Cyclic homology and path algebra resolutions

Dave Benson Mathematical Institute Oxford

It follows from a theorem of Loday and Quillen [6, Prop. 5.4] that one may calculate the cyclic homology of an algebra in characteristic zero by taking a semisimplicial resolution by free algebras, quotienting out commutators and then taking homology of the resulting complex. In this paper we explain how this is a special case of a more general method based on resolutions by path algebras of directed graphs. The Loday-Quillen result may be seen as the case where the graph has only one vertex.

In the case of a finite dimensional algebra over an algebraically closed field of characteristic zero, this is especially suited to computations, since there is an essentially unique minimal such resolution. Even if the field does not have characteristic zero, one can obtain some information in terms of a spectral sequence abutting to cyclic homology.

I would like to thank Dan Quillen for giving a series of lectures at the University of Oxford, which started off my interest in cyclic homology.

1 Path algebras

In this section, we recall some of the basic notions associated with path algebras.

Definition 1.1 A quiver is a directed graph, possibly with multiple arrows and loops.

If Q is a quiver and k is a field, we define the path algebra kQ as follows. It is an algebra over k, which as a vector space has a basis consisting of the paths • → • → ⋯ → • in Q. Multiplication is given on basis elements by composition of paths in reverse order (because we are dealing with left rather than right modules) if the paths are composable in this way, and zero otherwise. Thus for example corresponding to each vertex x there is a path of length zero giving rise to an idempotent basis element denoted e_{x}. A free algebra is an example of a path algebra, for a graph with only one vertex. The algebra kQ has an identity element if and only if Q has finitely many vertices. It is finitely generated if and only if Q has only finitely many vertices and arrows, and finite dimensional if and only if in addition it has no oriented cycles.

A representation of a quiver Q associates to each vertex x of Q a vector space V_{x}, and to each arrow x → y a linear transformation V_{x} → V_{y} between the corresponding vector spaces.

There is a natural one-one correspondence between representations of Q and kQmodules given as follows. Given a representation of Q, we form a kQ-module whose underlying vector space is ⨁_{x} V_{x}, and where the action of a basis element x_{1} → ⋯ → x_{2} is as the composite of the corresponding maps

⨁_{x} V_{x} → → V_{x_{1}} → ⋯ → V_{x_{2}} ↪ ⨁_{x} V_{x}.

So for example the action of the idempotent e_{x} is to project onto V_{x}.

Conversely, given a kQ-module V, we form a representation of Q by setting V_{x} = e_{x}V. If u is the basis element of kQ corresponding to an arrow x → y, then e_{y}u = u = ue_{x}, and so u maps V_{x} to V_{y}. These are the maps we use to define the representation of Q. It is clear that the above procedures are mutually inverse.

There is a simple kQ-module S_{x} of dimension one corresponding to each vertex x of Q. It consists of a one dimensional vector space at the vertex x, and a zero dimensional vector space at each other vertex. In case kQ is finite dimensional, these are the only simple kQ-modules, but otherwise there are others. For example, for a free algebra with at least two generators, there are simple modules of every dimension.

The projective module P_{x} = kQ.e_{x} corresponding to an idempotent e_{x} may be described as follows. The basis elements correspond to the paths in Q which begin with the vertex x, and the action of kQ is given by (reverse) composition of paths as before.

Definition 1.2 We write kQ_{(n)} for the linear span in kQ of the paths of length at least n. It is a two sided ideal in kQ.

As a (left) kQ-module, kQ_{(n)}/kQ_{(n+1)} is semisimple. It is a direct sum of simple modules of the form S_{x}, and the copies of a particular S_{x} correspond to the paths of length n ending at x. That is, e_{x}.kQ_{(n)}/kQ_{(n+1)} has a basis consisting of the paths of length n ending at x. For example, if kQ is finite dimensional then kQ_{(n)} = J^{n}(kQ).

Now a path of length m, x → ⋯ → y, induces an injective map

e_{x}.kQ_{(n)}/kQ_{(n+1)} → e_{y}.kQ_{(n+m)}/kQ_{(n+m+1)}

and the images of these maps for distinct paths are linearly independent.

