x] where Λ is concentrated in degree 0 and x has degree 1.

Theorem 1.2 Let k be a commutative ring, R a k–algebra and Λ a R–algebra. Then the following have natural W (R)–module structures. In fact they are Carf(R)– modules.

• Ω_{Λ[x]/k}^{n}/Ω_{Λ/k}^{n} if Λ is commutative.

• H_{dR}^{n}(Λ[x])/H_{dR}^{n}(Λ) if Λ is commutative.

• HH_{n}(Λ[x])/HH_{n}(Λ)

• HH_{dR}^{n}(Λ[x])/HH_{dR}^{n}(Λ)

• HC_{1}(Λ[x])/HC_{1}(Λ)

• HC_{n}(Λ[x])/HC_{n}(Λ) if Q ⊆ k or if R = k.

Here Carf(R) is a subring of the Cartier ring Cart(R) of natural endomorphisms of W (R). The elements of Carf(R) and Cart(R) are sums of the form ∑ V_{m}[r_{mn}]F_{n} where V_{m} is the Verschiebung, [r] is the homothety and F_{m} is the Frobenius operator. We prove that the groups mentioned in Theorem 1.2 are Carf(R)–modules by using decomposition theorems for the functors Ω^{∗}, HH_{∗} and HC_{∗} on polynomial extensions. Since W (R) is the subring of Carf(R) consisting of elements ∑ V_{m}[r_{m}]F_{m} it then follows that these groups are W (R)–modules.

It should be noted, however, that having a Carf(R)–module structure is a much stronger property than having a W (R) module structure. For example, for a general graded ring the groups in 1.1 are W (R)–modules but not Carf(R)–modules. As another example, if Q ⊆ k then the decomposition of HC_{n}(Λ[x]) given by [K-CCM] follows easily from the fact that HC_{n}(Λ[x]) is a Carf(R)–module.

The W (R)–module operations on Hochschild and cyclic homology are compatible with the module operations on K-theory.

Theorem 1.3 Let k, R and Λ be as in Theorem 1.1. The Dennis trace map D : K_{n}(Λ)/K_{n}(R) → HH_{n}(Λ)/HH_{n}(R) is a W (R)–module homomorphism. Further if Q ⊆ k then D factors through B : HC_{n−1}(Λ)/HC_{n−1}(R) → HH_{n}(Λ)/HH_{n}(R) as a W (R)–module homomorphism. If Λ is a polynomial ring the Dennis trace map is a linear map of Carf(R)–modules.

If Q ⊆ k then W (R) contains R as a subring so a W (R)–module is also an R– module. However the R–module structures on the the groups in Theorem 1.1 and Theorem 1.2 induced from W (R) will generally be different from the expected R– module structure. For example, for Kähler differentials we have the following explict description of the W (R) and R–module structures.

Theorem 1.4 If Λ = R ⊕ Λ_{1} ⊕ Λ_{2} ⊕ ⋯ is a commutative graded algebra over R then Ω_{Λ/k}^{n}/Ω_{R/k}^{n} is a W (R)–module where the operation is given as follows. Let a_{0}, a_{1}, …, a_{n} be homogeneous elements of A. Let ℓ = deg a_{0} +deg a_{1} +⋯+deg a_{n} > 0 and let ω = a_{0}da_{1} ∧ da_{2} ∧ ⋯ ∧ da_{n}. Then

(1 − rt^{m}) ∗ ω = mr^{ℓ/m}ω+

r^{ℓ/m−1}a_{0} ∑_{i=1}^{n}(deg a_{i})a_{i}da_{1} ∧ ⋯ ∧ da_{i−1} ∧ dr ∧ da_{i+1} ∧ ⋯ ∧ da_{n}

if m|ℓ and (1 − rt^{m}) ∗ ω = 0 otherwise. Furthermore, if k contains Q then Ω_{Λ}^{n}/Ω_{R}^{n} has an R–module structure induced by W (R) satisfying (for ω above)

r ∗ ω = rω + a_{0} ∑_{i=1}^{n} (\frac{deg a_{i}}{ℓ} ) a_{i}da_{1} ∧ ⋯ ∧ da_{i−1} ∧ dr ∧ da_{i+1} ∧ ⋯ ∧ da_{n}

It should be noted that any graded R–module A = A_{1} ⊕ A_{2} ⊕ ⋯ has a W (R)– module structure given by (for a_{i} ∈ A_{i})

(1 − rt^{m}) ∗ a_{i} = \{ \begin{matrix}mr^{i/m}a_{i} & if m|i \\ 0 & otherwise \end{matrix}

We call this W (R)–module structure the structure induced by “restriction of scalars along the ghost map.” The formulas in 1.4 show that the W (R)–module structure on Ω_{Λ}^{n}/Ω_{R}^{n} is not induced by the ghost map in general.

As an application of our results we improve and correct the computation of the Ktheory of a union of lines through the origin of affine m-space Ak^{m} contained in [W-LL].

For a functor F : Graded Rings → Abelian groups we write F̃(A) = ker(F (A) → F (A_{0})) ≈ F (A)/F (A_{0}) where A_{0} is the degree zero part of A. We then have

Theorem 1.5 Let A be the coordinate ring of b + 1 lines through the origin of Ak^{m} where k is a field of characteristic 0. Let C be the seminormalization of A. Then there are W (k)-modules V_{n}, W_{n} and exact sequences of W (k)-modules

0 → V_{n+1} → C/A ⊗ Ωk^{n} → K̃_{n}(A) → H̃C_{n−1}(A) → W_{n} → 0

and

0 → Ã ⊗ Ωk^{n−1} → W_{n} → V_{n} → 0

Section 2 contains a discussion of the rings W (R), Cart(R) and Carf(R) and modules over these rings. In Section 3 we discuss the Carf(R) and W (R)–module structures on Kähler differentials, using an explict description of the operations. Up to here no knowledge of cyclic or Hochschild homology is assumed of the reader. In Section 4 we construct the Verschiebung, homothety and Frobenius operators and show that they define Carf(R)–module structures on NHH_{∗}(Λ)/NHH_{∗}(R) and, in case Q ⊆ k, on NHC_{∗}(Λ)/NHC_{∗}(R). Section 5 treats the case of graded R–algebras and in Section 6 we prove, without the assumption that k contains Q, that NHC_{∗}(Λ)/NHC_{∗}(R) is at least a Carf(k)–module.

2 W (R) and Carf(R) - modules

Our method of constructing W (R)–modules is based on Cartier’s method [C1, C2] of constructing a W (R)–module structure on the set C of curves in a formal group; he embeds W (R) in a larger ring Cart(R) which acts naturally on C. We are going to use an intermediate ring Carf(R) because Cart(R) is too big for our purposes. In order to fix notation, we first recall some well-known facts about W (R) and Cart(R).

2.1 Witt Vectors

Let R be a commutative ring. The ring W (R) of (big) Witt vectors over R is a topologically complete commutative ring. Since we know of no definitive literature for W (R), we begin with an elementary description of W (R)^{1}. Its underlying topological abelian group is the subgroup (1 + tR[[t]])^{∗} of units of the power series ring R[[t]], the topology being induced from the t-adic topology on R[[t]]. The multiplicative unit of W (R) is (1 − t). Using ∗ for the ring product, the multiplicative structure of W (R) is completely determined by the formula

1W (R) differs somewhat from the ring W^{(p)}(R) first used by E. Witt in 1936 in Kummer theory, and later by I.S. Cohen in 1946 in ring theory and Dieudonné in 1955 in Lie theory, all in the case when R is a perfect field of characteristic p. The formulation we give appeared in P. Cartier’s 1967 paper [C1]. The authors refer the reader to [Bloch, Bour, Grays, Haz, Lang, Laz, W-MVS] for more details.

(1 − rt^{m}) ∗ (1 − st^{n}) = (1 − r^{n/d}s^{m/d}t^{mn/d})^{d}, d = gcd(m, n)  (1)

This is because every element w ∈ W (R) has a unique convergent expansion w(t) = ∏_{m=1}^{∞}(1 − r_{m}t^{m}) with r_{m} ∈ R. For historical reasons the r_{m} are called the Witt coordinates of w(t) and w(t) is sometimes represented as the sequence of coordinates (r_{1}, r_{2}, …). See [Lang] for this point of view; for another interpretation of the multiplication in W (R) see [Grays].

Here are some basic remarks about the ring structure of W (R). Let p be a prime number. W (F_{p}) is a countable product of copies of Ẑ_{p}, so if p = 0 in R then W (R) is a Ẑ_{p}–algebra. In contrast, if \frac{1}{p} ∈ R then \frac{1}{p} ∈ W (R) because (1 − t)^{1/p} ∈ 1 + tR[[t]]. If Q ⊆ R then not only is Q ⊆ W (R) but W (R) isomorphic to the R–algebra ∏_{i=1}^{∞} R via the ghost map (below) and the diagonal ring homomorphism λ_{t} : R → W (R) is given by the power series expansion

λ_{t}(r) = (1 − t)^{r} = 1 − rt + \binom{r}{2} t^{2} + ⋯ + (−1)^{i} \binom{r}{i} t^{i} + ⋯

For any commutative ring R there is a ring homomorphism gh : W (R) → ∏_{i=1}^{∞} R called the ghost map; the n^{th} component of gh is the map gh_{n} : W (R) → R defined by

gh_{n}(∏(1 − r_{d}t^{d})) = ∑_{d|n} dr_{d}^{n/d}.

See for example [Grays, p. 253], [Lang, p. 234]. The ghost map is sometimes defined by identifying ∏_{i=1}^{∞} R with the R–module Ω(R[[t]]/R) of continuous Kähler differentials (the i^{th} coordinate idempotent e_{i} corresponds to t^{i−1}dt, see Example 2.7 below), via the formula [Bour, Ex.1.42]

∑ gh_{i}(w)t^{i−1}dt = −\frac{d}{dt} log(w)dt.

From this formula it is easy to see that if R is a Q–algebra then gh is an isomorphism and that gh(λ_{t}) : R → ∏ R is the diagonal inclusion, i.e., gh(λ_{t}(r)) = (r, r, r, …) The following lemma/definition is an easy exercise.

Lemma 2.1 Let A be an R–module and n ≥ 1. Write At^{n} = {at^{n} : a ∈ A} for the W (R)–module induced from A by restriction of scalars by gh_{n} : W (R) → R. Then the W (R)–module structure is determined by the formula:

(1 − r_{d}t^{d}) ∗ at^{n} = \{ \begin{matrix}dr_{d}^{n/d}at^{n} & if d|n \\ 0 & otherwise \end{matrix}

If R is a Q–algebra, the restriction of scalars along λ_{t} : R → W (R) recovers the original R–module structure on A.

