Theory and Applications of Categories, Vol. 3, No. 1, 1997, pp. 1–23.

HIGHER DIMENSIONAL PEIFFER ELEMENTS IN SIMPLICIAL COMMUTATIVE ALGEBRAS

Z.ARVASI AND T.PORTER

Transmitted by Ronald Brown

ABSTRACT. Let E be a simplicial commutative algebra such that E_{n} is generated by degenerate elements. It is shown that in this case the nth term of the Moore complex of E is generated by images of certain pairings from lower dimensions. This is then used to give a description of the boundaries in dimension n − 1 for n = 2, 3, and 4.

Introduction

Simplicial commutative algebras occupy a place somewhere between homological algebra, homotopy theory, algebraic K-theory and algebraic geometry. In each sector they have played a significant part in developments over quite a lengthy period of time. Their own internal structure has however been studied relatively little. The present article is one of a series in which we will study the n-types of simplicial algebras and will apply the results in various, mainly homological, settings. The pleasing, and we believe significant, result of this study is that simplicial algebras lend themselves very easily to detailed general calculations of structural maps and thus to a determination of a remarkably rich amount of internal structure. These calculations can be done by hand in low dimensions, but it seems likely that more general computations should be possible using computer aided calculations.

R.Brown and J-L.Loday [5] noted that if the second dimension G_{2} of a simplicial group G, is generated by the degenerate elements, that is, elements coming from lower dimensions, then the image of the second term NG_{2} of the Moore complex (NG, ∂) of G by the differential, ∂, is [Kerd_{1}, Kerd_{0}] where the square brackets denote the commutator subgroup. An easy argument then shows that this subgroup of NG_{1} is generated by elements of the form (s_{0}d_{0}(y)x(s_{0}d_{0}y^{−1}))(yx^{−1}y^{−1}) and that it is thus exactly the Peiffer subgroup of NG_{1}, the vanishing of which is equivalent to ∂_{1} : NG_{1} → NG_{0} being a crossed module.

It is clear that one should be able to develop an analogous result for other algebraic structures and in the case of commutative algebras, it is not difficult to see, cf. Arvasi [2] and section 3 (below), that if E is a simplicial algebra in which the subalgebra, E_{2}, is generated by the degenerate elements then the corresponding image is the ideal Kerd_{1}Kerd_{0} in NE_{1} and that it is generated by the elements (x − s_{0}d_{0}x)y which give the analogous Peiffer ideal in the theory of crossed modules of algebras, (cf. Porter [14]). The vanishing of these elements is important in the construction of the cotangent complex of Lichtenbaum and Schlessinger, [13], and the simplicial version of the cotangent complex of Quillen [15], André [1] and Illusie [12]. It is natural to hope for higher dimensional analogues of this result and for an analysis and interpretation of the structure of the resulting elements in NE_{n}, n ≥ 2.

Received by the editors 3 October 1996 and, in revised form, 7 January 1997.

Published on 24 January 1997

1991 Mathematics Subject Classification : 18G30, 18G55, 16E99 .

Key words and phrases: Simplicial commutative algebra, boundaries, Moore complex .

\begin{matrix}c \\ ◯ \end{matrix} Z.Arvasi and T.Porter 1997. Permission to copy for private use granted.

We generalise the complete result for commutative algebras to dimensions 2, 3 and 4 and get partial results in higher dimensions. The methods we use are based on ideas of Conduché, [8] and techniques developed by Carrasco and Cegarra [7]. In detail, this gives the following:

Let E be a simplicial commutative algebra with Moore complex NE and for n > 1, let D_{n} be the ideal generated by the degenerate elements in dimension n. If E_{n} = D_{n}, then

∂_{n}(NE_{n}) = ∂_{n}(I_{n}) for all n > 1

where I_{n} is an ideal in E_{n} (generated by a fairly small set of elements which will be explicitly given later on).

If n = 2, 3 or 4, then the ideal of boundaries of the Moore complex of the simplicial algebra E can be shown to be of the form

∂_{n}(NE_{n}) = ∑_{I,J} K_{I}K_{J}

for ∅ ≠ I, J ⊂ [n − 1] = {0, 1, …, n − 1} with I ∪ J = [n − 1], where

K_{I} = ⋂_{i∈I} Kerd_{i} and K_{J} = ⋂_{j∈J} Kerd_{j}.

This gives internal criteria for the vanishing of the higher Peiffer elements which yield conditions for various crossed algebra structures on the Moore complex. In general however for n > 4, we can only prove

∑_{I,J} K_{I}K_{J} ⊆ ∂_{n}(NE_{n})

but suspect the opposite inclusion holds as well.

These results are quite technical, being internal to the theory of simplicial algebras themselves. It is known [3], [14] that simplicial algebras lead to crossed modules and crossed complexes of algebras, that free crossed modules are related to Koszul complex constructions and higher dimensional analogues have been proposed by Ellis [9] for use in homotopical and homological algebra. In a sequel to this paper it will be shown how technical results found here facilitate the study of these aspects of crossed higher dimensional algebra, in particular by examining a suitable way of defining free ‘crossed algebras’ of various types.

Acknowledgement

This work was partially supported by the Royal Society ESEP programme in conjunction with TÜBİTAK, the Scientific and Technical Research Council of Turkey.

1. Definitions and preliminaries

In what follows ‘algebras’ will be commutative algebras over an unspecified commutative ring, k, but for convenience are not required to have a multiplicative identity.

A simplicial (commutative) algebra E consists of a family of algebras {E_{n}} together with face and degeneracy maps d_{i} = d_{i}^{n} : E_{n} → E_{n−1}, 0 ≤ i ≤ n, (n ≠ 0) and s_{i} = s_{i}^{n} : E_{n} → E_{n+1}, 0 ≤ i ≤ n, satisfying the usual simplicial identities given in André [1] or Illusie [12] for example. It can be completely described as a functor E: Δ^{op} →CommAlg_{k} where Δ is the category of finite ordinals [n] = {0 < 1 < ⋯ < n} and increasing maps.

Quillen [15] and Illusie [12] both discuss the basic homotopical algebra of simplicial algebras and their application in deformation theory. André [1] gives a detailed examination of their construction and applies them to cohomology via the cotangent complex construction. Another essential reference from our point of view is Carrasco’s thesis, [6], where many of the basic techniques used here were developed systematically for the first time and the notion of hypercrossed complex was defined.

The following notation and terminology is derived from [6] and the published version, [7], of the analogous group theoretic case.

For the ordered set [n] = {0 < 1 < … < n}, let α_{i}^{n} : [n + 1] → [n] be the increasing surjective map given by

α_{i}^{n}(j) = \{ \begin{matrix}j & if & j ≤ i \\ j − 1 & if & j > i. \end{matrix}

Let S(n, n − r) be the set of all monotone increasing surjective maps from [n] to [n − r]. This can be generated from the various α_{i}^{n} by composition. The composition of these generating maps is subject to the following rule α_{j}α_{i} = α_{i−1}α_{j}, j < i. This implies that every element α ∈ S(n, n − r) has a unique expression as α = α_{i_{1}} ∘ α_{i_{2}} ∘ … ∘ α_{i_{r}} with 0 ≤ i_{1} < i_{2} < … < i_{r} ≤ n − 1, where the indices i_{k} are the elements of [n] such that {i_{1}, …, i_{r}} = {i : α(i) = α(i + 1)}. We thus can identify S(n, n − r) with the set {(i_{r}, …, i_{1}) : 0 ≤ i_{1} < i_{2} < … < i_{r} ≤ n−1}. In particular, the single element of S(n, n), defined by the identity map on [n], corresponds to the empty 0-tuple ( ) denoted by ∅_{n}. Similarly the only element of S(n, 0) is (n − 1, n − 2, …, 0). For all n ≥ 0, let

S(n) = ⋃_{0≤r≤n} S(n, n − r).

We say that α = (i_{r}, …, i_{1}) < β = (j_{s}, …, j_{1}) in S(n)

if i_{1} = j_{1}, …, i_{k} = j_{k} but i_{k+1} > j_{k+1} (k ≥ 0) or
if i_{1} = j_{1}, …, i_{r} = j_{r} and r < s.

This makes S(n) an ordered set. For instance, the orders in S(2) and in S(3) are respectively:

\begin{aligned}S(2) & = & \{∅_{2} < (1) < (0) < (1, 0)\}; \\ S(3) & = & \{∅_{3} < (2) < (1) < (2, 1) < (0) < (2, 0) < (1, 0) < (2, 1, 0)\}. \end{aligned}

We also define α ∩ β as the set of indices which belong to both α and β.

The Moore complex

The Moore complex NE of a simplicial algebra E is defined to be the differential graded module (NE, ∂) with

(NE)_{n} = ⋂_{i=0}^{n−1} Kerd_{i}

and with differential ∂_{n} : NE_{n} → NE_{n−1} induced from d_{n} by restriction.