If F is any free kQ-module (not necessarily finitely generated) with a chosen basis, then it inherits a filtration F_{(n)} from the above filtration on kQ. Again if x → ⋯ → y is a path of length m then the induced map

e_{x}.F_{(n)}/F_{(n+1)} → e_{y}.F_{(n+m)}/F_{(n+m+1)}

is injective, and the images of these maps for distinct paths are linearly independent.

Theorem 1.3 Suppose Q has only finitely many vertices. Then every submodule of a free kQ-module is isomorphic to a direct sum of modules of the form kQ.e_{x}, so that kQ is a hereditary algebra. The Krull-Schmidt theorem holds for finitely generated projective kQ-modules.

Proof Suppose P is a submodule of a free kQ-module F. Define F_{(n)} as above, and P_{(n)} = P ∩ F_{(n)}. Suppose P = P_{(t)} ⊃ P_{(t+1)}. Then

P_{(t)}/P_{(t+1)} = ⨁_{x} e_{x}.P_{(t)}/P_{(t+1)}

and so P_{(t)}/P_{(t+1)} is a direct sum of modules of the form S_{x}. Choose a basis {u_{x,α}} of each e_{x}.P_{(t)}/P_{(t+1)} and inverse images ũ_{x,α} ∈ P. By the discussion preceding the statement of the theorem, there is a surjective map

ϕ : P → ⨁_{x,α} P_{x,α},

where each P_{x,α} ≅ kQ.e_{x} and the map sends ũ_{x,α} ↦ e_{x} ∈ P_{x,α}. We construct such a map inductively as follows. In P_{(t+1)}/P_{(t+2)}, we choose a kQ-complement to the images of the u_{x,α} under the action of the paths of length at least one. We choose a kQ-basis {v_{y,β}} for this complement, and lift to elements ṽ_{y,β} of P_{(t+1)}. These elements will be sent to zero by ϕ. Now in P_{(t+2)}/P_{(t+3)}, the map ϕ is determined on the images of the u_{x,α} under the action of the paths of length two, and of the v_{y,β} under the action of the paths of length one, so we choose a kQ-complement to these, choose a basis and lift as before, and send these new elements to zero. Continuing this way, we construct ϕ on the whole of P.

Since ⨁_{x,α} P_{x,α} is projective, the map ϕ splits, and so

P ≅ P′ ⊕ ⨁_{x,α} P_{x,α}

with P′ = P′_{(t+1)}. Continuing this way, we eventually write P as a direct sum of modules isomorphic to some kQ.e_{x}. In particular, every submodule of a projective module is projective, and so kQ is hereditary.

To prove the Krull-Schmidt theorem for finitely generated projective kQ-modules, we note that

kQ/kQ_{(1)} ⊗_{kQ} ⨁_{x,α} P_{x,α} = ⨁_{x,α} S_{x,α}

with S_{x,α} ≅ S_{x}. Thus if the module is finitely generated, the number of copies of of each P_{x,α} is finite, and hence well defined by applying the Krull-Schmidt theorem for finitely generated kQ/kQ_{(1)}-modules. □

Corollary 1.4 Suppose Q is a quiver with finitely many vertices. Then K_{0}(kQ) is a free abelian group whose rank is equal to the number of vertices in Q.

The following corollary is well known.

Corollary 1.5 Every projective module over a free algebra (over a field) is free.

We now explain why path algebras are important in representation theory. It follows from Morita theory (See for example Bass [1, Chapter II]) that to study the representations of a finite dimensional algebra over an algebraically closed field, it suffices to consider the case where every irreducible module is one dimensional. Namely the given algebra is Morita equivalent to the endomorphism ring of the direct sum of one copy of each projective indecomposable module. This is called the basic algebra of the given algebra. Note that by Loday and Quillen [6, Cor. 1.7], cyclic homology is Morita invariant, and so it does not hurt us to assume from now on that we are only dealing with basic algebras.

Definition 1.6 Suppose Λ is a finite dimensional algebra over an algebraically closed field k. Let S_{1}, …, S_{r} be the isomorphism classes of simple Λ-modules with projective covers P_{i} = Λe_{i}. The Ext-quiver Q(Λ) has vertices x_{1}, …, x_{r} corresponding to these simple modules, and the number of arrows from x_{i} to x_{j} is dim_{k} Ext_{Λ}^{1}(S_{i}, S_{j}), namely the dimension of

Hom_{Λ}(P_{j}, Rad(P_{i}))/Hom_{Λ}(P_{j}, Rad^{2}(P_{i})) = e_{j}J(Λ)e_{i}/e_{j}J^{2}(Λ)e_{i}.