Definition 2.2 Let A be an R–module. We shall write tA[t] and tA[[t]] for the direct sum ∐_{i=1}^{∞} At^{i} and the direct product ∏_{i=1}^{∞} At^{i} of the W (R)–modules At^{i} defined in 2.1 above. We shall write elements of these modules as sums ∑ a_{i}t^{i}. If r ∈ R we can also define a W (R)–module endomorphism [r] on At^{n}, tA[t] and tA[[t]], called the homothety, by the formula [r](at^{i}) = ar^{i}t^{i}.

We will call a W (R)–module M continuous if it satisfies the rule that for every v ∈ M, ann_{W (R)}(v) is an open ideal in W (R). That is, there is a k such that (1 − rt)^{m} ∗ v = 0 for all r ∈ R and all m ≥ k. Note that tA[t] is a continuous W (R)–module but that tA[[t]] is not.

Example 2.3 Let M_{1}, M_{2}, … be R-modules. Then ⨁ M = M_{1}x ⊕ M_{2}x^{2} ⊕ M_{3}x^{3} ⋯ is a continuous W (R)–module. This is because the category of continuous W (R)– modules is closed under direct sums (and every colimit for that matter). The individual M_{i}x^{i} are continuous because if d > i then (1 − rt^{d}) ∗ M_{i}x^{i} = 0.

We note that if R is of characteristic p then any continuous W (R)–module is a p-group. This comes from the fact that p^{n} = (1 − t)^{p^{n}} = (1 − t^{p^{n}}).

When Q ⊆ R it is well-known that gh : W (R) → ∏_{i=1}^{∞} R is a ring isomorphism. The element ε_{i} = exp(−t^{i}/i) of W (R) is idempotent and corresponds to the i^{th} coordinate idempotent of ∏_{i=1}^{∞} R.

Theorem 2.4 Let Q ⊆ R and M be a continuous W (R)–module. Setting M_{i} = ε_{i}M, M is isomorphic to the module ⨁ M_{i}x^{i} of 2.3. Moreover, for each i M_{i} is an eigenspace with eigenvalue r^{i} for each of the endomorphisms (1 − rt)∗ of M. In particular if r ∈ Q, r ∉ {±1, 0}, this is the eigenspace decomposition of M for the operator (1 − rt)∗.

Proof As w ∗ ε_{i} = gh_{i}(w)ε_{i} in ε_{i}W (R) ≈ R it is clear that M_{i} = M_{i}x^{i}. Since ε_{i} ∗ ε_{j} = 0 for i ≠ j, the evident map ⨁ M_{i}x^{i} → M is an injection. Since M is a continuous W (R)–module, given v ∈ M we can pick k so that ε_{i} ∗ M = 0 for i > k. It is then seen that v = 1 ∗ v = ∑_{i=1}^{k} ε_{i}v is a finite sum of elements of the submodules M_{i}. Finally, the assertions about eigenspaces follow from the observation that gh_{i}(1 − rt) = r^{i}.

2.2 The rings Cart(R)

In [C1], Cartier constructed the non-commutative ring Cart(R) as follows. Consider the functor W^{+} from the category of commutative R-algebras to abelian groups, sending the algebra Λ to the abelian group (1+tΛ[[t]])^{∗} underlying W (Λ). Cart(R) is the ring of natural endomorphisms of this functor. As such it clearly contains W (R) as a subring. It also contains the homothety [r], Verschiebung V_{m}, and Frobenius F_{m} maps defined as follows. For w = w(t) ∈ W (R),

V_{m}(w) = w(t^{n})
[r](w) = w(rt)
F_{m}(w) = Norm w( ^{m} \sqrt{t})

the norm being taken from R[[ ^{m} \sqrt{t}]] to R[[t]]. In particular F_{m}(1 − rt) = (1 − r^{m}t) and F_{m} is actually a ring endomorphism of W (R) (see [Bour, Ex.1.47]). Cartier shows in [C1] that every element u ∈ Cart(R) has a unique convergent row-finite expansion u = ∑ V_{m}[r_{mn}]F_{n} for m, n ≥ 1. “Row-finite” refers to the matrix (r_{mn}): for each m only finitely many of the r_{mn} are non-zero. For example, the inclusion W (R) ⊆ Cart(R) is determined by the formula

∏_{m=1}^{∞}(1 − r_{m}t^{m}) = ∑_{m=1}^{∞} V_{m}[r_{m}]F_{m}.

(The reader should beware that the Cart(R)–module operations on W (R) described above are not left multiplication.)

In order to give a presentation for Cart(R), recall [AT, p. 258] that there are integer polynomials p_{d}(x, y) of degree d in x and y, such that

x^{m} + y^{m} = ∑_{d|m} dp_{d}(x, y)^{m/d}

for each m = 1, 2, · · ·. They may be determined recursively. For instance,

p_{1}(x, y) = x + y , p_{2}(x, y) = −xy
p_{3}(x, y) = −(x^{2}y + xy^{2}) , p_{4}(x, y) = −(x^{3}y + 2x^{2}y^{2} + xy^{3}).

Cartier also showed in [C2] (see also [Haz, Laz]) that the ring Cart(R) satisfies the following identities, which completely determine the ring structure on Cart(R):

Theorem 2.5 (Cartier’s Identities) Let R be a commutative ring. The following identities hold in Cart(R) for all m, k ≥ 1 and all r, s ∈ R.

(i) [1] = V_{1} = F_{1} is the multiplicative identity.
(ii) [r][s] = [rs], V_{k}V_{m} = V_{km}, F_{k}F_{m} = F_{km}
(iii) [r]V_{m} = V_{m}[r^{m}], F_{m}[r] = [r^{m}]F_{m}
(iv) F_{m}V_{m} is multiplication by m
(v) If (k, m) = 1 then V_{m}F_{k} = F_{k}V_{m}
(vi) [r] + [s] = ∑_{m≥1} V_{m}[p_{m}(r, s)]F_{m}

It should be noted that identities (ii), (iv) and (v) together say that F_{k}V_{m} = dV_{m/d}F_{k/d} where d = (m, k). In particular,

F_{k}(1 − rt^{m}) = dV_{m/d}[r^{k/d}]F_{mk/d} = (1 − r^{k/d}t^{m/d})^{d}F_{k/d}

Thus W (R) is not in the center of Cart(R).

Any Cart(R)–module will be a W (R)–module, but Cart(R)–modules tend to be too big for our purposes. For example, W (R) = 1 + tR[[t]] is a cyclic Cart(R)– module by definition. Using the ghost map, we can construct the following family of Cart(R)–modules.

Example 2.6 If A is any R–module, the W (R)–module structure on tA[[t]] described in 2.2 extends to a left Cart(R)–module structure via the formulas

V_{m}(at^{i}) = at^{mi}
[r](at^{i}) = ar^{i}t^{i}
F_{m}(at^{i}) = \{ \begin{matrix}mat^{i/m} & if m|i \\ 0 & otherwise \end{matrix}

We leave the verification of these facts to the reader, noting that since F_{n}(at^{i}) = 0 whenever n > i the formula for ∑ V_{m}[r_{mn}]F_{n} is well defined.

Note that if Λ is a commutative R–algebra containing Q the logarithm W (Λ) → tΛ[[t] is an isomorphism of Cart(R)–modules.

Example 2.7 [Bloch, p. 195] Let A be an R–module and write Ω(A[[t]]/A) for the product ∏_{i=1}^{∞} Ae_{i} of copies of A. (The generators e_{i} are often written as t^{i−1}dt.) Define operations V_{m}, [r], F_{m} by the formulas

V_{m}(ae_{i}) = mae_{mi}

[r](ae_{i}) = ar^{i}e_{i}
F_{m}(ae_{i}) = \{ \begin{matrix}ae_{i/m} & if m|i \\ 0 & otherwise \end{matrix}

Because of the row-finiteness condition on Cart(R), the formula

(∑_{m,n} V_{m}[r_{mn}]F_{n})(∑_{i} a_{i}e_{i}) = ∑_{m,n,i} mr_{mn}^{i}a_{in}e_{mi} = ∑_{k} ∑_{m|k} ∑_{n} mr_{mn}^{k/m}a_{kn/m}e_{k}

is well defined. By direct observation, Cartier’s identities 2.5 are satisfied. Therefore Ω(A[[t]]/A) is a Cart(R)–module.

Any Cart(R)–module has an underlying W (R)–module structure via the inclusion W (R) ⊂ Cart(R). The two distinct Cart(R)–modules Ω(A[[t]]/A) and tA[[t]] have isomorphic underlying W (R)–module structures, via the correspondence e_{i} ↔ t^{i}.

Using the explicit formulas for V_{m}, [r], and F_{m} on W (R) one checks (as in [Bloch]) that the ghost map gh : W (Λ) → Ω(Λ[[t]]/Λ) is a Cart(R)–module homomorphism whenever Λ is a commutative R–algebra. Of course, if Q ⊆ R it is an isomorphism, and the formulas for V_{m}, [r], and F_{m} on Ω(Λ[[t]]/Λ) are obtained by transferring the formulas for W (Λ).

2.3 The ring Carf(R)

We define Carf(R) to be the subset of Cart(R) consisting of elements ∑ V_{m}[r_{mn}]F_{n} in which the matrix (r_{mn}) is both row and column-finite. That is, for each n only finitely many of the r_{mn} are non-zero. Since Carf(R) contains W (R) and hence the element “–1” = (1 − t)^{−1}, it is easy to see from Cartier’s identities 2.5 that Carf(R) is a subring of Cart(R) containing W (R).

The ring Carf(R) is a topological ring in which the left ideals I_{N} = {∑ V_{m}[r_{mn}]F_{n} : n ≥ N} form a basic family of neighborhoods of 0, and W (R) ⊂ Carf(R) is an inclusion of topological rings. We say that a left Carf(R)–module A is continuous provided that for every a ∈ A annCarf(R)(a) is an open left ideal of Carf(R), i.e. F_{n}(a) = 0 for all but finitely many n. By restriction of scalars, a continuous Carf(R)– module is also a continuous W (R)–module.

Theorem 2.8 Suppose that A is an abelian group equipped with endomorphisms F_{m}, [r] and V_{m}, (m ≥ 1, r ∈ R) satisfying Cartier’s identities 2.5 (i) — (vi). Suppose also for each a ∈ A (vii) F_{n}(a) = 0 for all but finitely many n Then these operations make A into a continuous Carf(R)–module, and a fortiori into a continuous W (R)–module.