The Moore complex has the advantage of being smaller than the simplicial algebra itself and being a differential graded module is of a better known form for manipulation. Its homology gives the homotopy groups of the simplicial algebra and thus in specific cases, e.g. a truncated free simplicial resolution of a commutative algebra, gives valuable higher dimensional information on syzygy-like elements.

The Moore complex, NE, carries a hypercrossed complex structure (see Carrasco [6]) which allows the original E to be rebuilt. We recall briefly some of those aspects of this reconstruction that we will need later.

The Semidirect Decomposition of a Simplicial Algebra

The fundamental idea behind this can be found in Conduché [8]. A detailed investigation of it for the case of a simplicial group is given in Carrasco and Cegarra [7]. The algebra case of that structure is also given in Carrasco’s thesis [6].

Given a split extension of algebras

[figure: 0; K; E; d; R; s; 0]

we write E ≅ K ⋊s(R), the semidirect product of the ideal, K, with the image of R under the splitting s.

1.1. Proposition. If E is a simplicial algebra, then for any n ≥ 0

\begin{aligned}E_{n} & ≅ & (… (NE_{n} ⋊ s_{n−1}NE_{n−1}) ⋊ … ⋊ s_{n−2} … s_{0}NE_{1})⋊ \\ & & (… (s_{n−2}NE_{n−1} ⋊ s_{n−1}s_{n−2}NE_{n−2}) ⋊ … ⋊ s_{n−1}s_{n−2} … s_{0}NE_{0}). \end{aligned}

Proof. This is by repeated use of the following lemma.

1.2. Lemma. Let E be a simplicial algebra. Then E_{n} can be decomposed as a semidirect product:

E_{n} ≅ Kerd_{n}^{n} ⋊ s_{n−1}^{n−1}(E_{n−1}).

Proof. The isomorphism can be defined as follows:

\begin{aligned}θ : & E_{n} & ⟶ & Kerd_{n}^{n} ⋊ s_{n−1}^{n−1}(E_{n−1}) \\ & e & ⟼ & (e − s_{n−1}d_{n}e, s_{n−1}d_{n}e). \end{aligned}

The bracketting and the order of terms in this multiple semidirect product are generated by the sequence:

E_{1} ≅ NE_{1} ⋊ s_{0}NE_{0}
E_{2} ≅ (NE_{2} ⋊ s_{1}NE_{1}) ⋊ (s_{0}NE_{1} ⋊ s_{1}s_{0}NE_{0})
E_{3} ≅ ((NE_{3} ⋊ s_{2}NE_{2}) ⋊ (s_{1}NE_{2} ⋊ s_{2}s_{1}NE_{1}))⋊
((s_{0}NE_{2} ⋊ s_{2}s_{0}NE_{1}) ⋊ (s_{1}s_{0}NE_{1} ⋊ s_{2}s_{1}s_{0}NE_{0})).

and

E_{4} ≅ (((NE_{4} ⋊ s_{3}NE_{3}) ⋊ (s_{2}NE_{3} ⋊ s_{3}s_{2}NE_{2}))⋊
((s_{1}NE_{3} ⋊ s_{3}s_{1}NE_{2}) ⋊ (s_{2}s_{1}NE_{2} ⋊ s_{3}s_{2}s_{1}NE_{1})))⋊
s_{0}(decomposition of E_{3}).

Note that the term corresponding to α = (i_{r}, …, i_{1}) ∈ S(n) is

s_{α}(NE_{n−#α}) = s_{i_{r}...i_{1}}(NE_{n−#α}) = s_{i_{r}}...s_{i_{1}}(NE_{n−#α}),

where #α = r. Hence any element x ∈ E_{n} can be written in the form

x = y + ∑_{α∈S(n)} s_{α}(x_{α}) with y ∈ NE_{n} and x_{α} ∈ NE_{n−#α}.

Crossed Modules of Commutative Algebras

Recall from [14] the notion of a crossed module of commutative algebras. Let k be a fixed commutative ring and let R be a k-algebra with identity. A crossed module of commutative algebras, (C, R, ∂), is an R-algebra C, together with an action of R on C and an R-algebra morphism

∂ : C ⟶ R,

such that for all c, c′ ∈ C, r ∈ R,

CM1) ∂(r · c) = r∂c CM2) ∂c · c′ = cc′.

The second condition (CM2) is called the Peiffer identity.

A standard example of a crossed module is any ideal I in R giving an inclusion map I → R, which is a crossed module. Conversely, given any crossed module ∂ : C → R, the image I = ∂C of C is an ideal in R.

2. Hypercrossed Complex Pairings and Boundaries in the Moore Complex

The following lemma is noted by Carrasco [6].

2.1. Lemma. For a simplicial algebra E, if 0 ≤ r ≤ n let \overline{NE}_{n}^{(r)} = ⋂_{i≠r} Kerd_{i} then the

mapping

φ : NE_{n} ⟶ \overline{NE}_{n}^{(r)}

in E_{n}, given by

φ(e) = e − ∑_{k=0}^{n−r−1}(−1)^{k+1}s_{r+k}d_{n}e,

is a bijection.

This easily implies:

2.2. Lemma. Given a simplicial algebra E, then we have the following

d_{n}(NE_{n}) = d_{r}(\overline{NE}_{n}^{(r)}).

2.3. Proposition. Let E be a simplicial algebra, then for n ≥ 2 and nonempty I, J ⊆ [n − 1] with I ∪ J = [n − 1]

(⋂_{i∈I} Kerd_{i})( ⋂_{j∈J} Kerd_{j}) ⊆ ∂_{n}NE_{n}.

Proof. For any J ⊂ [n − 1], J ≠ ∅, let r be the smallest element of J. If r = 0, then replace J by I and restart and if 0 ∈ I ∩ J, then redefine r to be the smallest nonzero element of J. Otherwise continue.

Letting e_{0} ∈ ⋂_{j∈J} Kerd_{j} and e_{1} ∈ ⋂_{i∈I} Kerd_{i}, one obtains

d_{i}(s_{r−1}e_{0}s_{r}e_{1}) = 0 for i ≠ r

and hence s_{r−1}e_{0}s_{r}e_{1} ∈ \overline{NE}_{n}^{(r)}. It follows that

e_{0}e_{1} = d_{r}(s_{r−1}e_{0}s_{r}e_{1}) ∈ d_{r}(\overline{NE}_{n}^{(r)}) = d_{n}NE_{n} by the previous lemma,

and this implies

(⋂_{i∈I} Kerd_{i})( ⋂_{j∈J} Kerd_{j}) ⊆ ∂_{n}NE_{n}.

Writing the abbreviations,

K_{I} = ⋂_{i∈I} Kerd_{i} and K_{J} = ⋂_{j∈J} Kerd_{j}

then 2.3 implies:

2.4. Theorem. For any simplicial algebra E and for n ≥ 2

∑_{I,J} K_{I}K_{J} ⊆ ∂_{n}NE_{n}

for ∅ ≠I, J ⊂ [n − 1] and I ∪ J = [n − 1].

Truncated Simplicial Algebras and n-type Simplicial Algebras.

By a n-truncated simplicial algebra of order n or n-type simplicial algebra, we mean a simplicial algebra E′ obtained by killing dimensions of order > n in the Moore complex NE of some simplicial algebra, E.

2.5. Corollary. Let E be a simplicial algebra and let E′ be the corresponding n-type simplicial algebra, so we have a canonical morphism E ⟶ E′. Then E′ satisfies the following property:

For all nonempty sets of indices ( I ≠ J) I, J ⊂ [n − 1] with I ∪ J = [n − 1],

( ⋂_{j∈J} Kerd_{j}^{n−1})(⋂_{i∈I} Kerd_{i}^{n−1}) = 0.

Proof. Since ∂_{n}NE′_{n} = 0, this follows from proposition 2.3.

Hypercrossed complex pairings

We recall from Carrasco [6] the construction of a family of k-linear morphisms. We define a set P (n) consisting of pairs of elements (α, β) from S(n) with α ∩ β = ∅, where α = (i_{r}, …, i_{1}), β = (j_{s}, ..., j_{1}) ∈ S(n). The k-linear morphisms that we will need,

{C_{α,β} : NE_{n−#α} ⊗ NE_{n−#β} ⟶ NE_{n} : (α, β) ∈ P (n), n ≥ 0}

are given as composites by the diagrams

[figure: NE_{n−#α} ⊗ NE_{n−#β}; s_{α}⊗s_{β}; C_{α,β}; NE_{n}; E_{n} ⊗ E_{n}; μ; E_{n}; p]

where ⊗ is the tensor product of k-modules,

s_{α} = s_{i_{r}} … s_{i_{1}} : NE_{n−#α} ⟶ E_{n} , s_{β} = s_{j_{s}} … s_{j_{1}} : NE_{n−#β} ⟶ E_{n},

p : E_{n} → NE_{n} is defined by composite projections p = p_{n−1} … p_{0} with

p_{j} = 1 − s_{j}d_{j} with j = 0, 1, … n − 1

and where μ : E_{n} ⊗ E_{n} → E_{n} denotes multiplication. Thus

C_{α,β}(x_{α} ⊗ y_{β}) = pμ(s_{α} ⊗ s_{β})(x_{α} ⊗ y_{β})
= p(s_{α}(x_{α})s_{β}(y_{β}))
= (1 − s_{n−1}d_{n−1}) … (1 − s_{0}d_{0})(s_{α}(x_{α})s_{β}(y_{β})).