The following proposition is due to Gabriel [4, p. 9].

Proposition 1.7 Suppose Λ is a finite dimensional basic algebra over an algebraically closed field k, and let Q = Q(Λ) be its Ext-quiver. Then there is a surjective map of algebras ϕ : kQ → → Λ such that the kernel of ϕ is contained in the ideal of paths of length at least two. In particular, in case kQ is finite dimensional, this is equal to the square of the radical, and ϕ induces a bijection between the simple Λ-modules and the simple kQ-modules, and between the blocks of Λ and the blocks of kQ.

Definition 1.8 A system of linear relations on a quiver Q is a two sided ideal R contained in the ideal of paths of length at least two. This is the same as assigning to each pair of vertices x and y in Q a subspace of the space of paths of length at least two from x to y, called the relations, in such a way that composing a relation on either side with any path gives another relation.

We call the pair (Q, R) a quiver with relations. The path algebra of a quiver with relations is the algebra kQ/R.

The importance of the above definition is that by the proposition, every finite dimensional basic algebra over an algebraically closed field is isomorphic to the path algebra of a quiver with relations. The algebra is hereditary if and only if there are no relations, and so finite dimemsional hereditary algebras are the same as path algebras of quivers with finitely many vertices and arrows, and with no oriented cycles.

Whether or not kQ/R is finite dimensional, as long as Q has only finitely many vertices, we may describe the projective module P_{x} corresponding to a vertex idempotent e_{x} as before. It consists of the space of paths starting at x, modulo the relations between these paths given by restricting the system of relations R.

When we come to calculate the Hochschild and cyclic homology of path algebras, we shall need to understand algebras of the form kQ ⊗ kQ^{op} for a quiver Q. First note that (kQ)^{op} = k(Q^{op}), so that there is no confusion of notation. In general, a tensor product of path algebras can be expressed as a quotient of a path algebra by a system of linear relations as follows. If Q and Q′ are quivers, we form a quiver Q × Q′ whose vertex set is the direct product of the vertex sets of Q and Q′, and with an arrow (x, z) → (y, z) for each arrow x → y in Q and vertex z in Q′, and an arrow (x, z) → (x, t) for each vertex x in Q and arrow z → t in Q′. Let R be the ideal in k(Q × Q′) given by the commutativity relations

(x, z) → (y, z) → (y, t) = (x, z) → (x, t) → (y, t),

one for each arrow x → y in Q and each arrow z → t in Q′. Then

kQ ⊗ kQ′ ≅ k(Q × Q′)/R.

Thus the projective kQ⊗kQ^{op}-module corresponding to a pair (x, y) has a basis consisting of pairs of paths

(w → ⋯ → x, y → ⋯ → z).

2 Hochschild and cyclic homology of path algebras

Let A = kQ be the path algebra of a quiver Q with finitely many vertices. Let E be the finite dimensional commutative semisimple subalgebra generated by the vertex idempotents e_{x}, and let I be the ideal generated by the paths of positive length. Thus as a vector space we have A = E ⊕ I. Denote by C_{∗}(A) the complex of cyclic chains on A (i.e., the total complex of the double complex describe by Loday and Quillen [6, Section 1]). Let C_{∗}(A, E) be the quotient complex

0 → C_{∗}(E) → C_{∗}(A) → C_{∗}(A, E) → 0

so that taking homology we have a long exact sequence

⋯ → HC_{n}(E) → HC_{n}(A) → HC_{n}(A, E) → ⋯

Since there is a splitting

C_{∗}(A) → C_{∗}(E)

coming from the surjective algebra homomorphism A → E, this long exact sequence breaks up as a disconnected set of short exact sequences, and so we have

HC_{∗}(A) = HC_{∗}(E) ⊕ HC_{∗}(A, E).