The proof of the above Theorem is straightforward, once we notice that each element of Carf(R) has a unique representation of the form ∑ V_{m}[r_{mn}]F_{n} which is both row and column-finite, and that only finitely many of the terms V_{m}[r_{mn}]F_{n} are non-zero on any given a ∈ A. Example 2.9 Let A be an R–module, then defining V_{m}, [r] and F_{m} on tA[t] by the formulas of 2.6 satisfies the identities of 2.8 and thus tA[t] is a continuous Carf(R)– module. Example 2.10 (Differentials) Let A be an R–module. We shall write Ω(A[t]/A) for the direct sum ∐_{i=1}^{∞} Ae_{i} of copies of A on generators e_{i}. (The generators e_{i} are often written as t^{i−1}dt. This is a subgroup of the Cart(R)–module Ω(A[[t]]/A) = ∏ Ae_{i} of 2.7, and is closed under the operations V_{m}, [r] and F_{m} of loc. cit. Since Cartier’s identities hold in Ω(A[[t]]/A) and 2.8(vii) is clearly satisfied, 2.8 shows that Ω(A[t]/A) is a continuous (left) Carf(R)–module. Our notation comes from the fact that if Λ is a commutative ring and we write t^{i−1}dt for e_{i} then Ω(Λ[t]/Λ) ≈ Ω_{Λ[t]/Λ} is the usual Λ–module of Kähler differentials of the polynomial ring Λ[t] over Λ.

Note that the underlying W (R)–modules tA[t] of 2.9 and Ω(A[t]/A) of 2.10 are isomorphic via the correspondence t^{i} ↦ e_{i}. If R contains Q then tA[t] ≈ Ω(A[t]/A) as Carf(R)–modules via the correspondence t^{i} ↦ ie_{i}. Example 2.11 Define the de Rham differential D : tA[t] → Ω(A[t]/A) by the formula D(t^{i}) = it^{i−1}dt = ie_{i}. This is a Carf(R)–module homomorphism for every R– module A. Therefore the kernel and cokernel are continuous Carf(R)–modules. When Λ is a commutative ring, the kernel is the de Rahm cohomology group H_{dR}^{0}(Λ[t]/Λ) and the cokernel is the cyclic homology group HC_{1}^{Λ}(Λ[t]). (In both cases we are taking the ground ring k = Λ. We will discuss both of these groups in more detail later in this paper). For example, the following are continuous Carf(Z)–modules:

• HC_{1}(Z[t]) = Z/2Ze_{2} ⊕ Z/3Ze_{3} ⊕ ⋯ = ∐_{ℓ≥2}^{∞} Z/ℓZe_{ℓ};

• HC_{1}(Z/p[t]) = Z/pZe_{p} ⊕ Z/pZe_{2p} ⊕ ⋯ = ∐_{ℓ=1}^{∞} Z/pZe_{ℓp};

• H_{dR}^{0}(Z/p[t] / Z/pZ) = t^{p}Z/pZ[t^{p}].

We end this section with a characterization of Carf(R)–modules when Q ⊆ R.

Lemma 2.12 If Q ⊆ R then R is the center of both Carf(R) and Cart(R) via the inclusions R ⟶^{λ_{t}} W (R) ⊆ Carf(R) ⊆ Cart(R). In particular, V_{m} and F_{m} act Rlinearly on any Carf(R)–module.

Proof If Λ is a commutative R–algebra then a direct calculation shows that the endomorphisms λ_{t}(r) = (r, r, …) commute with V_{m}, [s] and F_{m} on ∏_{i=1}^{∞} Λ ≈ W (Λ) (via the ghost map). By definition of Cart(R) this means that λ_{t}(r) is in the center of Cart(R) for each r ∈ R. Now suppose Θ = ∑ V_{m}[r_{mn}]F_{n} is in the center of Carf(R) or Cart(R); by adding λ_{t}(−r_{11}) to Θ we may assume that r_{11} = 0. Choose m, n minimizing m + n subject to r_{mn} ≠ 0; computing the commutators [F_{m}, Θ] and [Θ, V_{n}] we obtain a contradiction unless Θ = 0. Theorem 2.13 Let M be a continuous Carf(R)–module where Q ⊆ R. Then M = ∐_{i=1}^{∞} M_{i} where M_{i} is the eigenspace for r^{i} under the endomorphism [r] and is a Rsubmodule. Further V_{m} : M_{i} → M_{mi} and F_{m} : M_{im} → M_{i} are R-module isomorphisms. In particular M_{i} ≈ M_{1} for all i ≥ 1. Finally for a ∈ M_{i} we have

r ∗ a = \frac{1}{i}V_{i}[r]F_{i}(a).  (2)

Proof F_{i}V_{i} = i by the Cartier identity (iv). On the other hand, V_{i}F_{i} = V_{i}[1]F_{i} is a Witt vector with gh_{i}(V_{i}F_{i}) = i so V_{i}F_{i}ε_{i} = iε_{i} and hence V_{i}F_{i} restricted to M_{i} is also i. This proves the isomorphisms. In addition this shows r ∗ a = r ∗ \frac{1}{i}V_{i}F_{i}(a) = \frac{1}{i}V_{i}(r ∗ F_{i}(a)) = \frac{1}{i}V_{i}([r]F_{i}(a)) = \frac{1}{i}V_{i}[r]F_{i}(a) since F_{i}(a) ∈ M_{1} which is the eigenspace of r under [r]. Corollary 2.14 Let M be a continuous left Carf(R)–module, Q ⊆ R. Then as a Carf(R)–module M is isomorphic to the module tA[t] of Example 2.9 where A = M_{1}. Proof Map tA[t] → M by at^{i} ↦ V_{i}(a). By the above theorem this gives an isomorphism (tA[t])_{i} → M_{1} → M_{i} of R–modules and hence an R–module isomorphism of graded R–modules tA[t] → M. A direct calculation using the Cartier identities and formula (2) shows that this is a Carf(R)–module isomorphism.

3 Module structures on Ω_{Λ}^{n}

In this section we give an explicit brute force description of the Carf(R) and W (R)- module structures on Kähler differentials.

In this section k will be a commutative ring (usually a field or Z ), R and Λ will be commutative k–algebras. We will write Ω_{Λ}^{n} for the module of Kähler differentials Ω_{Λ/k}^{n} and NΩ_{Λ}^{n} = ker(Ω_{Λ[x]}^{n} ⟶^{x↦0} Ω_{Λ}^{n}). We will let Ω_{Λ}^{0} = Λ and we note that NΩ_{Λ}^{0} = xΛ[x] has the Carf-module structure described in Example 2.9.

Let Λ = R[x], it is well known [Mats, 26.J] that Ω_{Λ} = (Λ ⊗_{R} Ω_{R}) ⊕ Λdx. Since Ω_{k[x]}^{2} = 0 we have Ω_{Λ}^{n} = (Λ ⊗_{R} Ω_{R}^{n}) ⊕ (Λ ⊗_{R} Ω_{R}^{n−1} ∧ dx) (see also [Kunz, Cor. 4.10]). Taking the kernel as x ↦ 0 we have NΩ_{R}^{n} = xR[x]Ω_{R}^{n} ⊕ R[x]Ω_{R}^{n−1} ∧ dx.

Definition 3.1 Let Λ be a commutative k–algebra and n ≥ 1. Define endomorphisms V_{m}, [r] and F_{m} on NΩ_{Λ}^{n} by

V_{m}(x^{i}ω) = x^{im}ω
[r](x^{i}ω) = r^{i}x^{i}ω
F_{m}(x^{i}ω) = \{ \begin{matrix}mx^{i/m}ω & if m|i \\ 0 & otherwise \end{matrix}

and

V_{m}(x^{i−1}ν ∧ dx) = x^{im−m}ν ∧ d(x^{m}) = mx^{im−1}ν ∧ dx

[r](x^{i−1}ν ∧ dx) = r^{i−1}x^{i−1}ν ∧ d(rx) = r^{i}x^{i−1}ν ∧ dx + r^{i−1}x^{i}ν ∧ dr
F_{m}(x^{i−1}ν ∧ dx) = \{ \begin{matrix}x^{i/m−1}ν ∧ dx & if m|i \\ 0 & otherwise \end{matrix}

where r ∈ Λ, ω ∈ Ω_{Λ}^{n} and ν ∈ Ω_{Λ}^{n−1}.

Theorem 3.2 Let Λ be a commutative k–algebra. The above endomorphisms make NΩ_{Λ}^{n} a Carf(Λ)–module. This makes NΩ_{Λ}^{n} a W (R)–module by the formulas

(1 − rt^{m}) ∗ x^{i}ω = mr^{i/m}x^{i}ω
(1 − rt^{m}) ∗ x^{i−1}ν ∧ dx = mr^{i/m}x^{i−1}ν ∧ dx + r^{i/m−1}x^{i}ν ∧ dr

if m|i and 0 otherwise. Finally, there is a short exact sequence

0 → xΩ_{Λ}^{n}[x] → NΩ_{Λ}^{n} → Ω_{Λ}^{n−1} ⊗ Ω(k[x]/k) → 0

of Carf(Λ)–modules, where the left and right terms have the Carf(Λ)–module structures of 2.2 and 2.10 respectively.

Remark 3.3 If Q ⊆ k it is easily seen that if M = NΩ_{Λ}^{n} then M_{i} is generated by forms x^{i}ω and x^{i−1}ν ∧ dx where ω ∈ Ω_{Λ}^{n} and ν ∈ Ω_{Λ}^{n−1}. However by Theorem 2.13 the Λ–module operation on M_{i} is given by

r ∗ (x^{i}ω) = rx^{i}ω
r ∗ (x^{i−1}ν ∧ dx) = rx^{i−1}ν ∧ dx + \frac{1}{i}x^{i}ν ∧ dr

which is not the expected operation.

Proof We must verify the Cartier identities in Theorem 2.8. Since V_{m} and F_{m} respect the direct sum decomposition, the verification of the identities involving only V_{m}, F_{m} is straightforward. The identity [r][s] = [rs] is also straightforward and the identity [r]V_{m} = V_{m}[r^{m}] is similar to the other part of identity (iii). Thus we will demonstrate only the identity F_{m}[r] = [r^{m}]F_{m} of identity (iii) and identity (vi).

If m does not divide i then it is easily seen that F_{m}[r](x^{i}ω) = [r^{m}]F_{m}(x^{i}ω) = 0 and F_{m}[r](νx^{i−1} ∧ dx) = [r^{m}]F_{m}(νx^{i−1} ∧ dx) = 0. Thus we suppose that m|i. Calculating we have F_{m}[r](x^{i}ω) = F_{m}(r^{i}x^{i}ω) = mr^{i}x^{i/m}ω = m(r^{m})^{i/m}x^{i/m}ω = [r^{m}](mx^{i/m}ω) = [r^{m}]F_{m}(x^{i}ω). On the other summand

F_{m}[r](x^{i−1}ν ∧ dx)
= F_{m}(r_{i}x^{i−1}ν ∧ dx + r^{i−1}x^{i}ν ∧ dr)
= r^{i}νx^{i/m−1} ∧ dx + mr^{i−1}x^{i/m}ν ∧ dr.