We now define the ideal I_{n} to be that generated by all elements of the form

C_{α,β}(x_{α} ⊗ y_{β})

where x_{α} ∈ NE_{n−#α} and y_{β} ∈ NE_{n−#β} and for all (α, β) ∈ P (n).

Example. For n = 2, suppose α = (1), β = (0) and x, y ∈ NE_{1} = Kerd_{0}. It follows that

C_{(1)(0)}(x ⊗ y) = p_{1}p_{0}(s_{1}xs_{0}y)
= s_{1}xs_{0}y − s_{1}xs_{1}y
= s_{1}x(s_{0}y − s_{1}y)

and these give the generator elements of the ideal I_{2}.

For n = 3, the linear morphisms are the following

\begin{matrix}C_{(1,0)(2)}, & C_{(2,0)(1)}, & C_{(2,1)(0)}, \\ C_{(2)(0)}, & C_{(2)(1)}, & C_{(1)(0)}. \end{matrix}

For all x ∈ NE_{1}, y ∈ NE_{2}, the corresponding generators of I_{3} are:

\begin{aligned}C_{(1,0)(2)}(x ⊗ y) & = & (s_{1}s_{0}x − s_{2}s_{0}x)s_{2}y, \\ C_{(2,0)(1)}(x ⊗ y) & = & (s_{2}s_{0}x − s_{2}s_{1}x)(s_{1}y − s_{2}y), \\ C_{(2,1)(0)}(x ⊗ y) & = & s_{2}s_{1}x(s_{0}y − s_{1}y + s_{2}y); \end{aligned}

whilst for all x, y ∈ NE_{2},

\begin{aligned}C_{(1)(0)}(x ⊗ y) & = & s_{1}x(s_{0}y − s_{1}y) + s_{2}(xy), \\ C_{(2)(0)}(x ⊗ y) & = & (s_{2}x)(s_{0}y), \\ C_{(2)(1)}(x ⊗ y) & = & s_{2}x(s_{1}y − s_{2}y). \end{aligned}

In the following we analyse various types of elements in I_{n} and show that sums of them give elements that we want in giving an alternative description of ∂_{n}NE_{n} in certain cases.

2.6. Proposition. Let E be a simplicial algebra and n > 0, and D_{n} the ideal in E_{n} generated by degenerate elements. We suppose E_{n} = D_{n}, and let I_{n} be the ideal generated by elements of the form

C_{α,β}(x_{α} ⊗ y_{β}) with (α, β) ∈ P (n)

where x_{α} ∈ NE_{n−#α}, y_{β} ∈ NE_{n−#β}. Then

∂_{n}(NE_{n}) = ∂_{n}(I_{n}).

We defer the proof until we have some technical lemmas out of the way 2.7. Lemma. Given x_{α} ∈ NE_{n−#α}, y_{β} ∈ NE_{n−#β} with α = (i_{r}, …, i_{1}), β = (j_{s}, …, j_{1}) ∈ S(n). If α ∩ β = ∅ with α < β and u = s_{α}(x_{α})s_{β}(y_{β}), then

(i) if k ≤ j_{1}, then p_{k}(u) = u,

(ii) if k > j_{s} + 1 or k > i_{r} + 1, then p_{k}(u) = u,

(iii) if k ∈ {i_{1}, …, i_{r}, i_{r} + 1} and k = j_{l} + 1 for some l, then

p_{k}(u) = s_{α}(x_{α})s_{β}(y_{β}) − s_{k}(z_{k}),

for some z_{k} ∈ E_{n−1}, (iv) if k ∈ {j_{1}, …, j_{s}, j_{s} + 1} and k = i_{m} + 1 for some m, then

p_{k}(u) = s_{α}(x_{α})s_{β}(y_{β}) − s_{k}(z_{k}),

where z_{k} ∈ E_{n−1} and 0 ≤ k ≤ n − 1.

Proof. Assuming α < β and α ∩ β = ∅ which implies j_{1} < i_{1}. In the range 0 ≤ k ≤ j_{1},

p_{k}(u) = s_{α}(x_{α})s_{β}(y_{β}) − (s_{k}d_{k}s_{α}x_{α})(s_{k}d_{k}s_{β}y_{β})
= s_{α}(x_{α})s_{β}(y_{β}) − (s_{k}s_{i_{r}−1} … s_{i_{1}−1}d_{k}x_{α})(s_{k}d_{k}s_{β}y_{β})
= s_{α}(x_{α})s_{β}(y_{β}) since d_{k}(x_{α}) = 0.

Similarly if k > j_{s} + 1, or if k > i_{r} + 1.

If k ∈ {i_{1}, …, i_{r}, i_{r} + 1} and k = j_{l} + 1 for some l, then

\begin{aligned}p_{k}(u) & = & s_{α}(x_{α})s_{β}(y_{β}) − s_{k}[d_{k}(s_{α}(x_{α})s_{β}(y_{β}))] \\ & = & s_{α}(x_{α})s_{β}(y_{β}) − s_{k}(z_{k}) \end{aligned}

where z_{k} = s_{α′}(x_{α′})s_{β′}(y_{β′}) ∈ E_{n−1} for new strings α′, β′ as is clear. The proof of (iv) is essentially the same so we will leave it out.

2.8. Lemma. If α ∩ β = ∅ and α < β, then

p_{n−1} … p_{0}(s_{α}(x_{α})s_{β}(y_{β})) = s_{α}(x_{α})s_{β}(y_{β}) − ∑_{k=1}^{n−1} s_{k}(z_{k})

where z_{k} ∈ E_{n−1}.

Proof. We prove this by using induction on n. Write u = s_{α}(x_{α})s_{β}(y_{β}). For n = 1, it is clear to see that the equality is verified. We suppose that it is true for n − 2. It then follows that

\begin{aligned}p_{n−1} … p_{0}(u) & = & p_{n−1}(u − ∑_{k=1}^{n−2} s_{k}(z_{k})) \\ & = & p_{n−1}(u) − p_{n−1}(∑_{k=1}^{n−2} s_{k}(z_{k})) \end{aligned}

as p_{n−1} is a linear map. Next look at p_{n−1}(u) = u − s_{n−1}(\underbrace{d_{n−1}u}_{z′} ) = u − s_{n−1}(z′) and

\begin{aligned}p_{n−1}(∑_{k=1}^{n−2} s_{k}(z_{k})) & = & ∑_{k=1}^{n−2} s_{k}(z_{k}) − s_{n−1}(\underbrace{∑_{k=1}^{n−2} d_{n−1}s_{k}(z_{k})}_{z′′} ) \\ & = & ∑_{k=1}^{n−2} s_{k}(z_{k}) − s_{n−1}(z′′). \end{aligned}

Thus

p_{n−1} … p_{0}(u) = u − ∑_{k=1}^{n−2} s_{k}(z_{k}) + s_{n−1}(\underbrace{z′′ − z′}_{z_{n−1}} )
= u − ∑_{k=1}^{n−2} s_{k}(z_{k}) + s_{n−1}(z_{n−1})
= u − ∑_{k=1}^{n−1} s_{k}(z_{k}).

as required. Note that: For x, y ∈ NE_{n−1}, it is easy to see that

p_{n−1} … p_{0}(s_{n−1}(x)s_{n−2}(y)) = s_{n−1}(x)(s_{n−2}y − s_{n−1}y)

and taking the image of this element by d_{n} gives

d_{n}[s_{n−1}(x)(s_{n−2}y − s_{n−1}y)] = x(s_{n−2}d_{n−1}y − y)

which gives a Peiffer type element of dimension n. 2.9. Lemma. Let x_{α} ∈ NE_{n−#α}, y_{β} ∈ NE_{n−#β} with α, β ∈ S(n), then

s_{α}(x_{α})s_{β}(y_{β}) = s_{α∩β}(z_{α∩β})

where z_{α∩β} has the form (s_{α′}x_{α})(s_{β′}y_{β}) and α′ ∩ β′ = ∅. Proof. If α ∩ β = ∅, then this is trivially true. Assume #(α ∩ β) = t, with t ∈ IN. Take α = (i_{r}, …, i_{1}) and β = (j_{s}, …, j_{1}) with α ∩ β = (k_{t}, …, k_{1}),

s_{α}(x_{α}) = s_{i_{r}} … s_{k_{t}} … s_{i_{1}}(x_{α}) and s_{β}(y_{β}) = s_{j_{s}} … s_{k_{t}} … s_{j_{1}}(y_{β}).