According to the example after Proposition 1.11 in [6], HC_{n}(k) = k for n ≥ 0 even and HC_{n}(k) = 0 for n odd. Since the Hochschild homology of a direct sum of rings is the direct sum of the Hochschild homologies, the same is true of cyclic homology and so we have

HC_{n}(E) = \{ \begin{matrix}k^{r} & n even \\ 0 & n odd \end{matrix}

where r is the number of vertices in Q. Thus we need only calculate HC_{∗}(A, E). We begin by calculating HH_{∗}(A, E). As above, we have

HH_{∗}(A) = HH_{∗}(E) ⊕ HH_{∗}(A, E)

HH_{∗}(E) = \{ \begin{matrix}k^{r} & n = 0 \\ 0 & n > 0. \end{matrix}

To calculate HH_{∗}(A), we use the following lemma.

Lemma 2.1 There is an exact sequence of A ⊗ A^{op}-modules

0 → X ⊕ Y →^{γ′} A ⊗ A →^{b′} A → 0

where the right hand map is given by multiplication

b′(a_{1} ⊗ a_{2}) = a_{1}a_{2}

and X and Y are projective A ⊗ A^{op}-modules given as follows. X is the A ⊗ A^{op}-module with basis elements a_{1} ⊗ a_{2} where a_{2} and a_{1} are paths which do not compose, and Y is the A ⊗ A^{op}-module with basis elements a_{1} ⊗ a_{2} ⊗ a_{3} where a_{3}, a_{2} and a_{1} are paths which do compose and a_{2} has length one. The map γ′ is given by

γ′(a_{1} ⊗ a_{2}) = a_{1} ⊗ a_{2}
γ′(a_{1} ⊗ a_{2} ⊗ a_{3}) = a_{1}a_{2} ⊗ a_{3} − a_{1} ⊗ a_{2}a_{3}.

Proof The fact that the sequence is exact is easily checked explicitly in terms of paths. The module X takes care of the paths which do not compose, while Y takes care of those which do. We see that X is the sum of the projective modules corresponding to vertices (x, y) of Q × Q^{op} with x ≠ y, while Y is the sum of the projective modules corresponding to vertices (x, y) of Q × Q^{op}, one for each arrow x → y in Q. □

We now use the resolution given in the above lemma to compute

HH_{∗}(A) = Tor_{n}^{A⊗A^{op}}(A, A).

To do this we tensor the complex

⋯ → 0 → X ⊕ Y → A ⊗ A^{op} → 0

with A over A ⊗ A^{op} and taking homology. This tensoring amounts to being allowed to move elements of A from the right end to the left end of a tensor. For example Y ⊗_{A⊗A^{op}} A = Y/ ∼ where

a_{1} ⊗ a_{2} ⊗ a_{3}a ∼ aa_{1} ⊗ a_{2} ⊗ a_{3}.

Thus we obtain the complex

0 → U ⊕ V →^{γ} A → 0

Here, U has a basis consisting of the tensors a ⊗ 1 with a an open path, namely one with distinct end points, so that a^{2} = 0. On the other hand, V has a basis consisting of the tensors a_{1} ⊗ a_{2} ⊗ 1 where a_{1} and a_{2} are paths with a_{1}a_{2} ≠ 0 and a_{2}a_{1} ≠ 0. The map γ is given by

γ(a ⊗ 1) = a
γ(a_{1} ⊗ a_{2} ⊗ 1) = a_{1}a_{2} − a_{2}a_{1}.

Denote by A^{(m)} the linear span in A of the closed paths (i.e., paths starting and ending at the same vertex) of length m, and A^{(−1)} the open paths, so that as a vector space

A = ⨁_{m=−1}^{∞} A^{(m)}.

Let σ denote the action on A^{(m)} (m ≥ 0) by an unsigned permutation rotating paths

σ(a_{1}a_{2} … a_{m}) = (a_{m}a_{1} … a_{m−1}).

Then the above calculation shows that the Hochschild homology of A is given by

HH_{0}(A) = ⨁_{m≥0} A^{(m)}/(1 − σ) = A/[A, A]
HH_{1}(A) = ⨁_{m≥1}(A^{(m)})^{σ}
HH_{n}(A) = 0 (n ≥ 2).

We have HH_{0}(E) = A^{(0)}, and so

HH_{0}(A, E) = ⨁_{m≥1} A^{(m)}/(1 − σ) = I/[A, A]
HH_{1}(A, E) = ⨁_{m≥1}(A^{(m)})^{σ}
HH_{n}(A, E) = 0 (n ≥ 2).

The following lemma parallels [6, Lemma 5.3], but see also Kassel [5, Section 3].