On the other hand, using the differentiation formula d(r^{m}) = mr^{m−1}dr

[r^{m}]F_{m}(x^{i}ν ∧ dx) = [r^{m}](x^{i/m−1}ν ∧ dx)
= (r^{m})^{i/m}x^{i/m−1}ν ∧ dx + (r^{m})^{i/m−1}x^{i/m}ν ∧ d(r^{m})
= r^{i}x^{i/m−1}ν ∧ dx + mr^{m−1}r^{i−m}x^{i/m} ∧ dr

= r^{i}x^{i/m−1}ν ∧ dx + mr^{i−1}x^{i/m} ∧ dr

For identity (vi) we have on the first summand

∑_{m≥1} V_{m}[p_{m}(r, s)]F_{m}(x^{i}ω)

= ∑_{m|i} V_{m}[p_{m}(r, s)](F_{m}(x^{i}ω)) = ∑_{m|i} V_{m}[p_{m}(r, s)](mx^{i/m}ω)
= ∑_{m|i} V_{m}(mp_{m}(r, s)^{i/m}x^{i/m}ω) = ∑_{m|i} mp_{m}(r, s)^{i/m}x^{i}ω
= (∑_{i|m} mp_{m}(r, s)^{i/m}) x^{i}ω = (r^{i} + s^{i})x^{i}ω
= r^{i}x^{i}ω + s^{i}x^{i}ω = [r]x^{i}ω + [s]x^{i}ω
= ([r] + [s])(x^{i}ω)

The second summand presents added difficulties in several ways. We first consider the case where Q ⊆ k. Then as above

∑_{m≥1} V_{m}[p_{m}(r, s)]F_{m}(x_{i−1}ν ∧ dx)

= ∑_{m≥1} V_{m}[p_{m}(r, s)]F_{m}(\frac{1}{i}ν ∧ dx^{i})
= ∑_{m|i} V_{m}[p_{m}(r, s)](\frac{1}{i}ν ∧ dx^{i/m})
= ∑_{m|i} \frac{1}{i}V_{m}(ν ∧ d(p_{m}(r, s)^{i/m}x^{i/m}))
= ∑_{m|i} \frac{m}{i}ν ∧ d(p_{m}(r, s)^{m/i}x^{i})

= \frac{1}{i}ν ∧ d (∑_{m|i} p_{m}(r, s)^{i/m}) x^{i}
= \frac{1}{i}ν ∧ d(r^{i} + s^{i})x^{i}
= \frac{1}{i}ν ∧ dr^{i}x^{i} + \frac{1}{i}ν ∧ ds^{i}x^{i}
= \frac{1}{i}[r]ν ∧ dx^{i} + \frac{1}{i}[s]ν ∧ dx^{i}
= [r](x^{i−1}ν ∧ dx) + [s](x^{i−1}ν ∧ dx)
= ([r] + [s])(x^{i−1}ν ∧ dx)

For the case where k does not contain Q we let R = Z[r′, s′] where r′, s′ are indeterminants and note that the result is true for NΩ_{R/Z}^{1} since this is a subgroup of NΩ_{R⊗Q}^{1} where the formula has just been verified. Then given x^{i−1}ν ∧ dx = ν ∧ x^{i−1}dx ∈ Ω_{Λ}^{n}

∑_{m≥1} V_{m}[p_{m}(r, s)]F_{m}(ν ∧ x^{i−1}dx)

= ∑_{m≥1} ν ∧ V_{m}[p_{m}(r′, s′)]F_{m}(x^{i−1}dx)
= ν ∧ ([r′] + [s′])(x^{i−1}dx) = ([r] + [s])(x^{i−1}ν ∧ dx)

This completes the proof.

Now let Λ = Λ_{0} ⊕ Λ_{1} ⊕ Λ_{2} ⊕ ⋯ be a commutative graded ring with Λ_{0} = R a k–algebra. An element a ∈ Λ_{i} is called homogeneous of degree i. As above we let Ω_{Λ}^{n} denote Ω_{Λ/k}^{n} and we will write Ω̃_{Λ}^{n} for Ω_{Λ}^{n}/Ω_{R}^{n}. If n = 0 we write Ω̃_{Λ}^{0} = Λ_{1} ⊕ Λ_{2} ⊕ ⋯ which has a W (R)–module structure given by Example 2.9.

Theorem 3.4 Let Λ = R ⊕ Λ_{1} ⊕ Λ_{2} ⊕ ⋯ be a commutative graded ring where R is a k–algebra and n ≥ 1. Then Ω̃_{Λ}^{n} is a W (R)–module where the operation is given as follows. Let a_{0}, a_{1}, …, a_{n} be homogeneous elements of Λ. Let ℓ = deg a_{0} + deg a_{1} + ⋯ + deg a_{n} > 0 and let ω = a_{0}da_{1} ∧ da_{2} ∧ ⋯ ∧ da_{n}. Then

(1 − rt^{m}) ∗ ω = r^{ℓ/m}mω +
r^{ℓ/m−1}a_{0} ∑_{i=1}^{n}(deg a_{i})a_{i}da_{1} ∧ ⋯ ∧ da_{i−1} ∧ dr ∧ da_{i+1} ∧ ⋯ ∧ da_{n}  (3)

if m|ℓ and (1 − rt^{m}) ∗ ω = 0 otherwise. Furthermore, if k contains Q then Ω̃_{Λ}^{n} has an R–module structure induced by W (R) satisfying (for ω above)

r ∗ ω = rω + a_{0} ∑_{i=1}^{n} (\frac{deg a_{i}}{ℓ} ) a_{i}da_{1} ∧ ⋯ ∧ da_{i−1} ∧ dr ∧ da_{i+1} ∧ ⋯ ∧ da_{n} (4)

Proof We have a ring map ι : Λ → Λ[x] given by ι(a) = ax^{i} for a homogeneous element of degree i. Then ι induces a group map ι_{•} : Ω_{Λ}^{n} → Ω_{Λ[x]}^{n}. It is easily seen that ι_{•} restricts to a map ι_{•} : Ω̃_{Λ}^{n} → NΩ_{Λ}^{n}. We denote by (1 − rt^{m})∗ the group endomorphism of Ω̃_{Λ}^{n} induced by formula (3) and by V_{m}[r]F_{m}∗ the group endomorphism of NΩ_{Λ}^{n} induced by left multiplication by this Witt vector. It may be verified by direct computation, using the formulas 3.1, that the following diagram commutes.

Ω̃_{Λ}^{n} ⟶^{ι_{•}} NΩ_{Λ}^{n}
(1 − rt^{m}) ∗ ↓ ↓V_{m}[r]F_{m}∗
Ω̃_{Λ}^{n} ⟶^{ι_{•}} NΩ_{Λ}^{n}

The map ι_{•} is a split injection whose inverse is given by the substitution x ↦ 1, so we may identify Ω̃_{Λ}^{n} with its image under ι_{•}. But from the diagram it is clear that the image is a W (R)–submodule of NΩ_{Λ}^{n}, so Ω̃_{Λ}^{n} is a W (R)–module under the operation given by formula (1). Formula (2) can similarly be checked inside NΩ_{Λ}^{n} using Theorem 2.13.

Exercise The reader should check directly that Equation (4) does in fact define a R–module operation on Ω̃_{A}^{n}. In particular she should show that this operation is well defined and that s ∗ (r ∗ ω) = sr ∗ ω.

Remark 3.5 Let R be the polynomial ring k[r] and Λ = R[x] = k[r, x]. In NΩ_{R}^{1} = Ω̃_{Λ}^{1} the W (R)–module structure is given by formula (1): if m|i then

(1 − rt^{m}) ∗ (x^{i−1}dx) = mr^{i}x^{i−1}dx + r^{i−1}x^{i}dr.

Since this is not divisible by m when 1/m ∉ k, we see from 2.1 that this W (R)– module structure cannot be induced from a family of R–modules via the ghost maps gh_{n} : W (R) → R.

Recall that for n ≥ 0 the differential d : Ω_{Λ}^{n} → Ω^{n+1} is defined by d(aω) = da ∧ ω. This defines additive morphisms NΩ_{Λ}^{n} → NΩ_{A}^{n+1} and, in the case that Λ is graded, Ω̃_{Λ}^{n} → Ω̃_{Λ}^{n+1}. Note that the case n = 0 is essentially Example 2.11. A direct calculation based on the formulas 3.1 and Theorem 3.4 shows Proposition 3.6 Let R be a k–algebra and Λ be a R–algebra. The differential d : NΩ_{Λ}^{n} → NΩ_{Λ}^{n+1} is a morphism of Carf(R)–modules. If Λ is graded with Λ_{0} = R then d : Ω̃^{n} → Ω̃^{n+1} is a morphism of W (R) modules. Furthermore, if Q ⊆ k and we use the R–module structures induced by W (R), then the differential is an R-linear homomorphism. Exercise The reader is encouraged to check directly from formula (2) of Theorem 3.4 that when Q ⊆ k then the differential is an R-linear homomorphism for the R– module structure induced by W (R). This computation will illuminate the purpose of the ℓ in the denominator of (2).

Recall also that the classical deRahm cohomology H_{dR}^{i}(Λ) is the cohomology of the complex

0 ⟶ Ω_{Λ}^{0} ⟶^{d} Ω_{Λ}^{1} ⟶^{d} Ω_{Λ}^{2} ⟶^{d} ⋯

We may then define NH_{dR}^{i}(Λ) = ker(H_{dR}^{i}(Λ[x]) → H_{dR}^{i}(Λ)) and, for graded Λ, H̃_{dR}^{i}(Λ) = ker(H̃_{dR}^{i}(Λ) → H̃_{dR}^{i}(Λ_{0})).

It should be noted that if Q ⊆ k then it is known (see [Rob]) that NH_{dR}^{i}(Λ) = 0 as is H̃_{dR}^{i}(Λ) if Λ is graded. Thus the above corollary is of interest only when k = Z or is of positive characteristic.

Corollary 3.7 With k, R and Λ as above, NH_{dR}^{i} is a Carf(R)–module. Further, if Λ is graded with Λ_{0} = R then H̃_{dR}^{i}(Λ) is a W (R)–module.

4 The Verschiebung, homothety and Frobenius

In this section we construct the “VHF” operators V_{m}, F_{m} and [r] and prove that they define a Carf(R)–module structure on NHH_{∗}(Λ) for every (not necessarily commutative) R–algebra Λ. If Q ⊆ k, we deduce that the VHF operators also define a Carf(R)–module structure on NHC_{∗}(Λ). As always, R is a k–algebra, and we are taking Hochschild and cyclic homology over k.

We shall need a certain map ı = ı_{m} from Λ[x] into M_{m}(Λ[x]), the ring of m × m matrices with entries in Λ[x]. One way to describe this map is that ı imbeds Λ into the diagonal of M_{m}(Λ[x]) and sends x to the matrix

( \begin{matrix}0 & 0 & ⋯ & 0 & x \\ 1 & 0 & ⋯ & 0 & 0 \\ 0 & 1 & ⋯ & 0 & 0 \\ ⋮ & ⋮ & ... & ⋮ & ⋮ \\ 0 & 0 & ⋯ & 1 & 0 \end{matrix} )

Note that under ı, x^{m} is sent to the diagonal matrix with diagonal entries x, so in a sense, x is being sent to its m-th root. Also note that the nonzero entries of ı(x) lie just below the diagonal; this includes x if we index the rows and columns modulo m.