Using repeatedly the simplicial axiom s_{e}s_{d} = s_{d}s_{e−1} for d < e until obtaining that s_{k_{t}} … s_{k_{1}} is at the beginning of the string, one gets the following

s_{α}(x_{α}) = s_{k_{t}...k_{1}}(s_{α′}x_{α}) and s_{β}(y_{β}) = s_{k_{t}...k_{1}}(s_{β′}y_{β}).

Multiplying these expressions together gives

s_{α}(x_{α})s_{β}(y_{β}) = s_{k_{t}} … s_{k_{1}}(s_{α′}x_{α})s_{k_{t}} … s_{k_{1}}(s_{β′}y_{β})
= s_{k_{t}...k_{1}}((s_{α′}x_{α})(s_{β′}y_{β}))
= s_{α∩β}(z_{α∩β}),

where z_{α∩β} = (s_{α′}x_{α})(s_{β′}y_{β}) ∈ E_{n−#(α∩β)} and where α\(α ∩ β) = α′, β\(α ∩ β) = β′. Hence α′ ∩ β′ = ∅. Moreover α′ < α and β′ < β as #α′ < #α and #β′ < #β.

Proof. (of Proposition 2.6) From proposition 1.3, E_{n} is isomorphic to

NE_{n} ⋊ s_{n−1}NE_{n−1} ⋊ s_{n−2}NE_{n−1} ⋊ … ⋊ s_{n−1}s_{n−2} … s_{0}NE_{0},

where NE_{n} = ⋂_{i=0}^{n−1}Kerd_{i} and NE_{0} = E_{0}. Hence any element x in E_{n} can be written in the

following form

x = e_{n} + s_{n−1}(x_{n−1}) + s_{n−2}(x′_{n−1}) + s_{n−1}s_{n−2}(x_{n−2}) + … + s_{n−1}s_{n−2} … s_{0}(x_{0}),

with e_{n} ∈ NE_{n}, x_{n−1}, x′_{n−1} ∈ NE_{n−1}, x_{n−2} ∈ NE_{n−2}, x_{0} ∈ NE_{0}, etc.

We start by comparing I_{n} with NE_{n}. We show NE_{n} = I_{n}. It is enough to prove that, equivalently, any element in E_{n}/I_{n} can be written

s_{n−1}(x_{n−1}) + s_{n−2}(x′_{n−1}) + s_{n−1}s_{n−2}(x_{n−2}) + … + s_{n−1}s_{n−2} … s_{0}(x_{0}) + I_{n}

which implies, for any b ∈ E_{n},

b + I_{n} = s_{n−1}(x_{n−1}) + s_{n−2}(x′_{n−1}) + … + s_{n−1}s_{n−2} … s_{0}(x_{0}) + I_{n}.

for some x_{n−1} ∈ NE_{n−1} etc.

If b ∈ E_{n}, it is a sum of products of degeneracies so first of all assume it to be a product of degeneracies and that will suffice for the general case.

If b is itself a degenerate element, it is obvious that it is in some semidirect factor s_{α}(E_{n−#α}). Assume therefore that provided an element b can be written as a product of k − 1 degeneracies it has the desired form mod I_{n}, now for an element b which needs k degenerate elements

b = s_{β}(y_{β})b′ with y_{β} ∈ NE_{n−#β}

where b′ needs fewer than k and so

b + I_{n} = s_{β}(y_{β})(b′ + I_{n})
= s_{β}(y_{β})(s_{n−1}(x_{n−1}) + s_{n−2}(x′_{n−1}) + … + s_{n−1}s_{n−2} … s_{0}(x_{0}) + I_{n})
= ∑_{α∈S(n)} s_{β}(y_{β})s_{α}(x_{α}) + I_{n}.

Next we ignore this summation and just look at the product

s_{α}(x_{α})s_{β}(y_{β}) (∗).

We check this product case by case as follows:

If α ∩ β = ∅, then by lemmas 2.7 and 2.8, there exists an element s_{α}(x_{α})s_{β}(y_{β}) −

∑_{k=1}^{n−1} s_{k}(z_{k}) in I_{n} with z_{k} ∈ E_{n−1} and k ∈ α so that

s_{α}(x_{α})s_{β}(y_{β}) ≡ ∑_{k=1}^{n−1} s_{k}(z_{k}) mod I_{n}.

If α ∩ β ≠ ∅, then one gets, from lemma 2.9, the following

s_{α}(x_{α})s_{β}(y_{β}) = s_{α∩β}(z_{α∩β})

where z_{α∩β} = (s_{α′}x_{α})(s_{β′}y_{β}) ∈ E_{n−t}, with t ∈ IN. Since α′ ∩ β′ = ∅, we can use lemma 2.8 to form an equality

s_{α′}(x_{α})s_{β′}(y_{β}) ≡ ∑_{k′=0}^{n−1} s_{k′}(z_{k′}) mod I_{n}

where z_{k′} ∈ E_{n−1}. It then follows that

\begin{aligned}s_{α∩β}(z_{α∩β}) & = & s_{α∩β}((s_{α′}x_{α})(s_{β′}y_{β})) \\ & ≡ & ∑_{k′=0}^{n−1} s_{α∩β}s_{k′}(z_{k′}) mod I_{n}. \end{aligned}

Thus we have shown that every product which can be formed in the required form is in I_{n}. Therefore ∂_{n}(I_{n}) = ∂_{n}(NE_{n}).

3. Products of Kernels Elements and Boundaries in the Moore Complex

By way of illustration of potential applications of the above proposition we look at the case of n = 2. Case n = 2

We know that any element e_{2} of E_{2} can be expressed in the form

e_{2} = b + s_{1}y + s_{0}x + s_{0}u

with b ∈ NE_{2}, x, y ∈ NE_{1} and u ∈ s_{0}E_{0}. We suppose D_{2} = E_{2}. For n = 1, we take α = (1), β = (0) and x, y ∈ NE_{1} =Kerd_{0}. The ideal I_{2} is generated by elements of the form

C_{(1)(0)}(x ⊗ y) = s_{1}x(s_{0}y − s_{1}y).

The image of I_{2} by ∂_{2} is known to be Kerd_{0}Kerd_{1} by direct calculation. Indeed,

\begin{aligned}d_{2}[C_{(1)(0)}(x ⊗ y)] & = & d_{2}[s_{1}x(s_{0}y − s_{1}y)] \\ & = & x(s_{0}d_{1}y − y) \end{aligned}

where x ∈ Kerd_{0} and (s_{0}d_{1}y − y)x ∈ Kerd_{1} and all elements of Kerd_{1} have this form due to lemma 2.1.

The bottom, ∂ : NE_{1} → NE_{0}, of the Moore complex of E is always a precrossed module, that is it satisfies CM1 where r ∈ NE_{0} operates on c ∈ NE_{1} via s_{0}, r · c = s_{0}(r)c. The elements ∂y · x − yx are called the Peiffer elements.

As ∂ is the restriction of ∂_{1} to NE1, these are precisely the d_{2}(C_{(1)(0)}(x ⊗ y)). In other words the ideal ∂I_{2} is the ‘Peiffer ideal’ of the precrossed module d_{1} : NE_{1} → NE_{0}, whose vanishing is equivalent to this being a crossed module. The description of ∂I_{2} as Kerd_{0}Kerd_{1} gives that its vanishing in this situation is module-like behaviour since a module, M, is an algebra with MM = 0. Thus if (NE, ∂) yields a crossed module this fact will be reflected in the internal structure of E by the vanishing of Kerd_{0}Kerd_{1}. Because the image of this C_{(1)(0)}(x ⊗ y) is the Peiffer element determined by x and y, we will call the C_{α,β}(x ⊗ y) in general higher dimensional Peiffer elements and will seek similar internal conditions for their vanishing.

We have seen that in all dimensions

∑_{I,J} K_{I}K_{J} ⊆ ∂_{n}(NE_{n}) = ∂I_{n}

and we will show shortly that this inclusion is an equality, not only in dimension 2 (as above), but in dimensions 3 and 4. The arguments are calculatory and do not generalise in an obvious way to higher dimensions although similar arguments can be used to get partial results there.