Lemma 2.2 The Connes boundary operator B : HH_{0}(A, E) → HH_{1}(A, E) is given by the norm map

N_{σ} = ∑_{i=0}^{m−1} σ^{i} : A^{(m)}/(1 − σ) → (A^{(m)})^{σ}.

Proof To prove this lemma, we must compare our resolution of A as an A ⊗ A^{op}-module with the Hochschild resolution. Namely, we have a commutative diagram

⋯ →^{b′} A^{4} →^{b′} A^{3}  →^{b′} A^{2}  →^{b′} A → 0
↓  ↓j′  ∥  ∥
⋯ → 0 → X ⊕ Y →^{γ′} A ⊗ A →^{b′} A → 0

where the map j′ is given as follows. If a_{1}, …, a_{m} are paths of length one, then

j′(a ⊗ 1 ⊗ a′) = 0
j′(a ⊗ a_{1} … a_{m} ⊗ a′) = a ⊗ a_{1} … a_{m}a′ ∈ X if aa_{1} = 0,
aa_{1} … a_{m} ⊗ a′ ∈ X if a_{m}a′ = 0,
and ∑_{i=1}^{m} aa_{1} … a_{i−1} ⊗ a_{i} ⊗ a_{i+1} … a_{m}a ∈ Y
if aa_{1} ≠ 0 and a_{m}a′ ≠ 0.

On truncating and tensoring over A ⊗ A^{op} with A, we obtain the following diagram, which is a quasi-isomorphism of complexes.

⋯ →^{b} A^{3} →^{b} A^{2}  →^{b} A
↓  ↓j  ∥
⋯ → 0 → U ⊕ V →^{γ} A

Here, the map j is given by

j(a ⊗ 1) = 0
j(a ⊗ a_{1} … a_{m}) = aa_{1} … a_{n} ⊗ 1 ∈ U if a_{n}a = 0
and ∑_{i=1}^{m} a_{i+1} … a_{m}aa_{1} … a_{i} if a_{n}a ≠ 0.

Thus we have

B(a_{1} … a_{m}) = j_{∗}(1 ⊗ a_{1} … a_{m})
= ∑_{i=1}^{m} a_{i+1} … a_{m}a_{1} … a_{i} ∈ (A^{(m)})^{σ}. □

Proposition 2.3 We have

HC_{n}(A, E) = ⨁_{m≥1} H_{n}(Z/m, A^{(m)})

where Z/m acts on A^{(m)} via σ.

Proof This now follows from the long exact sequence of Connes (see for example [6, Theorem 1.6] □

In case k has characteristic zero, this implies that

HC_{n}(A, E) = \{ \begin{matrix}⨁_{m≥1} A^{(m)}/(1 − σ) = I/[A, A] & if n = 0 \\ 0 & otherwise. \end{matrix}

We conjecture that the a similar answer should hold for hereditary algebras which are not path algebras. An example of such an algebra can be found in Dlab and Ringel [3].

In case k has characteristic p, however, the kernel and cokernel of the norm map consist of the paths which are p^{th} powers:

Lemma 2.4 There is an exact sequence

0 → Φ^{p}A^{(m)}/(1 − σ) → A^{(mp)}/(1 − σ) ⟶^{N_{σ}} (A^{(mp)})^{σ} → Φ^{p}(A^{(m)})^{σ} → 0,

where Φ^{p} denotes the Frobenius map. For m coprime to p the map N_{σ} : A^{(m)}/(1 − σ) → (A^{(m)})^{σ} is an isomorphism.

Proof The elements in the kernel of N_{σ} are those of the form

(a_{1}, …, a_{m}, a_{1}, …, a_{m}, …, a_{1}, …, a_{m})

(p repetitions), while the cokernel is represented by elements which are a sum of the distinct images under σ of such an element. □

Thus we obtain a Frobenius twist in cyclic homology

HC_{n}(A, E) = \{ \begin{matrix}⨁_{m≥1} A^{(m)}/(1 − σ) = I/[A, A] & if n = 0 \\ ⨁_{m≥1} Φ^{p}A^{(m)}/(1 − σ) = Φ^{p}(I/[A, A]) & for n even, n > 0 \\ ⨁_{m≥1} Φ^{p}(A^{(m)})^{σ} & for n odd. \end{matrix}

3 Path algebra resolutions

We now indicate how to build semisimplicial resolutions of algebras by path algebras. We call these path resolutions. For basic information on semisimplicial objects, see for example May [7] and Dold and Puppe [2].