Alternatively we can describe ı as follows. We set y = x^{m} and regard Λ[x] as the free Λ[y]–algebra on basis {x^{1}, x^{2}, …, x^{m−1}}. Then left multiplication of Λ[x] by an element corresponds to an endomorphism of the free Λ[y]–module on m generators, i.e. to an m × m matrix. Renaming the dummy variable x instead of y, ı sends Λ[x] to M_{m}(Λ[x]).

Now let Λ be a R–algebra and recall that we are writing HH_{∗}(Λ) = HH_{∗}^{k}(Λ, Λ) and HC_{∗}(Λ) = HC_{∗}^{k}(Λ). Recall that these are the HH_{∗} and HC_{∗} of the bar complex C_{•}(Λ), considered as a mixed complex in the sense of [K-CCM].

Let NC_{•}(Λ) denote the kernel of the map C_{•}(Λ[x]) ⟶^{x↦0} C_{•}(Λ). By naturality the Hochschild and cyclic homology of NC_{•}(Λ) is NHH_{∗}(Λ) and NHC_{∗}(Λ).

Since Λ is Morita equivalent to M_{m}(Λ) there is a natural map of mixed complexes (anachronistically due to R.K. Dennis [DI, 3.7]) C_{•}(M_{m}(Λ)) ⟶^{trace} C_{•}(Λ) given by the formula

g^{0}[g^{1}|g^{2}| ⋯ |g^{n}] ↦ ∑_{i_{0},...,i_{n}=1}^{n} g_{i_{0}i_{1}}^{0}[g_{i_{1}i_{2}}^{1}| ⋯ |g_{i_{n}i_{0}}^{n}]  (5)

where g^{0}, g^{1}, …, g^{n} ∈ M_{m}(Λ). The trace map induces isomorphisms HH_{∗}(M_{m}(Λ)) ≈ HH_{∗}(Λ), HC_{∗}(M_{m}(Λ)) ≈ HC_{∗}(Λ), etc. We remark that trace is an inverse to the map C_{•}(Λ) → C_{•}(M_{m}(Λ)) induced by sending an element f ∈ Λ to the m×m matrix with f in the first row, first column and zeros elsewhere.

Definition 4.1 (VHF Operators) Let Λ be a k–algebra, R be a commutative k– algebra contained in Λ.

• The Verschiebung V_{m} is the endomorphism of the mixed complexes C_{•}(Λ[x]) and NC_{•}(Λ) induced by the map Λ[x] → Λ[x] sending x to x^{m}.

• For r ∈ R the homothety [r] is the endomorphism of the mixed complexes C_{•}(Λ[x]) and NC_{•}(Λ) induced by the map Λ[x] → Λ[x] sending x to rx.

• The Frobenius F_{m} is the composition C_{•}(Λ[x]) ⟶^{ı_{∗}} C_{•}(M_{m}(Λ[x])) ⟶^{trace} C_{•}(Λ[x]), considered as an endomorphism of the mixed complexes C_{•}(Λ[x]) and NC_{•}(Λ). That is, F_{m} is the transfer map associated with the ring map sending x to x^{m}.

The induced endomorphisms of HH_{∗}(Λ[x]), NHH_{∗}(Λ), HC_{∗}(Λ[x]), NHC_{∗}(Λ), etc. will also be called V_{m}, [r] and F_{m}. We remark that there is a natural decomposition C_{•}(Λ[x]) = C_{•}(Λ) ⊕ NC_{•}(Λ); on the subcomplex C_{•}(Λ) the maps V_{m} and [r] are the identity, while F_{m} is multiplication by m. The interesting thing about these “VHF” operators is their action upon NHH_{∗} and NHC_{∗}.

Theorem 4.2 Let k be a commutative ring, R a commutative k–algebra and Λ an R– algebra. Then the VHF operators V_{m}, [r] and F_{m} induce a Carf(R)–module structure on NHH_{n}(Λ). Moreover, there is a short exact sequence of Carf(R)–modules:

0 → HH_{n}(Λ) ⊗_{k} xk[x] → NHH_{n}(Λ) → HH_{n−1}(Λ) ⊗k Ω(k[x]/k) → 0

Here the left and right terms have the Carf(R)–module structures given in 2.9 and 2.10, respectively. This sequence is split by the Künneth formula, but is not split as a sequence of Carf(R)–modules (or even as W (R)–modules) when Ω_{R/k} is non-zero.

To prove the preceeding theorem, we will need explicit formulas for the endomorphisms V_{m}, [r], and F_{m} on HH_{n}(Λ[x]). For this, we need to decompose HH_{∗}(Λ[x]) using the shuffle product and the Künneth Formula. These give the following direct sum decomposition of HH_{n}(Λ[x]), which is well-known [Mac, X, Thm 7.4] or [CE] when k is a field.

Proposition 4.3 (Künneth Formula) Let k be a commutative ring, Λ any k– algebra. Then the shuffle product # : C_{•}(Λ) ⊗ C_{•}(k[x]) → C_{•}(Λ[x]) is a quasiisomorphism, so as a k[x]–module

HH_{n}(Λ[x]) = HH_{n}(Λ) ⊗k k[x] ⊕ HH_{n−1}(Λ) ⊗k k[x]dx

Proof The bar complex C_{•}(k[x]) used to compute HH_{∗}(k[x]) is free over k as is the resulting homology. Thus the result follows from the Künneth Tensor Formula [Mac, p. 166].

By the Künneth Formula, HH_{n}(Λ[x]) is generated by cycles of the form α#x^{i} and β#x^{i−1}dx, where α is a cycle in C_{n}(Λ) and β is a cycle in C_{n−1}(Λ). Note that we are identifying dx with [x] (via the isomorphism HH_{1}(k[x]) ≈ Ω_{k[x]/k}), so as to not confuse the latter with the homothety [r].

Theorem 4.4 Let k be a commutative ring, R a commutative k–algebra and Λ an R–algebra. Let α be a cycle in HH_{n}(Λ) and β be a cycle in HH_{n−1}(Λ). Then in HH_{n}(Λ[x]) we have the following formulas for V_{m}, [r], F_{m} where i ≥ 1, r ∈ R.

V_{m}(α#x^{i}) = α#x^{im}
[r](α#x^{i}) = r^{i}α#x^{i}
F_{m}(α#x^{i}) = \{ \begin{matrix}mα#x^{i/m} & if m|i \\ 0 & otherwise \end{matrix}

and

V_{m}(β#x^{i−1}dx) = β#x^{m(i−1)}dx^{m} = mβ#x^{im−1}dx
[r](β#x^{i−1}dx) = β#(rx)^{i−1}d(rx) = r^{i}β#x^{i−1}dx + (β#r^{i−1}dr)#x^{i}
F_{m}(β#x^{i−1}dx) = \{ \begin{matrix}β#x^{i/m−1}dx & if m|i \\ 0 & otherwise \end{matrix}

The first step in proving the above theorems is the special case Λ = k. Recall from [HKR] that there is a map Ω_{R/k}^{n} → HH_{n}(R) given by

a_{0}da_{1} ∧ da_{2} ∧ ⋯ ∧ da_{n} ↦ ∑_{σ∈Σ_{n}} sgn(σ)a_{0}[a_{σ^{−1}(1)}| ⋯ |a_{σ^{−1}(n)}]

where Σ_{n} is the symmetric group on n elements. If R is smooth over k or if n = 0, 1 then it is classical that this map is an isomorphism. If k contains Q then it is also well known that the image is a direct summand of HH_{n}(R). It is clear that these results extend to the case of NΩ_{R/k}^{n} → NHH_{n}(R).

Theorem 4.5 The VHF operators commute with the map ϕ : NΩ_{R/k}^{n} → NHH_{n}(R). Here the VHF operators are defined as in 4.1 for HH_{∗} and as in 3.1 for NΩ_{R}^{n}.

Proof The result is clear for V_{m} and [r] since these endomophisms are induced by ring homomorphisms (x ↦ x^{m} and x ↦ rx, respectively).

As in §3 every element of NΩ_{R} can be written as a sum of differentials x^{i}ω and x^{i−1}ν ∧ dx where ω ∈ Ω_{R}^{n} and ν ∈ Ω_{R}^{n−1}. For the first case, ϕ(x^{i}ω) is a sum of chains of the form b_{0}x^{i}[b_{1}| ⋯ |b_{n}] where the b_{j} ∈ R. To apply F_{m} to this we first apply ı and then the trace. But ı(b_{j}) is a diagonal matrix so all non-zero terms of the trace must also take the diagonal of the lefthand factor in formula (1). But ı(b_{0}x^{i}) has diagonal terms only if i is a multiple of m and then each diagonal entry is b_{0}x^{i/m}. Thus F_{m}(b_{0}x^{i}[b_{1}| ⋯ |b_{n}]) = mb_{0}x^{i/m}[b_{1}| ⋯ |b_{n}] from which it follows easily that F_{m}(ϕx^{i}ω) = ϕ(mx^{i/m}ω) = ϕF_{m}(x^{i}ω). The second case is similar in that each term of ϕ(x^{i−1}ν ∧ dx) has a x^{i−1} and x in two positions, and no other powers of x. The nonzero terms in ı(x) lie just below the diagonal (modulo m), and the terms in ı(x^{i−1}) lying just above the diagonal (modulo m) are zero unless i is a multiple of m, when they are x^{i/m}. Corollary 4.6 (The case Λ = k) The isomorphism ϕ : NΩk^{n} ≈ NHH_{n}(k) commutes with V_{m}, F_{m} and [r] for r ∈ k. The Carf(k)–module structures on xk[x] = NHH_{0}(k) and Ω(k[x]/k) = NHH_{1}(k) which were described in 2.9 and 2.10 are induced by the natural VHF operators defined in this section.

Recall that the shuffle product # is a chain complex map. As such it gives rise to maps HH_{p}(Λ) ⊗ HH_{q}(R[x]) → HH_{p+q}(Λ[x]) whenever Λ is an R–algebra and R is a commutative k–algebra [Mac, Chap X, also p 313]. The following is immediate from the definitions of the VHF operators. Lemma 4.7 Let Λ be a R–algebra where R is a commutative k–algebra. Let p = 0, 1 and q = 1 − p. Then the following formulas hold in NHH_{p+q}(Λ) for α ∈ HH_{p}(Λ), β ∈ NHH_{q}(R) , m > 0 and r ∈ R.