4. Case n = 3

The analogue of the ‘Kerd_{0}Kerd_{1}’ result here is:

4.1. Proposition.

∂_{3}(NE_{3}) = ∑_{I,J} K_{I}K_{J} + K_{{0,1}}K_{{0,2}} + K_{{0,2}}K_{{1,2}} + K_{{0,1}}K_{{1,2}}

where I ∪ J = [2], I ∩ J = ∅ and

K_{{0,1}}K_{{0,2}} = (Kerd_{0} ∩ Kerd_{1})(Kerd_{0} ∩ Kerd_{2})
K_{{0,2}}K_{{1,2}} = (Kerd_{0} ∩ Kerd_{2})(Kerd_{1} ∩ Kerd_{2})
K_{{0,1}}K_{{1,2}} = (Kerd_{0} ∩ Kerd_{1})(Kerd_{1} ∩ Kerd_{2})

Proof. By proposition 2.8, we know the generator elements of the ideal I_{3} and ∂_{3}(I_{3}) = ∂_{3}(NE_{3}). The image of all the listed generator elements of the ideal I_{3} is summarised in the following table.

α  β  I, J
1 (1,0) (2)  {2} {0,1}
2 (2,0) (1)  {1} {0,2}
3 (2,1) (0)  {0} {1,2}
4 (2) (1)  {0,1} {0,2}
5 (2) (0)  {0,1} {1,2}+{0,1} {0,2}
6 (1) (0) {0,2} {1,2}+{0,1} {1,2}+{0,1} {0,2}

The explanation of this table is the following:

∂_{3}C_{α,β}(x ⊗ y) is in K_{I}K_{J} in the simple cases corresponding to the first 4 rows. In row 5, ∂_{3}C_{(2)(0)}(x ⊗ y) ∈ K_{{0,1}}K_{{1,2}} + K_{{0,1}}K_{{0,2}} and similarly in row 6, the higher Peiffer element is in the sum of the indicated K_{I}K_{J}.

To illustrate the sort of argument used we look at the case of α = (1, 0) and β = (2), i.e. row 1. For x ∈ NE_{1} and y ∈ NE_{2},

\begin{aligned}d_{3}[C_{(1,0)(2)}(x ⊗ y)] & = & d_{3}[(s_{1}s_{0}x − s_{2}s_{0}x)s_{2}y] \\ & = & (s_{1}s_{0}d_{1}x − s_{0}x)y \end{aligned}

and so

d_{3}[C_{(1,0)(2)}(x ⊗ y)] = (s_{1}s_{0}d_{1}x − s_{0}x)y ∈ Kerd_{2}(Kerd_{0} ∩ Kerd_{1}).

We have denoted Kerd_{2}(Kerd_{0} ∩ Kerd_{1}) by K_{{2}}K_{{0,1}} where I = {2} and J = {0, 1}. Rows 2, 3 and 4 are similar. For Row 5, α = (2), β = (0) with x, y ∈ NE_{2} = Kerd_{0} ∩ Kerd_{1},

\begin{aligned}d_{3}[C_{(2)(0)}(x ⊗ y)] & = & d_{3}[s_{2}xs_{0}y] \\ & = & xs_{0}d_{2}y. \end{aligned}

We can assume, for x, y ∈ NE_{2},

x ∈ Kerd_{0} ∩ Kerd_{1} and y + s_{0}d_{2}y − s_{1}d_{2}y ∈ Kerd_{1} ∩ Kerd_{2}

and, multiplying them together,

x(y + s_{0}d_{2}y − s_{1}d_{2}y) = xy + xs_{0}d_{2}y − xs_{1}d_{2}y
= x(y − s_{1}d_{2}y) + xs_{0}d_{2}y
= d_{3}[C_{(2)(1)}(x ⊗ y)] + d_{3}[C_{(2)(0)}(x ⊗ y)]

and so

\begin{aligned}d_{3}[C_{(2)(0)}(x ⊗ y)] & ∈ & K_{\{0,1\}}K_{\{1,2\}} + d_{3}[C_{(2)(1)}(x ⊗ y)] \\ & ⊆ & K_{\{0,1\}}K_{\{1,2\}} + K_{\{0,1\}}K_{\{0,2\}}. \end{aligned}

For Row 6, for α = (1), β = (0) and x, y ∈ NE_{2} = Kerd_{0} ∩ Kerd_{1},

\begin{aligned}d_{3}[C_{(1)(0)}(x ⊗ y)] & = & d_{3}[s_{1}xs_{0}y − s_{1}xs_{1}y + s_{2}xs_{2}y] \\ & = & s_{1}d_{2}xs_{0}d_{2}y − s_{1}d_{2}xs_{1}d_{2}y + xy \end{aligned}

We can take the following elements

(s_{0}d_{2}y − s_{1}d_{2}y + y) ∈ Kerd_{1} ∩ Kerd_{2} and (s_{1}d_{2}x − x) ∈ Kerd_{0} ∩ Kerd_{2}.

When we multiply them together, we get

(s_{0}d_{2}y − s_{1}d_{2}y + y)(s_{1}d_{2}x − x) = [s_{0}d_{2}ys_{1}d_{2}x − s_{1}d_{2}ys_{1}d_{2}x + yx]
−[xs_{0}d_{2}y] + [x(s_{1}d_{2}y − y)]
+[y(s_{1}d_{2}x − x)]
= d_{3}[C_{(1)(0)}(x ⊗ y)] − d_{3}[C_{(2)(0)}(x ⊗ y)]+
d_{3}[C_{(2)(1)}(x ⊗ y) + C_{(2)(1)}(y ⊗ x)]

and hence

d_{3}[C_{(1)(0)}(x ⊗ y)] ∈ K_{{0,2}}K_{{1,2}} + K_{{0,1}}K_{{1,2}} + K_{{0,1}}K_{{0,2}}.

So we have shown

∂_{3}I_{3} ⊆ ∑_{I,J} K_{I}K_{J} + K_{{0,1}}K_{{0,2}} + K_{{0,2}}K_{{1,2}} + K_{{0,1}}K_{{1,2}}.

The opposite inclusion can be verified by using proposition 2.3. Therefore

∂_{3}(NE_{3}) = Kerd_{2}(Kerd_{0} ∩ Kerd_{1}) + Kerd_{1}(Kerd_{0} ∩ Kerd_{2})+
Kerd_{0}(Kerd_{1} ∩ Kerd_{2}) + (Kerd_{0} ∩ Kerd_{1})(Kerd_{0} ∩ Kerd_{2})+
(Kerd_{1} ∩ Kerd_{2})(Kerd_{0} ∩ Kerd_{2}) + (Kerd_{1} ∩ Kerd_{2})(Kerd_{0} ∩ Kerd_{1})

This completes the proof of the proposition.

5. Illustrative Application: 2-Crossed Modules of Algebras

5.1. Definition. (cf. [10]) A 2-crossed module of k-algebras consists of a complex of C_{0} -algebras

[figure: C_{2}; ∂_{2}; C_{1}; ∂_{1}; C_{0}]

and ∂_{2}, ∂_{1} morphisms of C_{0}-algebras, where the algebra C_{0} acts on itself by multiplication such that

[figure: C_{2}; ∂_{2}; C_{1}]

is a crossed module in which C_{1} acts on C_{2} via C_{0}, (we require thus that for all x ∈ C_{2}, y ∈ C_{1} and z ∈ C_{0} that (xy)z = x(yz)), further, there is a C_{0}-bilinear function

{ ⊗ } : C_{1} ⊗_{C_{0}} C_{1} ⟶ C_{2},

called a Peiffer lifting, which satisfies the following axioms:

P L1 :  ∂_{2}{y_{0} ⊗ y_{1}} = y_{0}y_{1} − y_{0} · ∂_{1}(y_{1}),
P L2 :  {∂_{2}(x_{1}) ⊗ ∂_{2}(x_{2})} = x_{1}x_{2},
P L3  {y_{0} ⊗ y_{1}y_{2}} = {y_{0}y_{1} ⊗ y_{2}} + ∂_{1}y_{2} · {y_{0} ⊗ y_{1}},
P L4 : a)  {∂_{2}(x) ⊗ y} = y · x − ∂_{1}(y) · x,
b)  {y ⊗ ∂_{2}(x)} = y · x,
P L5 :  {y_{0} ⊗ y_{1}} · z = {y_{0} · z ⊗ y_{1}} = {y_{0} ⊗ y_{1} · z},

for all x, x_{1}, x_{2} ∈ C_{2}, y, y_{0}, y_{1}, y_{2} ∈ C_{1} and z ∈ C_{0}.

We denote such a 2-crossed module of algebras by {C_{2}, C_{1}, C_{0}, ∂_{2}, ∂_{1}}.

5.2. Proposition. Let E be a simplicial algebra with the Moore complex NE. Then the complex of algebras

[figure: NE_{2}/∂_{3}(NE_{3} ∩ D_{3}); ^{\overline{∂}^{2}}; NE_{1}; ∂_{1}; NE_{0}]

is a 2-crossed module of algebras, where the Peiffer map is defined as follows:

\begin{aligned}\{ ⊗ \} : & NE_{1} ⊗ NE_{1} & ⟶ & NE_{2}/∂_{3}(NE_{3} ∩ D_{3}) \\ & (y_{0} ⊗ y_{1}) & ⟼ & \overline{s_{1}y_{0}(s_{1}y_{1} − s_{0}y_{1})}. \end{aligned}

Here the right hand side denotes a coset in NE_{2}/∂_{3}(NE_{3} ∩ D_{3}) represented by the corresponding element in NE_{2}.