Since every algebra Λ is a quotient of a free algebra, it follows that every algebra is a quotient of a path algebra of a quiver with finitely many vertices. If the ideal being quotiented out is not contained in the ideal of paths of length at least two, then some of the generators are redundent, and so we may assume that it is. Thus we write Λ = A_{0}/R_{0} and A_{0} = kQ_{0} = E ⊕ I_{0} where I_{0} is the ideal of paths of length at least one, and R_{0} ⊆ I_{0}^{2}. Thus Λ = E ⊕ I_{0}/R_{0}. Choosing a set of generators for the ideal R_{0} and adjoining new arrows to Q_{0} corresponding to these, we obtain a new quiver Q_{1} with the same vertices as Q_{0}, but with more arrows, and maps d_{0}, d_{1} : A_{1} = kQ_{1} → A_{0} with Im(d_{0} − d_{1}) = R_{0}, namely d_{0} takes the new paths to the relations and d_{1} takes them to zero, while both maps are the identity on the old paths. There is also an obvious injective map s_{0} : A_{0} → A_{1}, satisfying d_{0}s_{0} = d_{1}s_{0} = id. At the next stage, we build Q_{2} again to have the same set of vertices as Q_{0}, with one copy of each arrow in Q_{0}, two copies of each arrow in Q_{1} (“degenerate 1-simplices”), and some new arrows corresponding to the relations between the arrows in Q_{0} and Q_{1} (i.e., the kernel of d_{0} − d_{1}). The semisimplicial relations between the s operators and the d operators tell us what to do with all the arrows in Q_{2} under the action of d_{0}, d_{1} and d_{2}, except for the new (non-degenerate) ones. We may choose, for example, to send these to generators for the kernel of d_{0} −d_{1} under the action of d_{0}, and to let d_{1} and d_{2} send them to zero. We continue in this way, at each stage the boundary map being determined on the degenerate simplices, and we choose non-degenerate simplices to be sent by d_{0} to what needs to be killed, and by all other d_{i} to zero. What we produce this way is a semisimplicial object in the category of path algebras, which is acyclic except in degree zero, where it augments onto Λ. Such an object is called a semisimplicial path algebra resolution of Λ.

⋯ R_{2} \begin{matrix}→ \\ ← \\ → \\ ← \\ → \end{matrix} I_{1}  \begin{matrix}→ \\ ← \\ → \end{matrix} I_{0} → I_{0}/R_{0}
∩ ∩ ∩ ∩
⋯ A_{2} \begin{matrix}→ \\ ← \\ → \\ ← \\ → \end{matrix} A_{1} \begin{matrix}→ \\ ← \\ → \end{matrix} A_{0} → Λ → 0
∪ ∪ ∪ ∪
⋯ E = E = E = E

Definition 3.1 The path algebra homology of Λ with respect to E is the homology H_{∗}^{path}(Λ, E) of the complex

⋯ → I_{2}/[A_{2}, A_{2}] ⟶^{d_{0}−d_{1}+d_{2}} I_{1}/[A_{1}, A_{1}] ⟶^{d_{0}−d_{1}} I_{0}/[A_{0}, A_{0}] → 0.

Of course, by the remarks we have already made, for a finite dimensional algebra Λ over an algebraically closed field, there is an essentially unique minimal semisimplicial path algebra resolution. This is in contrast to free resolutions, where choices must be made at each stage. But in any case, by the usual methods of comparison, the path algebra homology is independent of the choice of resolution, up to canonical isomorphism.

The following theorem is a modification of a theorem of Quillen.

Theorem 3.2 If Λ is an algebra over a field of characteristic zero, then there is a canonical isomorphism

H_{∗}^{path}(Λ, E) ≅ HC_{∗}(Λ, E).