V_{m}(α#β) = α#V_{m}(β)

[r](α#β) = α#[r](β)
F_{m}(α#β) = α#F_{m}(β)

At last we can give the proofs of our main theorems! Proof of Theorem 4.4 The formulas for V_{m}, F_{m} follow from 4.7, 4.6, 3.1 and examples 2.9, 2.10. Also using 4.7, we can regard x^{i} as a cycle in NHH_{0}(R) = xR[x] and x^{i−1}dx as a cycle in NHH_{1}(R) = Ω_{R[x]/k} so [r](α#x^{i}) = α#[r]x^{i} = α#(rx)^{i} = r^{i}α#x^{i} and V_{m}(β#x^{i−1}dx) = β#[r](x^{i−1}dx) = β#(r^{i}x^{i−1}dx + r^{i−1}x^{i}dr) = β#r^{i}x^{i−1}dx + β#r^{i−1}dr#x^{i} by 3.1. Proof of Theorem 4.2 We shall show that the Cartier identities of Theorem 2.8 hold. Since by 4.4, HH_{n}(Λ)⊗xk[x] is closed under the action of V_{m}, [r] and F_{m}, it will then be a Carf(R)–submodule. Now V_{m} and F_{m} respect the direct sum decomposition of the Künneth Formula, so identities (i-V,F), (ii-V,F), (iv), (v) and (vii) follow from the special case Λ = k and the results of Sections 2 and 3. Identities (i-[r]), (ii- [r]) and (iii-V) follow immediately from the definitions. Thus it is necessary only to demonstrate axioms (iii-F) and (vi). But again we use the formulas of Theorem 4.4 and the argument then follows closely that of Theorem 3.2.

Since W (R) is contained in Carf(R) the above theorems immediately yield:

Corollary 4.8 With notation as in 4.4, NHH_{n}(Λ) is a W (R)–module with operations:

(1 − rt^{m}) ∗ (α#x^{i}) = mr^{i/m}#x^{i}
(1 − rt^{m}) ∗ (β#x^{i−1}dx) = mr^{i/m}β#x^{i−1}dx + (β#r^{i/m−1}dr)#x^{i}

if m|i and 0 otherwise.

As an application, we can generalize the module structure of Corollary 3.6 to non-commutative de Rham cohomology. There are well known maps (see e.g. [LQ]) B : HC_{n}(Λ) → HH_{n+1}(Λ) and I : HH_{n}(Λ) → HC_{n}(Λ), which when composed give a map BI : HH_{n}(Λ) → HH_{n+1}(Λ). For example, the cokernel of BI : HH_{0}(Λ) → HH_{1}(Λ) is HC_{1}(Λ). Since (BI)^{2} = 0 we get a complex

0 ⟶ HH_{0}(Λ) ⟶^{BI} HH_{1}(Λ) ⟶^{BI} HH_{2}(Λ) ⟶^{BI} ⋯

The cohomology of this complex is called the non-commutative de Rahm cohomology of Λ (see eg. [LQ]). We will use the notation HH_{dR}^{∗}(Λ) to distinguish it from the classical de Rahm cohomology H_{dR}^{∗}(Λ) used in §3. Of course, if Λ is commutative and smooth over k then H_{dR}^{∗} and HH_{dR}^{∗} agree. Finally, we use the notation NHH_{dR}^{n}(Λ) = ker(HH_{dR}^{n}(Λ[x]) ⟶^{x↦0} HH_{dR}^{n}(Λ)). This is known to be a torsion abelian group. In particular, if Q ⊆ k then NHH_{dR}^{∗}(Λ) = 0.

Proposition 4.9 Let R be a commutative k–algebra and suppose that Λ is a (possibly non-commutative) R–algebra. The differential BI : NHH_{n}(Λ) → NHH_{n+1}(Λ) is a morphism of Carf(R)–modules. In particular,

• NHH_{dR}^{n}(Λ) is a Carf(R)–module.

• NHC_{1}(Λ) is a Carf(R)–module

Proof: The VHF endomorphisms are endomorphisms of mixed complexes. As such they commute with S, B, I and the composition BI. Hence BI commutes with every element of Carf(R).

We can now show that NHC_{∗}(Λ) is also a Carf(R)–module, at least if we assume that Q ⊆ k. It follows from [Goodw] that the SBI sequence breaks up into short exact sequences

0 ⟶ NHC_{n−1}(Λ) ⟶^{B} NHH_{n}(Λ) ⟶^{I} NHC_{n}(Λ) ⟶ 0

¿From this we see that NHC_{n−1}(Λ) is the kernel of NHH_{n}(Λ) ⟶^{BI} NHH_{n+1}(Λ). The following follows immediately from 4.9.

Theorem 4.10 Let Q ⊆ k and R, Λ as above. Then each NHC_{n}(Λ) is a Carf(R)– module, and

0 ⟶ NHC_{n−1}(Λ) ⟶^{B} NHH_{n}(Λ) ⟶^{I} NHC_{n}(Λ) ⟶ 0

is a short exact sequence of Carf(R)–modules.

As an illustration of the power of module structures, we give a quick proof of a calculation of Kassel [K-CCM, Example 3.3]. For this, we need the Leibniz formula [LQ, p.576], [FT] (see also [GRW, 5.5]):

BI(α#β) = BI(α)#β + (−1)^{p}α#BI(β),

where α ∈ HH_{p}(Λ), and β ∈ HH_{q}(k[x]). Identifying HH_{1}(k[x]) = Ω_{k[x]}, we also note that BI(dx) = BI(BI(x)) = 0.

Proposition 4.11 Let k be any commutative ring containing Q, and let Λ be any (not necessarily commutative) k–algebra. Then there is a natural isomorphism of Carf(k)–modules:

NHC_{n}(Λ) ≈ HH_{n}(Λ) ⊗ xk[x].

The module structure on the right is given in 2.9. Consequently,

HC_{n}(Λ[x]) = HC_{n}(Λ) ⊕ HH_{n}(Λ) ⊗ xk[x].

Proof By Theorem 4.10 above NHC_{n} is a Carf(k)–module, so by Corollary 2.14 it is enough to show that for some r ∈ k, r ≠ 0, ±1, the eigenspace M_{1} of r under the endomorphism (1 − rt)∗ is isomorphic to HH_{n}(Λ). Viewing NHC_{n}(Λ) as the image of BI : NHH_{n}(Λ) → NHH_{n+1}(Λ), BI is an epimorphism of Carf(k)–modules so M_{1} is the image of the corresponding eigenspace of NHH_{n}(Λ) which is seen to be generated by α#x, and β#dx, α ∈ HH_{n}(Λ) and β ∈ HH_{n−1}(Λ) by Theorem 4.4.

Now define Ψ : HH_{n}(Λ) → M_{1} by Ψ(α) = BI(α#x) = BI(α)#x + (−1)^{n}α#dx. But BI(β#dx) = BI(β)#dx = (−1)^{n}BI(BI(β)#x) = (−1)^{n}Ψ(BI(β)) so Ψ is surjective. Projecting onto the summand HH_{n}(Λ) ⊗ k[x]dx of NHH_{n+1}(Λ) (via the Künneth Formula) we see that Ψ is injective, completing the proof.

To conclude this section, we observe that the VHF operators commute with the Dennis Trace map K_{∗}(Λ[x]) → HH_{∗}(Λ[x]). [Igusa, 1.a]. This is because the Verschiebung NK_{n}(Λ) → NK_{n}(Λ) is induced by the map x → x^{m}, the homothety [r] is induced by x ↦ rx, and the Frobenius is the transfer on K-theory associated with this map (see [Bloch2, W-MVS]). We note that the Frobenius transfer (associated with x ↦ x^{m}) is given by composing the map on K-theory induced by ı with the isomorphism K_{n}(M_{m}(Λ)) ≈ K_{n}(Λ) induced by Morita invariance. Since these operations make NK_{n}(Λ) into a Carf(R)–module [W-MVS], we have the following result.

Proposition 4.12 Let Λ be a R algebra where R is a commutative k–algebra. Then the Dennis Trace map D : NK_{n}(Λ) → NHH_{n}(Λ) is a homomorphism of Carf(R)– modules.

5 The W (R)–module structure on HH_{∗} of graded

rings

In this section k will be a commutative ring, R a commutative k–algebra and Λ = Λ_{0} ⊕ Λ_{1} ⊕ Λ_{2} ⊕ ⋯ will be a graded unitary R–algebra, i.e. each graded piece Λ_{j} is a R–module.

Set H̃H_{n}(Λ) = HH_{n}(Λ)/HH_{n}(Λ_{0}). We will show that H̃H_{n}(Λ) is a W (R)– module. The example in [GRW, Thm. 3.11] shows that H̃H_{n}(Λ) need not be a Carf(R)–module, because Theorem 2.13 is violated for n ≥ 2.

Given a graded R–algebra Λ and an indeterminant x, we define a map f_{x} : Λ → Λ[x] by sending a_{k} ∈ Λ_{k} to a_{k}x^{k}. This induces a map HH_{n}(Λ) → HH_{n}(Λ[x]) which we also call f_{x}. If α is an cycle of HH_{n}(Λ) of weight k (see [GRW, Def. 1.1]), using the decomposition of the Künneth Formula we see f_{x}(α) = α′#x^{k} + β#x^{k−1}dx for some α′ ∈ HH_{n}(Λ) and some β ∈ HH_{n−1}(Λ). Now x ↦ 1 is a left inverse to f_{x} so (since d(1) = 0) α′ = α. Writing ξ_{x}(α) = β we have a homomorphism ξ : HH_{n}(Λ) → HH_{n−1}(Λ). We remark that ξ is very dependent on the grading of Λ.

For example, if Λ is the polynomial ring Λ_{0}[t], and t is in degree 1, the map f_{x} is the substitution t ↦ xt. It is easy to calculate directly that for ω ∈ HH_{∗}(Λ_{0})

ξ_{x}(t^{k}ω) = 0, but ξ_{x}(ω#t^{k−1}dt) = t^{k}ω.

Lemma 5.1 With notation as above, the map ξ : HH_{n}(Λ) → HH_{n−1}(Λ) defined by

f_{x}(α) = α#x^{k} + ξ(α)#x^{k−1}dx

satisfies the equation ξ^{2} = 0.

Proof We consider Λ[y] as a graded ring with Λ graded in the usual way and y in degree 0. Consider the following commutative diagram

Λ ⟶^{f_{y}} Λ[y]
f_{x}↓ ↓f_{x}
Λ[x] ⟶^{x↦xy} Λ[x, y].