Proof. We will show that all axioms of a 2-crossed module are verified. It is readily checked that the morphism \overline{∂}_{2} : NE_{2}/∂_{3}(NE_{3} ∩ D_{3}) → NE_{1} is a crossed module. (In the following calculations we display the elements omitting the overlines.)

PL1:

\begin{aligned}\overline{∂}_{2}\{y_{0} ⊗ y_{1}\} & = & ∂_{2}(s_{1}y_{0}(s_{1}y_{1} − s_{0}y_{1})) \\ & = & y_{0}y_{1} − y_{0} · ∂_{1}y_{1}. \end{aligned}

PL2: From ∂_{3}(C_{(1)(0)}(x_{1} ⊗ x_{2})) = s_{1}d_{2}(x_{1})s_{0}d_{2}(x_{2}) − s_{1}d_{2}(x_{1})s_{1}d_{2}(x_{2}) + x_{1}x_{2}, one obtains

\begin{aligned}\{\overline{∂}_{2}(x_{1}) ⊗ \overline{∂}_{2}(x_{2})\} & = & s_{1}d_{2}x_{1}(s_{1}d_{2}x_{2} − s_{0}d_{2}x_{2}) \\ & ≡ & x_{1}x_{2} mod ∂_{3}(NE_{3} ∩ D_{3}). \end{aligned}

PL3:

{y_{0} ⊗ y_{1}y_{2}} = s_{1}y_{0}[s_{1}(y_{1}y_{2}) − s_{0}(y_{1}y_{2})]
= s_{1}y_{0}[s_{1}y_{1}(s_{1}y_{2} − s_{0}y_{2})] + [s_{1}y_{0}(s_{1}y_{1} − s_{0}y_{1})]s_{0}y_{2}
= s_{1}(y_{0}y_{1})(s_{1}y_{2} − s_{0}y_{2}) + {y_{0} ⊗ y_{1}}s_{0}y_{2}

but ∂_{3}(C_{(1,0)(2)}(y ⊗ x)) = (s_{1}s_{0}d_{1}y − s_{0}y)x, so this implies

{y_{0} ⊗ y_{1}y_{2}} ≡ s_{1}(y_{0}y_{1})(s_{1}y_{2} − s_{0}y_{2}) + s_{1}s_{0}d_{1}(y_{2}){y_{0} ⊗ y_{1}} mod ∂_{3}(NE_{3} ∩ D_{3})
= {y_{0}y_{1}⊗y_{2}} + ∂_{1}y_{2} · {y_{0} ⊗ y_{1}} by the definition of the action.

PL4: a)

\{\overline{∂}_{2}(x) ⊗ y\} = s_{1}∂_{2}x(s_{1}y − s_{0}y),

but

∂_{3}(C_{(2,0)(1)}(y ⊗ x)) = (s_{0}y − s_{1}y)s_{1}d_{2}x − (s_{0}y − s_{1}y)x ∈ ∂_{3}(NE_{3} ∩ D_{3})

and

∂_{3}(C_{(1,0)(2)}(y ⊗ x)) = (s_{1}s_{0}d_{1}y − s_{0}y)x ∈ ∂_{3}(NE_{3} ∩ D_{3}),

so then

\{\overline{∂}_{2}(x) ⊗ y\} ≡ s_{1}(y)x − s_{0}(y)x  mod ∂_{3}(NE_{3} ∩ D_{3})
= y · x − ∂_{1}(y) · x  by the definition of the action,

b) since ∂_{3}(C_{(2,1)(0)}(y ⊗ x)) = s_{1}y(s_{0}d_{2}x − s_{1}d_{2}x) + s_{1}(y)x,

\{y ⊗ \overline{∂}_{2}(x)\} = s_{1}y(s_{1}∂_{2}x − s_{0}∂_{2}x)
≡ s_{1}(y)x mod ∂_{3}(NE_{3} ∩ D_{3})
= y · x by the definition of the action.

PL5:

{y_{0} ⊗ y_{1}} · z = (s_{1}y_{0}(s_{1}y_{1} − s_{0}y_{1})) · z
= s_{1}(s_{0}(z)y_{0})(s_{1}y_{1} − s_{0}y_{1})
= s_{1}(y_{0} · z)(s_{1}y_{1} − s_{0}y_{1}) by the definition of the action
= {y_{0} · z ⊗ y_{1}}.

Clearly the same sort of argument works for

{y_{0} · z ⊗ y_{1}} = {y_{0} ⊗ y_{1} · z}

with x, x_{1}, x_{2} ∈ NE_{2}/∂_{3}(NE_{3} ∩ D_{3}), y, y_{0}, y_{1}, y_{2} ∈ NE_{1} and z ∈ NE_{0}. This completes the proof of the proposition.

This only used the higher dimensional Peiffer elements. A result in terms of K_{I}K_{J} vanishing can also be given:

5.3. Proposition. If in a simplicial algebra E, K_{I}K_{J} = 0 in the following cases: I ∪J = [2], I ∩ J = ∅; I = {0, 1} J = {0, 2} or I = {1, 2}; and I = {0, 2}, J = {1, 2}, then

NE_{2} ⟶ NE_{1} ⟶ NE_{0}

can be given the structure of a 2-crossed module.

6. The case n = 4

With dimension 4, the situation is more complicated, but is still manageable.

6.1. Proposition.

∂_{4}(NE_{4}) = ∑_{I,J} K_{I}K_{J}

where I ∪ J = [3], I = [3] − {α}, J = [3] − {β} and (α, β) ∈ P (4).

Proof. There is a natural isomorphism

E_{4} ≅ NE_{4} ⋊ s_{3}NE_{3} ⋊ s_{2}NE_{3} ⋊ s_{3}s_{2}NE_{2} ⋊ s_{1}NE_{3}⋊

s_{3}s_{1}NE_{2} ⋊ s_{2}s_{1}NE_{2} ⋊ s_{3}s_{2}s_{1}NE_{1} ⋊ s_{0}NE_{3}⋊ s_{3}s_{0}NE_{2} ⋊ s_{2}s_{0}NE_{2} ⋊ s_{3}s_{2}s_{0}NE_{1}⋊ s_{1}s_{0}NE_{2} ⋊ s_{3}s_{1}s_{0}NE_{1} ⋊ s_{3}s_{2}s_{1}s_{0}NE_{0}.

We firstly see what the generator elements of the ideal I_{4} look like. One gets

\begin{aligned}S(4) & = & \{∅_{4} < (3) < (2) < (3, 2) < (1) < (3, 1) < (2, 1) < (3, 2, 1) < (0) < \\ & & (3, 0) < (2, 0) < (3, 2, 0) < (1, 0) < (3, 1, 0) < (3, 2, 1, 0)\}. \end{aligned}

The bilinear morphisms are the following:

C_{(3,2,1)(0)} C_{(3,2,0)(1)} C_{(3,1,0)(2)} C_{(2,1,0)(3)}
C_{(3,2)(1,0)} C_{(3,1)(2,0)} C_{(3,0)(2,1)} C_{(3,2)(1)}
C_{(3,2)(0)} C_{(3,1)(2)} C_{(3,1)(0)} C_{(3,0)(2)}
C_{(3,0)(1)} C_{(2,1)(3)} C_{(2,1)(0)} C_{(2,0)(3)}
C_{(2,0)(1)} C_{(1,0)(3)} C_{(1,0)(2)} C_{(3)(2)}
C_{(3)(1)} C_{(3)(0)} C_{(2)(1)} C_{(2)(0)}

C_{(1)(0)}.