Proof We form a double complex

⋮  ⋮  ⋮
↓  ↓  ↓
C_{2}(A_{0}, E) ← C_{2}(A_{1}, E) ← C_{2}(A_{2}, E) ← ⋯
↓ ↓ ↓
C_{1}(A_{0}, E) ← C_{1}(A_{1}, E) ← C_{1}(A_{2}, E) ← ⋯
↓ ↓ ↓
C_{0}(A_{0}, E) ← C_{0}(A_{1}, E) ← C_{0}(A_{2}, E) ← ⋯

The vertical maps in this double complex are the usual boundary maps in the cyclic theory, with the signs changed on every other column. The horizontal maps are the alternating sums of the maps induced by the semisimplicial structure. For example, the second row

C_{1}(A_{0}, E) ← C_{1}(A_{1}, E) ← C_{1}(A_{2}, E) ← ⋯

is the sum of the complex in the first row and the complex obtained from the complex

A_{0} ⊗ A_{0} ⟵^{d_{0}⊗d_{0}−d_{1}⊗d_{1}} A_{1} ⊗ A_{1} ⟵^{d_{0}⊗d_{0}−d_{1}⊗d_{1}+d_{2}⊗d_{2}} A_{2} ⊗ A_{2} ← ⋯

(which is exact by the Künneth theorem) by quotienting out the corresponding complex for E.

Using the Künneth theorem as above, we see that all the rows of this double complex are exact, and so the spectral sequence starting with the horizontal differential collapses at E^{2}. The E^{1} term is the complex C_{∗}(Λ, E) down the left hand wall, and zero elsewhere, and the differential d_{1} is the boundary map in cyclic homology, and so the spectral sequence abuts to HC_{∗}(Λ, E).

On the other hand, each HC_{n}(A_{s}, E) = 0 for n ≥ 1 by the calculation performed in the previous section, and so the columns of the double complex are also exact. Thus the spectral sequence starting with the vertical differential also collapses at E^{2}. This time, the E^{1} term has HC_{0}(A_{s}, E) = I_{s}/[A_{s}, A_{s}] along the bottom row, and zero elsewhere, and the differential d^{1} is the map used in the definition of H_{∗}^{path}(Λ, E), and so this spectral sequence abuts to H_{∗}^{path}(Λ, E). □

Corollary 3.3 Suppose Λ is a quotient of a finite dimensional hereditary algebra over a field of characteristic zero by a nilpotent ideal. Then HC_{n}(Λ) = 0 for n > 0.

Proof In the minimal path algebra resolution of Λ, there can never be any cycles in the quivers involved, and so H_{n}^{path}(Λ, E) = 0 for n > 0. □

More generally, the argument used in the above theorem shows that even if we are not in characteristic zero, there is a spectral sequence whose E^{2} term is the cyclic homology of the path algebras involved.

Theorem 3.4 If Λ is an algebra over a field of characteristic p, then there is a spectral sequence converging to HC_{∗}(Λ, E) and with

E_{st}^{1} = \{ \begin{matrix}⨁_{m≥1} A_{s}^{(m)}/(1 − σ) = I_{s}/[A_{s}, A_{s}] & if t = 0 \\ ⨁_{m≥1} Φ^{p}A_{s}^{(m)}/(1 − σ) = Φ^{p}(I_{s}/[A_{s}, A_{s}]) & for t even, t > 0 \\ ⨁_{m≥1} Φ^{p}(A_{s}^{(m)})^{σ} & for t odd. \end{matrix}

It would be interesting to understand the differentials in this spectral sequence.

References

[1] H. Bass. Algebraic K-theory. Benjamin, Inc., New York, 1968.

[2] A. Dold and D. Puppe. Homologie Nicht-Additiver Funktoren. Anwendungen, Ann. Inst. Fourier Grenoble 11 (1961), 201–312.

[3] V. Dlab and C. M. Ringel. The representations of tame hereditary algebras. In Representation Theory of Algebras, Proc. Philadelphia Conf., ed. R. Gordon, Marcel Dekker 1978.

[4] P. Gabriel. Auslander-Reiten sequences and representation-finite algebras. Representation theory I, Ottawa 1979. Springer Lecture Notes in Mathematics 831, Springer-Verlag, Berlin/New Nork 1980.

[5] C. Kassel. L’homologie cyclique des algèbres enveloppantes. Invent. Math. 91 (1988), 221–251.

[6] J.-L. Loday and D. Quillen. Cyclic homology and the Lie algebra homology of matrices. Comment. Math. Helvetici 59 (1984), 565–591.

[7] J. P. May. Simplicial objects in algebraic topology. Univ. of Chicago Press, 1967.