Calculating (writing h for the bottom map) we have for ω a homogeneous form of degree k in HH_{n}(Λ)

hf_{x}(ω) = h(ω#x^{k} + ξ(ω)#x^{k−1}dx)

= ω#(xy)^{k} + ξ(ω)#(xy)^{k−1}d(xy)
= ω#x^{k}y^{k} + ξ(ω)#x^{k}y^{k−1}dy + ξ(ω)#x^{k−1}y^{k}dx  (6)

On the other hand

f_{x}f_{y}(ω) = f_{x}(ω#y^{k} + ξ(ω)#y^{k−1}dy)

= ω#y^{k}x^{k} + ξ(ω)#x^{k}y^{k−1}dy + ξ(ω#y^{k} + ξ(ω)#y^{k−1}dy)#x^{k−1}dx
= ω#y^{k}x^{k} + ξ(ω)#x^{k}y^{k−1}dy + ξ(ω#y^{n})#x^{k−1}dx
+ ξ(ξ(ω)#y^{k−1}dy)#x^{k−1}dx  (7)

Now to calculate the last two terms in (7) we have

f_{x}(ω#y^{k}) = f_{x}(ω)#f_{x}(y^{k})
= (ω#x^{k} + ξ(ω)#x^{k−1}dx)#y^{k} = ω#x^{k}y^{k} + ξ(ω)#x^{k−1}y^{k}dx (8)

since y has degree 0, so that ξ(ω#y^{k}) = ξ(ω) y^{k}. Also

f_{x}(ξ(ω)#y^{k−1}dy) = f_{x}(ξ(ω))#f_{x}(y^{k−1}dy)
= (ξ(ω)x^{k} + ξ(ξ(ω))x^{k−1}#dx)#y^{k−1}dy
= ξ(ω)x^{k}y^{k−1}dy + ξ^{2}(ω)#x^{k−1}y^{k−1}dx ∧ dy  (9)

so that ξ(ξ(ω)#y^{k−1}dy) = ξ^{2}(ω)#y^{k−1}dy. Substituting (8) and (9) into (7) and comparing with (6) we see that ξ^{2}(ω)#y^{k−1}x^{k−1}dy ∧ dx = 0. But this term lives in the last summand of the Künneth Formula decomposition

HH_{n}(Λ[x, y]) = HH_{n}(Λ) ⊗_{k} k[x, y] ⊕ HH_{n−1}(Λ) ⊗_{k} Ω_{k[x,y]} ⊕ HH_{n−2}(Λ) ⊗_{k} Ω_{k[x,y]}^{2}.

Since Ω_{k[x,y]}^{2} = k[x, y]dx ∧ dy is a free k–module it follows that ξ^{2}(ω) = 0. This completes the proof of the lemma.

We can now prove our main theorem of this section. Theorem 5.2 Let Λ be a graded R–algebra where R is a commutative k–algebra. Then there is a W (R)–module structure on H̃H_{n}(Λ) such that the inclusion f_{x} : H̃H_{n}(Λ) → NHH_{n}(Λ) is a W (R)–module map. In particular, if ω is a homogeneous cycle of weight k then

(1 − rt^{m}) ∗ ω = mr^{k/m}ω + r^{k/m−1}ξ(ω)#dr  (10)

if m divides k and 0 otherwise. Proof We will show the following diagram commutes

H̃H_{n}(Λ) ⟶^{f_{x}} NHH_{n}(Λ)
∗↓ ↓∗
H̃H_{n}(Λ) ⟶^{f_{x}} NHH_{n}(Λ)

where ∗ denotes multiplication by (1 − rt^{m}) given by formula (10) on the left and given by Corollary 4.8 on the right. Then, since NHH_{n}(Λ) is a W (R)–module by 4.8 and f_{x} is injective it follows that H̃H_{n}(Λ) is a W (R)–module.

To this end, let ω be a homogeneous cycle in H̃H_{n} of weight k. If m does not divide k then both multiplications by (1 − rt^{m}) give 0. If m|k then as in the proof of Lemma 5.1 we calculate

ξ(mr^{k/m}ω + r^{k/m−1}ξ(ω)#dr) = ξ(mr^{k/m}ω) + ξ(r^{k/m−1}ξ(ω)#dr)

= mr^{k/m}ξ(ω) + r^{k/m−1}ξ^{2}(ω)#dr = mr^{k/m}ξ(ω)

using the fact that r is of degree 0 and ξ^{2}(ω) = 0. From this the easy calculation

(1 − rt^{m}) ∗ f_{x}(ω) = mr^{k/m}ω#x^{k} + mr^{k/m}ξ(ω)#x^{k−1}dx + r^{k/m−1}ξ(ω)#x^{k}dr
= f_{x}(1 − rt^{m}) ∗ ω

completes the proof.

Remark 5.3 If Λ is commutative the map Ω_{Λ/k}^{n} → HH_{n}(Λ) induces a map (which we call ϕ as in 4.5) Ω̃_{Λ/k}^{n} → H̃H_{n}(Λ). This map is a map of W (R)–modules. In fact, we can mimic the definition of ξ in the context of Ω̃_{Λ/k}^{n}. For if ω = a_{0}da_{1} ∧ da_{2} ∧ ⋯ ∧ da_{n} where e_{i} = deg(a_{i}) and k = e_{0} + ⋯ + e_{n} then

f_{x}(ω) = a_{0}x^{e_{0}}d(a_{1}x^{e_{1}}) ∧ ⋯ ∧ d(a_{n}x^{e_{n}})
= x^{k}ω + a_{0}x^{k−e_{i}} ∑_{i=1}^{n} a_{i}da_{1} ∧ ⋯ ∧ d(x^{e_{i}}) ∧ ⋯ ∧ da_{n}
= x^{k}ω + a_{0}x^{k−1} ∑_{i=1}^{n} e_{i}a_{i}da_{1} ∧ ⋯ ∧ dx ∧ ⋯ ∧ da_{n}
= x^{k}ω + a_{0}x^{k−1} ∑_{i=1}^{n}(−1)^{n−i}e_{i}a_{i}da_{1} ∧ ⋯ ∧ da_{i−1} ∧ da_{i+1} ∧ ⋯ ∧ da_{n} ∧ dx

Thus we may write

ξ(ω) = a_{0} ∑_{i=1}^{n}(−1)^{n−i}e^{i}a_{i}da_{1} ∧ ⋯ ∧ d̂a_{i} ∧ ⋯ ∧ da_{n}

Hence formula (1) of Theorem 3.4 could have been written

(1 − rt^{m}) ∗ ω = r^{k/m}mω + r^{k/m−1}ξ(ω) ∧ dr

which mirrors Theorem 5.2. In particular, if Λ is smooth over k then the formulas in Theorems 3.4 and 5.2 agree.

Our technique for obtaining the W (R)–module structure on H̃H_{∗}(Λ), i.e. imbedding H̃H_{∗}(Λ) in NHH_{∗}(Λ), is analagous to the method used in [W-MGR, §1] and thus we obtain

Theorem 5.4 With k, R, and Λ as in Theorem 5.2 the Dennis trace map D : K̃_{n}(Λ) → H̃H_{n}(Λ) is a W (R)–module homomorphism.

When Λ is a graded k–algebra, the map BI induces BI : H̃H_{n}(Λ) → H̃H_{n+1}(Λ). The homology of the resulting complex will be denoted H̃H_{dR}^{∗}(Λ). Alternatively we could define H̃H_{dR}^{∗}(Λ) to be the quotient HH_{dR}^{∗}(Λ)/HH_{dR}^{∗}(Λ_{0}), as in §3 and §4. We then have

Theorem 5.5 Let R be a commutative k algebra and Λ an graded R–algebra. Then BI : H̃H_{n}(Λ) → H̃H_{n+1}(Λ) is a W (R)–module map. Consequently, H̃H_{dR}^{n}(Λ) is a W (R)–module for each n.

Proof Since B and I are natural and hence commute with the embedding f_{x} of H̃H_{n}(Λ) in NHH_{n}(Λ), this follows from the corresponding result for NHH. This was proven in 4.9.

Finally we derive the graded analogues of the results in the last section about NHC when Q ⊆ k. If Λ is a graded k–algebra, H̃C_{n}(Λ) will denote the quotient HC_{n}(Λ)/HC_{n}(Λ_{0}). As in loc. cit., it follows from [Goodw] that the SBI sequence breaks up into short exact sequences

0 ⟶ H̃C_{n−1}(Λ) ⟶^{B} H̃H_{n}(Λ) ⟶^{I} H̃C_{n}(Λ) ⟶ 0.

¿From this we see that H̃C_{n−1}(Λ) is the kernel of the map H̃H_{n}(Λ) ⟶^{BI} H̃H_{n+1}(Λ)). Since BI is a W (R)–module map we have Corollary 5.6 Let Q ⊆ k and R, Λ as above, with Λ a graded R–algebra. Then H̃C_{n}(Λ) = HC_{n}(Λ)/HC_{n}(Λ_{0}) is a W (R)–module.

6 The W (k) and Carf(k)–module structure on HC_{∗}

in positive characteristic

In this section, we give a mixed complex C_{•}(Λ) ⊗ NΩk^{•} of Carf(k)–modules whose Hochschild and cyclic homology are NHH(Λ) and NHC(Λ). Under the induced Carf(k)–module structure on NHH(Λ) and NHC(Λ), we show that V_{m}, F_{m} and [r] with r ∈ k correspond to the VHF operators on C_{•}(Λ[x]). This will establish the following result:

Theorem 6.1 Let Λ be a k–algebra. The VHF operators on C_{•}(Λ[x]) defined in §4 make NHH_{∗}(Λ) and NHC_{∗}(Λ) into Carf(k)–modules.

Recall from [K-CCM] that a mixed complex (M, b, B) of k–modules is a graded k–module {M_{i} : i ≥ 0}, together with endomorphisms b and B of degree −1 and +1 respectively, which satisfy

b^{2} = B^{2} = Bb + bB = 0

Associated to a mixed complex are its Hochschild homology HH_{∗}(M) = H_{∗}(M, b), its cyclic homology HC_{∗}(M) = H_{∗}(_{B}M, d) (see [K-CCM, p. 198]) and related groups such as HP_{∗}(M) etc.

Example 6.2 If R is a commutative k–algebra, (Ω_{R}^{•}, 0, d) is a mixed complex of R–modules, where d : Ω_{R}^{n} → Ω_{R}^{n+1} is the de Rahm differential. By Theorem 3.2 the mixed subcomplex (NΩ_{R}^{•}, 0, d) of (Ω_{R[x]}^{•}, 0, d) is even a mixed complex of k⊗Carf(R)– modules.

Example 6.3 If Λ is any k–algebra, the “reduced bar construction” (see [CE, p.

176] or [Mac, X.2]) is the complex C_{•}(Λ) with

C_{n}(Λ) = Λ ⊗_{k} (\overbrace{Λ/k ⊗ ⋯ ⊗ Λ/k}^{n times})

This is a mixed complex, but not a cyclic module in the sense of Connes. It is traditional to use the notation a_{0}[a_{1}|a_{2}| ⋯ |a_{n}] for a generator of C_{n}(Λ). The endomorphism B is the usual one given in [LQ, K-CCM] etc.

Example 6.4 The tensor product of two mixed complexes of k–modules is determined in the usual way. In particular, we can combine the above two examples to construct (C_{•}(Λ) ⊗_{k} NΩ_{k}^{•}, b ⊗ 1, B ⊗ 1 ± 1 ⊗ d), which is not only a mixed complex of k–modules, but is also a mixed complex of Carf(k)–modules.