For x_{1}, y_{1} ∈ NE_{1}, x_{2}, y_{2} ∈ NE_{2} and x_{3}, y_{3} ∈ NE_{3}, the generator elements of the ideal I_{4} are

1) C_{(3,2,1)(0)}(x_{1} ⊗ y_{3}) = s_{3}s_{2}s_{1}x_{1}(s_{0}y_{3} − s_{1}y_{3} + s_{2}y_{3} − s_{3}y_{3})
2) C_{(3,2,0)(1)}(x_{1} ⊗ y_{3}) = (s_{3}s_{2}s_{0}x_{1} − s_{1}s_{2}s_{1}x_{1})(s_{1}y_{3} − s_{2}y_{3} + s_{3}y_{3})
3) C_{(3,1,0)(2)}(x_{1} ⊗ y_{3}) = (s_{3}s_{1}s_{0}x_{1} − s_{2}s_{2}s_{0}x_{1})(s_{2}y_{3} − s_{3}y_{3})
4) C_{(2,1,0)(3)}(x_{1} ⊗ y_{3}) = (s_{2}s_{1}s_{0}x_{1} − s_{3}s_{1}s_{0}x_{1})s_{3}y_{3}
5) C_{(3,2)(1,0)}(x_{2} ⊗ y_{2}) = (s_{1}s_{0}x_{2} − s_{2}s_{0}x_{2} + s_{3}s_{0}x_{2})s_{3}s_{2}y_{2}
6) C_{(3,1)(2,0)}(x_{2} ⊗ y_{2}) = (s_{3}s_{1}x_{2} − s_{3}s_{0}x_{2} + s_{2}s_{0}x_{2} − s_{1}s_{1}x_{2})
(s_{3}s_{1}y_{2} − s_{3}s_{2}y_{2})
7) C_{(3,0)(2,1)}(x_{2} ⊗ y_{2}) = (s_{2}s_{1}x_{2} − s_{3}s_{1}x_{2})(s_{3}s_{0}y_{2} − s_{1}s_{2}y_{2} + s_{2}s_{2}y_{2})
8)  C_{(3,2)(1)}(x_{2} ⊗ y_{3}) = s_{3}s_{2}x_{2}(s_{1}y_{3} − s_{2}y_{3} + s_{3}y_{3})
9)  C_{(3,2)(0)}(x_{2} ⊗ y_{3}) = s_{3}s_{2}x_{2}s_{0}y_{3}
10)  C_{(3,1)(2)}(x_{2} ⊗ y_{3}) = (s_{2}y_{3} − s_{3}y_{3})(s_{3}s_{1}x_{2} − s_{2}s_{2}x_{2})
11)  C_{(3,1)(0)}(x_{2} ⊗ y_{3}) = s_{3}s_{1}x_{2}(s_{0}y_{3} − s_{1}y_{3}) + s_{3}s_{2}x_{2}(s_{2}y_{3} − s_{3}y_{3})
12)  C_{(3,0)(2)}(x_{2} ⊗ y_{3}) = s_{3}s_{0}x_{2}(s_{2}y_{3} − s_{3}y_{3})
13)  C_{(3,0)(1)}(x_{2} ⊗ y_{3}) = s_{1}y_{3}(s_{3}s_{0}x_{2} − s_{1}s_{2}x_{2}) + s_{2}s_{2}x_{2}(s_{2}y_{3} − s_{3}y_{3})
14)  C_{(2,1)(3)}(x_{2} ⊗ y_{3}) = (s_{2}s_{1}x_{2} − s_{3}s_{1}x_{2})s_{3}y_{3}
15)  C_{(2,1)(0)}(x_{2} ⊗ y_{3}) = s_{2}s_{1}x_{2}(s_{0}y_{3} − s_{1}y_{3} + s_{2}y_{3}) + s_{3}s_{1}x_{2}s_{3}y_{3}
16)  C_{(2,0)(3)}(x_{2} ⊗ y_{3}) = (s_{2}s_{0}x_{2} − s_{3}s_{0}x_{2})s_{3}y_{3}
17)  C_{(2,0)(1)}(x_{2} ⊗ y_{3}) = (s_{2}s_{0}x_{2} − s_{1}s_{1}x_{2})(s_{1}y_{3} − s_{2}y_{3})+
(s_{3}s_{1}x_{2} − s_{3}s_{0}x_{2})s_{3}y_{3}
18)  C_{(1,0)(3)}(x_{2} ⊗ y_{3}) = s_{1}s_{0}x_{2}s_{3}y_{3}
19)  C_{(1,0)(2)}(x_{2} ⊗ y_{3}) = (s_{1}s_{0}x_{2} − s_{2}s_{0}x_{2})s_{2}y_{3} + s_{3}s_{0}x_{2}s_{3}y_{3}
20)  C_{(3)(2)}(x_{3} ⊗ y_{3}) = s_{3}x_{3}(s_{2}y_{3} − s_{3}y_{3})
21)  C_{(3)(1)}(x_{3} ⊗ y_{3}) = s_{3}x_{3}s_{1}y_{3}
22)  C_{(3)(0)}(x_{3} ⊗ y_{3}) = s_{3}x_{3}s_{0}y_{3}
23)  C_{(2)(1)}(x_{3} ⊗ y_{3}) = s_{2}x_{3}(s_{1}y_{3} − s_{2}y_{3}) + s_{3}(x_{3}y_{3})
24)  C_{(2)(0)}(x_{3} ⊗ y_{3}) = s_{2}x_{3}s_{0}y_{3}
25)  C_{(1)(0)}(x_{3} ⊗ y_{3}) = s_{1}x_{3}(s_{0}y_{3} − s_{1}y_{3}) + s_{2}(x_{3}y_{3}) − s_{3}(x_{3}y_{3})

By proposition 2.8, we have ∂_{4}(NE_{4}) = ∂_{4}(I_{4}). We take an image by ∂_{4} of each C_{α,β}, where α, β ∈ P (4). We summarise the images of all generator elements, which are listed early on, in the subsequent table.

α  β I, J
1 (3,2,1) (0) {0}{1,2,3}
2 (3,2,0) (1) {1}{0,2,3}
3 (3,1,0) (2) {2}{0,1,3}
4 (2,1,0) (3) {3}{0,1,2}
5 (3,2) (1,0) {0,1}{2,3}
6 (3,1) (2,0) {0,2}{1,3}
7 (3,0) (2,1) {1,2}{0,3}
8 (3,2) (1) {0,1}{0,2,3}
9 (3,2) (0) {0,1}{1,2,3}+{0,1}{0,2,3}
10 (3,1) (2) {0,2}{0,1,3}
11 (3,1) (0) {0,2}{1,2,3}+{0,2}{0,1,3}+{0,1}{1,2,3}+{0,1}{0,2,3}
12 (3,0) (2) {1,2}{0,1,3}+{0,2}{0,1,3}
13 (3,0) (1) {1,2}{0,2,3}+ {0,1}{0,2,3}+{1,2}{0,1,3}+{0,2}{0,1,3}
14 (2,1) (3) {0,3}{0,1,2}
15 (2,1) (0) {0,3}{1,2,3}+{0,3}{0,1,2}+{0,2}{1,2,3}+{0,2}{0,1,3}
16 (2,0) (3) {1,3}{0,1,2}+{0,3}{0,1,2}
17 (2,0) (1) {1,3}{0,2,3}+{0,3}{0,1,2}+{1,3}{0,1,2}+{1,2}{0,2,3}+
{0,2}{0,1,3}+{1,2}{0,1,3}
18 (1,0) (3) {2,3}{0,1,2}+{1,3}{0,1,2}
19 (1,0) (2) {2,3}{0,1,3}+{1,2}{0,1,3}+{1,3}{0,1,2}+{2,3}{0,1,2}
20 (3)  (2) {0,1,2}{0,1,3}
21 (3)  (1) {0,1,2}{0,2,3}+{0,1,2}{0,1,3}
22 (3)  (0) {0,1,2}{1,2,3}+{0,1,2}{0,2,3}+{0,1,2}{0,1,3}
23 (2)  (1) {0,1,3}{0,2,3}+{0,1,2}{1,2,3}+{0,1,2}{0,2,3}+
{0,1,2}{0,1,3}
24 (2)  (0) {0,1,3}{1,2,3}+{0,1,3}{0,2,3}+{0,1,2}{1,2,3}+
{0,1,2}{0,2,3}+{0,1,2}{0,1,3}
25 (1)  (0) {0,2,3}{1,2,3}+{0,1,3}{1,2,3}+{0,1,3}{0,2,3}+
{0,1,2}{1,2,3}+{0,1,2}{0,2,3}+{0,1,2}{0,1,3}

As the proofs are largely similar to those for n = 3 we leave most to the reader, limiting ourselves to one or two of the more complex cases by way of illustration.

Row: 17

\begin{aligned}d_{4}[C_{(2,0)(1)}(x_{2} ⊗ y_{3})] & = & (s_{2}s_{0}d_{2}x_{2} − s_{1}s_{1}d_{2}x_{2})(s_{1}d_{3}y_{3} − s_{2}d_{3}y_{3}) \\ & & +y_{3}(s_{1}x_{2} − s_{0}x_{2}). \end{aligned}

Take elements

a = (s_{2}s_{0}d_{2}x_{2} − s_{0}x_{2} + s_{1}x_{2} − s_{2}s_{1}d_{2}x_{2}) ∈ K_{{1,3}} and b = (s_{1}d_{3}y_{3} − s_{2}d_{3}y_{3} + y_{3}) ∈ K_{{0,2,3}},

d_{4}[C_{(2,0)(1)}(x_{2} ⊗ y_{3})] = ab − d_{4}[C_{(2,1)(3)}(x_{2} ⊗ y_{3})]+
d_{4}[C_{(2,0)(3)}(x_{2} ⊗ y_{3})] + d_{4}[C_{(3,0)(1)}(x_{2} ⊗ y_{3})]−
d_{4}[C_{(3,1)(2)}(x_{2} ⊗ y_{3})] + d_{4}[C_{(3,0)(2)}(x_{2} ⊗ y_{3})]
∈ K_{{1,3}}K_{{0,2,3}} + K_{{0,3}}K_{{0,1,2}}+
K_{{1,3}}K_{{0,1,2}} + K_{{1,2}}K_{{0,2,3}}+
K_{{0,2}}K_{{0,1,3}} + K_{{1,2}}K_{{0,1,3}}

by other results from earlier rows.