In order to show that HH and HC of the previous mixed complex agree with NHH(Λ) and NHC(Λ), we need to compare it to the mixed complex NC_{•}. For this, we need to recall the homotopy notion of morphism for mixed complexes. There are two notions of morphisms of mixed complexes: strict morphisms (map commuting with b and B) and strongly homotopy maps [K-CCM, 2.2]. If M and N are mixed complexes a strongly homotopy map G : M → N is a collection of graded k–module maps G^{(i)} : M_{•} → N_{•+2i}, i ≥ 0 such that G^{(0)} is a chain map ([G^{(0)}, b] = 0) and

[G^{(i+1)}, b] + [G^{(i)}, B] = 0 for all i ≥ 0

(In [K-CCM] the brackets stand for graded commutators but as the G^{(i)} have even degree here they are simply commutators.) The composition H∘G of strongly homotopy maps is again a strongly homotopy map: (HG)^{(n)} = ΣH^{(i)}G^{(j)}.

By [K-CCM, 2.3] a strongly homotopy map not only induces the obvious map G_{∗} : HH_{∗}(M) → HH_{∗}(N) but also maps G_{∗} : HC_{∗}(M) → HC_{∗}(N), etc.

We will call a map G : M → N (strict or strongly homotopy) a quasi-isomorphism if the maps HH_{∗}(M) → HH_{∗}(N) induced by G^{(0)} are isomorphisms. It is well known that the induced maps G_{∗} : HC_{∗}(M) → HC_{∗}(N) etc. are also isomorphisms whenever G is a quasi-isomorphism of mixed complexes.

Example 6.5 (homothety) If r ∈ k, the map R[x] → R[x] sending x to rx defines a strict endomorphism [r] of the mixed complex NΩ_{R}^{•} of Example 6.2. If we want to define a homothety [r] for other elements r of R, we need a slightly different construction, owing to the fact that dr need not vanish.

Consider the group Ω_{R}^{•} as a functor of R and apply this to the map k[x] → k[x, r] sending x ↦ rx, where r is an indeterminate. We let [r] denote the resulting map Ω_{k[x]}^{•} → Ω_{k[x,r]}^{•} = Ω_{k[r]}^{•} ⊗ Ω_{k[x]}^{•} . In particular we have

[r](x^{i}) = r^{i} ⊗ x^{i}
[r](x^{i−1}dx) = (rx)^{i−1}d(rx) = r^{i} ⊗ x^{i−1}dx + r^{i−1}dr ⊗ x^{i}

A simple calculation (left to the reader) shows that [r] is a strict morphism of the mixed complexes of Example 6.2 (Cf. 3.1). Composing with k[r] → R yields another strict morphism [r] : Ω_{k[x]}^{•} → Ω_{R}^{•} ⊗ Ω_{k[x]}^{•}.

Example 6.6 By [Rine], [HK] or [K-CCM, 2.4] there is a strongly homotopy map ∇ : C_{•}(Λ) ⊗ C_{•}(Λ′) → C_{•}(Λ ⊗ Λ′) of mixed complexes, natural in Λ and Λ′, such that ∇^{(0)} is the shuffle map, ∇^{(1)} is Rinehart’s cyclic shuffle map and ∇^{(i)} = 0 for i > 1. Since the shuffle map is a quasi-isomorphism, so is ∇.

Example 6.7 For the polynomial ring k[x] there is a strict morphism of mixed complexes η : C_{•}(k[x]) → Ω_{k[x]}^{•} with η_{0} the canonical isomorphism C_{0}(k[x]) ≈ k[x] = Ω_{k[x]}^{0} and η_{1}(f ⊗ g) = f dg. Since it induces an isomorphism on HH_{∗}, it is a quasiisomorphism. There is no strict morphism going the other way, however; one needs a strongly homotopy map for that.

Lemma 6.8 The chain map g_{j} : Ω_{k[x]}^{j} → C_{j}(k[x]) defined by the formulas g_{0}(x^{i}) = x^{i}, g_{1}(x^{i−1}dx) = x^{i−1}[x] extends to a strongly homotopy map

G : Ω_{k[x]}^{•} → C_{•}(k[x])

such that ηG is the identity on Ω_{k[x]}^{•}.

Proof Because b : C_{1}(k[x]) → C_{0}(k[x]) is zero, g = G^{(0)} is a chain map. Because G^{(i+1)}b = 0, (b = 0 on Ω_{k[x]}^{•}) we need G_{0}^{(j)} : k[x] → C_{2j}(k[x]) and G_{1}^{(j)} : Ω_{k[x]}^{1} → C_{2j+1}(k[x]) such that

bG_{0}^{(j+1)} = G_{1}^{(j)}B − BG_{0}^{(j)} and bG_{1}^{(j+1)} = −BG_{1}^{(j)}  (11)

For G_{0}^{(1)} we noted in §4 that the map Ω_{k[x]}^{1} → C_{1}(k[x]) → HH_{1}(k[x]) given by x^{i} dx^{j} ↦ x^{i}[x^{j}] is an isomorphism. Thus it follows that the cycles G_{1}^{(0)}B(x^{i}) = ix^{i−1}[x] and BG_{0}^{(j)}(x^{i}) = [x^{i}] map to the same element in HH_{1}(k[x]) and hence the image of G_{1}^{(j)}B − BG_{0}^{(j)} is contained in the image of b, thus it is possible to define G_{0}^{(1)} satisfying the first identity of Equation (11).

We now argue by induction and assume that G^{(j)} is defined and satisfies Equation (11). Then b(G_{1}^{(j)}B − BG_{0}^{(j)}) = bG_{1}^{(j)}B − bBG_{0}^{(j)} = bG_{1}^{(j)}B + BbG_{0}^{(j)} = −BG_{1}^{(j−1)}B + B(G_{1}^{(j−1)}B − BG_{0}^{(j−1)}) = 0. Now since the complex C_{•}(k[x]) is exact for • ≥ 2 we can define G_{0}^{(j+1)} to satisfy Equation (11). Likewise bBG_{1}^{(j)} = −BbG_{1}^{(j)} = BBG_{1}^{(j−1)} = 0 so we can likewise define G_{1}^{(j+1)}.

Remark 6.9 Because g is a chain map, ηg is the identity on Ω_{k[x]}^{•} and gη is chain homotopic to the identity on C_{•}(k[x]) by a homotopy ϕ, (Ω_{k[x]}^{•}, C_{•}(k[x]), g, η, ϕ) is a “deformation retract” in the sense of [K-LP]. Since η is a strict morphism, Lemma 6.8 is seen to be an application of [K-LP, Cor. 7.2].

Theorem 6.10 For every k–algebra Λ, there are natural quasi-isomorphisms of mixed complexes of k-modules

C_{•}(Λ) ⊗ Ω_{k[x]}^{•} ⟵^{1⊗η} C_{•}(Λ) ⊗ C_{•}(k[x]) ⟶^{∇} C_{•}(Λ[x])
C_{•}(Λ) ⊗ NΩ_{k}^{•} ⟵^{1⊗η} C_{•}(Λ) ⊗ NC_{•}(k) ⟶^{∇} NC_{•}(Λ)

Since C_{•}(Λ)⊗NΩk^{•} is a mixed complex of Carf(k)–modules by Theorem 3.2, it follows that there is a Carf(k)–module structure on NHH_{∗}(Λ), NHC_{∗}(Λ), etc.

Proof This follows from 6.6, 6.8 and the Künneth formula.

To finish the proof of Theorem 6.1, we need only show that this structure comes from the VHF operators of §4, which were defined on the mixed complex C_{•}(Λ[x]).

Lemma 6.11 The Verschiebung endomorphisms V_{m} on NHH_{∗}(Λ), and NHC_{∗}(Λ) are induced by the strict map 1 ⊗ V_{m} on C_{•}(Λ) ⊗ NΩk^{•}

Proof If y = x^{m}, naturality of ∇ and η, 6.6 and 6.8 yield a commutative diagram of mixed complexes and strongly homotopy maps:

C_{•}(Λ) ⊗ Ω_{k[y]}^{•} ⟵^{1⊗η} C_{•}(Λ) ⊗ C_{•}(k[y]) ⟶^{∇} C_{•}(Λ[y])
1 ⊗ V_{m}↓ ↓1 ⊗ V_{m} ↓V_{m}
C_{•}(Λ) ⊗ Ω_{k[x]}^{•} ⟵^{1⊗η} C_{•}(Λ) ⊗ C_{•}(k[x]) ⟶^{∇} C_{•}(Λ[x])

Therefore the vertical maps all induce the same maps on NHH_{∗}, NHC_{∗} etc.

Lemma 6.12 If r ∈ k the homothety [r] on NHH_{∗}(Λ) and NHC_{∗}(Λ) are induced by the strict map 1 ⊗ [r] on C_{•}(Λ) ⊗ NΩk^{•}. Proof Consider the following diagram of mixed complexes of k–modules.

C_{•}(Λ) ⊗ Ω_{k[y]}^{•} ⟵^{1⊗η} C_{•}(Λ) ⊗ C_{•}(k[y]) ⟶^{∇} C_{•}(Λ[y])
1 ⊗ [r]↓ ↓1 ⊗ [r] ↓[r]
C_{•}(Λ) ⊗ Ω_{k[x]}^{•} ⟵^{1⊗η} C_{•}(Λ) ⊗ C_{•}(k[x]) ⟶^{∇} C_{•}(Λ[x])

The vertical maps [r] are the strict morphisms of 4.1 and 3.1. The left square commutes by direct inspection. The right square commutes by naturality of Rinehart’s strongly homotopy map ∇. Lemma 6.13 The Frobenius endomorphism F_{m} on NHH_{∗}(Λ) and NHC_{∗}(Λ) is induced by the strict map 1 ⊗ F_{m} on C_{•}(Λ) ⊗ NΩk^{•}. Proof Consider the following diagram of chain complexes:

[figure: C_{•}(Λ) ⊗ Ω_{k[x]}^{•}; 1 ⊗ η; C_{•}(Λ) ⊗ C_{•}(k[x]); ∇; C_{•}(Λ[x]); C_{•}(Λ) ⊗ C_{•}(M_{m}(k[y])); ∇; C_{•}(M_{m}(Λ[y])); C_{•}(Λ) ⊗ Ω_{k[y]}^{•}; 1 ⊗ η; C_{•}(Λ) ⊗ C_{•}(k[y]); ∇; C_{•}(Λ[y]); 1 ⊗ F_{m}; 1 ⊗ ι; ι; trace; trace]

The left hand square commutes by a brute force calculation and the upper right by the naturality of ∇. The lower right square commutes by the explicit definition of ∇.

(We learned this calculation from Kassel; see [K-CCB, II.5.4] and [HK].)

These last three lemmas complete the proof of Theorem 6.1.

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