Row: 20

d_{4}[C_{(3)(2)}(x_{3} ⊗ y_{3})] = x_{3}(s_{2}d_{3}y_{3} − y_{3})
∈ (Kerd_{0} ∩ Kerd_{1} ∩ Kerd_{2})(Kerd_{0} ∩ Kerd_{1} ∩ Kerd_{3})
= K_{{0,1,2}}K_{{0,1,3}}.

Row: 21

d_{4}[C_{(3)(1)}(x_{3} ⊗ y_{3})] = x_{3}s_{1}d_{3}(y_{3}).

Take elements x_{3} ∈ NE_{3} = K_{{0,1,2}} and (s_{1}d_{3}y_{3}−s_{2}d_{3}y_{3}+y_{3}) ∈ K_{{0,2,3}}. When multiplying them together, one gets

\begin{aligned}d_{4}[C_{(3)(1)}(x_{3} ⊗ y_{3})] & ∈ & d_{4}[C_{(3)(2)}(x_{3} ⊗ y_{3})] + K_{\{0,1,2\}}K_{\{0,2,3\}} \\ & ⊆ & K_{\{0,1,2\}}K_{\{0,2,3\}} + K_{\{0,1,2\}}K_{\{0,1,3\}}. \end{aligned}

Row: 23

d_{4}[C_{(2)(1)}(x_{3} ⊗ y_{3})] = s_{2}d_{3}x_{3}(s_{1}d_{3}y_{3} − s_{2}d_{3}y_{3}) + x_{3}y_{3}.

Take elements a = (s_{1}d_{3}y_{3} − s_{2}d_{3}y_{3} + y_{3}) ∈ K_{{0,2,3}} and b = (s_{2}d_{3}x_{3} − x_{3}) ∈ K_{{0,1,3}}. Putting them together, we obtain

\begin{aligned}d_{4}[C_{(2)(1)}(x_{3} ⊗ y_{3})] & = & ab + d_{4}[C_{(3)(1)}(x_{3} ⊗ y_{3})] − d_{4}[C_{(3)(2)}(x_{3} ⊗ y_{3}) + C_{(3)(2)}(y_{3} ⊗ x_{3})] \\ & ∈ & K_{\{0,1,3\}}K_{\{0,2,3\}} + K_{\{0,1,2\}}K_{\{0,2,3\}} + K_{\{0,1,2\}}K_{\{0,1,3\}}. \end{aligned}

Finally

Row: 25

d_{4}[C_{(1)(0)}(x_{3} ⊗ y_{3})] = s_{1}d_{3}(x_{3})(s_{0}d_{3}y_{3} − s_{1}d_{3}y_{3}) + s_{2}d_{3}(x_{3}y_{3}) − x_{3}y_{3},

and

a = (s_{1}d_{3}x_{3} − s_{2}d_{3}x_{3} + x_{3}) ∈ K_{{0,2,3}} and b = (s_{2}d_{3}y_{3} − s_{1}d_{3}y_{3} + s_{0}d_{3}y_{3} − y_{3}) ∈ K_{{1,2,3}},

then one has

d_{4}[C_{(1)(0)}(x_{3} ⊗ y_{3})] = d_{4}[C_{(3)(1)}(y_{3} ⊗ x_{3}) + C_{(3)(1)}(x_{3} ⊗ y_{3})]−
d_{4}[C_{(3)(0)}(x_{3} ⊗ y_{3})] + d_{4}[C_{(2)(0)}(x_{3} ⊗ y_{3})]−
d_{4}[C_{(2)(1)}(x_{3} ⊗ y_{3}) + C_{(2)(1)}(y_{3} ⊗ x_{3})]−
d_{4}[C_{(3)(2)}(x_{3} ⊗ y_{3}) + C_{(3)(2)}(y_{3} ⊗ x_{3})] + ab
∈ K_{{0,1,2}}K_{{0,1,3}} + … + K_{{0,2,3}}K_{{1,2,3}}.

So we have shown that for each d_{4}C_{α,β}(x⊗y) ∈ ∑_{I,J} K_{I}K_{J} and hence ∂_{4}(I_{4}) ⊆ ∑_{I,J} K_{I}K_{J}. The opposite inclusion to this is again given by proposition 2.3.

To summarise we have:

6.2. Theorem. Let n = 2, 3, or 4 and let E be a simplicial algebra with Moore complex NE in which E_{n} = D_{n}, Then

∂_{n}(NE_{n}) = ∑_{I,J} K_{I}K_{J}

for any I, J ⊆ [n − 1] with I ∪ J = [n − 1], I = [n − 1] − {α} and J = [n − 1] − {β}, where (α, β) ∈ P (n).

In more generality we can observe that only elements of NE_{n} ∩ D_{n} were used.

6.3. Theorem. If for any simplicial algebra E with Moore complex NE,

∂_{n}(NE_{n} ∩ D_{n}) = ∑_{I,J} K_{I}K_{J} with n = 2, 3, 4.

6.4. Remark. In general for n > 4, we have only managed to prove

∑_{I,J} K_{I}K_{J} ⊆ ∂_{n}(NE_{n}).

To prove the opposite inclusion, we have a general argument for I ∩ J = ∅ and I ∪ J = [n − 1], but for I ∩ J ≠ ∅, we as yet do not see the pattern. One should be able to extend this result by means of computer algebra software such as AXIOM or MAPLE, and this may help reveal what structure is lying behind the observed behaviour in low dimensions, but the overall pattern is still mysterious.

References

[1] M.André. Homologie des algèbres commutatives. Die Grundlehren der Mathematischen Wissenchaften, 206 Springer-Verlag (1970).

[2] Z.Arvasi. Applications in commutative algebra of the Moore complex of a simplicial algebra. Ph.D. Thesis, University of Wales, (1994).

[3] Z.Arvasi and T.Porter. Simplicial and crossed resolutions of commutative algebras. J. Algebra, 181, (1996) 426-448.

[4] H.J.Baues. Combinatorial homotopy and 4-dimensional complexes. Walter de Gruyter, (1991).

[5] R.Brown and J-L.Loday. Van Kampen theorems for diagram of spaces. Topology, 26 (1987) 311-335.

[6] P.Carrasco. Complejos hipercruzados, cohomologia y extensiones. Ph.D. Thesis, Univ. de Granada, (1987).

[7] P.Carrasco and A.M.Cegarra. Group-theoretic algebraic models for homotopy types. Journal Pure Appl. Algebra, 75 (1991) 195-235.

[8] D.Conduché. Modules croisés généralisés de longueur 2. Journal Pure Appl. Algebra, 34 (1984) 155-178.

[9] G.J.Ellis. Higher dimensional crossed modules of algebras. Journal Pure Appl. Algebra, 52 (1988) 277-282.

[10] A.R.Grandjeán and M.J.Vale. 2-Modulos cruzados en la cohomologia de André-Quillen. Memorias de la Real Academia de Ciencias, 22 (1986) 1-28.

[11] D.Guin-Walery and J.L.Loday. Obstructions à l’excision en K-théorie algèbrique. Springer Lecture Notes in Math. 854 (1981) 179-216.

[12] L.Illusie. Complex cotangent et deformations I, II. Springer Lecture Notes in Math., 239 (1971) II 283 (1972).

[13] S. Lichtenbaum and M. Schlessinger, The cotangent complex of a morphism, Trans. Amer. Math. Soc, 128 (1967), 41-70.

[14] T. Porter, Homology of commutative algebras and an invariant of Simis and Vasconcelos, J. Algebra, 99 (1986) 458-465.

[15] D. Quillen, On the homology of commutative rings, Proc. Sympos. Pure Math, 17 (1970) 65-87.

[16] J.H.C. Whitehead. Combinatorial homotopy II. Bull. Amer. Math. Soc., 55 (1949) 213-245.

Department of Mathematics, Faculty of Science, Osmangazi University, Eskişehir, Turkey

and

School of Mathematics University of Wales, Bangor Gwynedd LL57 1UT, UK.

Email: zarvasi@vm.baum.anadolu.edu.tr and t.porter@bangor.ac.uk

This article may be accessed via WWW at http://www.tac.mta.ca/tac/ or by anony-

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