Z. Arvasi and T. Porter. April 21, Abstract. In this paper we give a construction of free 2-crossed modules.

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1 Freeness Conditions for 2-Crossed Modules of Commutative Algebras Z. Arvasi and T. Porter April 21, 1998 Abstract In this paper we give a construction of free 2-crossed modules. By the use of a `step-by-step' method based on the work of Andre, we will give a description of crossed algebraic models for the steps in the construction of a free simplicial resolution of an algebra. algebras. This involves the introduction of the notion of a free 2-crossed module of Keywords: Free 2-crossed modules, free simplicial algebras. A M S Classication: 18D35 18G30 18G50 18G55. Introduction Andre [?] uses simplicial methods to investigate homological properties of commutative algebras. Other techniques that can give related results include those using the Koszul complex. Any simplicial algebra yields a crossed module derived from the Moore complex [?] and any nitely generated free crossed module C! R of commutative algebras was shown in [?] to have C = R n =d( V2 R n ); i.e. the 2 nd Koszul complex term modulo the 2-boundaries. Higher dimensional analogues of crossed modules of commutative algebras have been dened: 2-crossed modules by Grandjean and Vale [?] and crossed n-cubes of algebras by Ellis [?]. It would not be sensible to expect a strong link between free 2-crossed modules or free crossed squares and Koszul-like constructions since the former record quadratic information which is less evidently there in the Koszul complex. Nevertheless it seems to be useful to try to dene what freeness of such `gadgets' should mean - for instance, to ask `free on what?' - and the aim of this note is to give a plausible solution to that problem. Our solution goes via free simplicial algebras as used by Andre. In [?] we gave a functor from simplicial algebras to 2-crossed modules, and that can be used here. For the alternative models, crossed squares, freeness will be analysed in a separate paper. 1

2 2 Z. Arvasi and T. Porter Acknowledgement This work was partially supported by the Royal Society ESEP programme in conjunction with T UB _ ITAK, the Scientic and Technical Research Council of Turkey. 1 Preliminaries All algebras will be commutative and will be over the same xed but unspecied ground ring. Truncated simplicial algebras Denoting the usual category of nite ordinals by, we obtain for each k 0 a subcategory k determined by the objects [j] of with j k. A simplicial algebra is a functor from the opposite category op to Alg; a k-truncated simplicial algebra is a functor from ( op ) to Alg. We denote the category of k-truncated simplicial algebras by Tr k ksimpalg: We recall from [?] some facts about the skeleton functor. There is a truncation functor tr k from the category of simplicial algebras, denoted by SimpAlg, to that of k-truncated simplicial algebras, given by restriction. This admits a right adjoint cosk k : Tr k SimpAlg?! SimpAlg called the k-coskeleton functor, and a left adjoint sk k : Tr k SimpAlg?! SimpAlg; called the k-skeleton functor. These functors can be given neat explicit descriptions, cf. Duskin, [?]. By an ideal chain complex of algebras, (X; d) we mean one in which each Imd i+1 is an ideal of X i. Given any ideal chain complex (X; d) of algebras and an integer n the truncation, t n] X; of X at level n will be dened by (t n] X) i = 8 >< >: X i if i < n X i =Imd n+1 if i = n 0 if i > n: The dierential d i of t n] X is that of X for i < n; d n is induced from the n th dierential of X and all others are zero. Recall that given a simplicial algebra E, the Moore complex of E is the chain complex dened by (NE) n = n?1 \ i=0 Kerd n i n : NE n! NE n?1 induced from d n n by restriction.

3 Free 2-Crossed Modules 3 The n th homotopy module n (E) of E is the n th homology of the Moore complex of E, i.e., n (E) = Hn T T = n Kerd n i =dn+1( n n+1 Kerd n+1 ): i i=0 We say that the Moore complex NE of a simplicial algebra is of length k if NE n = 0 for all n k + 1, so that a Moore complex of length k is also of length r for r k: The following lemma is due to the second author in the case of simplicial groups. We give an obvious analogue for the commutative algebra version, Lemma 1.1 Let tr k (E) be a k-truncated simplicial algebra, and cosk k (tr k (E)), the algebratheoretic k-coskeleton of tr k (E) (i.e. calculated within Alg). Then there is a natural epimorphism from N(cosk k (tr k (E))) to t k] NE with acyclic kernel. Thus cosk k (tr k (E)) and t k] (NE) have the same weak homotopy type as the natural epimorphism induces isomorphisms on all homotopy modules. i=0 Proof: Following Conduche [?], the Moore complex of cosk k (tr k (E)) is given by: N(cosk k (tr k (E))) l = 0 if l > k + 1 N(cosk k (tr k (E))) k+1 = Ker(@ k : NE k?! NE k?1 ) N(cosk k (tr k (E))) l = NE l if l k: The natural epimorphism gives on Moore complexes N(cosk k (tr k (E))) : 0 k+1 - NE k - NE k?1 -????? t k] (NE) : NE k =@NE k+1 - NE k?1 - and it is immediate that the kernel is acyclic as required. 2 The following notation and terminology is derived from [?] and the published version, [?], of the analogous group theoretic case. For the ordered set [n] = f0 < 1 < : : : < ng, let n : [n + 1]! [n] be the increasing i surjective map given by ( j if j i n i (j) = j? 1 if j > i: 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 n by composition. The composition of these i generating maps is subject to the following rule: j i = i?1 j ; j < i: This implies

4 4 Z. Arvasi and T. Porter that every element 2 S(n; n? r) has a unique expression as = : : : i 1 i2 i r with 0 i 1 < i 2 < : : : < i r n? 1, where the indices i k are the elements of [n] such that fi 1 ; : : :; i r g = fi : (i) = (i + 1)g: We thus can identify S(n; n? r) with the set f(i r ; : : :; i 1 ) : 0 i 1 < i 2 < : : : < i r n? 1g: In particular, the single element of S(n; n); dened 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) = S(n; n? r): 0rn We say that = (i r ; : : :; i 1 ) < = (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: S(2) = f; 2 < (1) < (0) < (1; 0)g; S(3) = f; 3 < (2) < (1) < (2; 1) < (0) < (2; 0) < (1; 0) < (2; 1; 0)g: We also dene \ as the intersection of and, that is the element of S(n) determined by the set of indices which belong to both and. The Semidirect Decomposition of a Simplicial Algebra The fundamental idea behind this can be found in Conduche [?]. A detailed investigation of it for the case of a simplicial group is given in Carrasco and Cegarra [?]. The algebra case of that structure is also given in Carrasco's thesis [?]. Proposition 1.2 If E is a simplicial algebra, then for any n 0 E n = (: : :(NEn o s n?1 NE n?1 ) o : : : o s n?2 : : :s 0 NE 1 )o (: : :(s n?2 NE n?1 o s n?1 s n?2 NE n?2 ) o : : : o s n?1 s n?2 : : : s 0 NE 0 ): Proof: This is by repeated use of the following lemma. 2 Lemma 1.3 Let E be a simplicial algebra. Then E n can be decomposed as a semidirect product: E n = Kerd n n o s n?1 n?1 (E n?1): Proof: The isomorphism can be dened as follows: : E n?! Kerd n n o s n?1 n?1 (E n?1) e 7?! (e? s n?1 d n e; s n?1 d n e):

5 Free 2-Crossed Modules 5 2 The bracketting and the order of terms in this multiple semidirect product are generated by the sequence: and E 1 = NE1 o s 0 NE 0 E 2 = (NE2 o s 1 NE 1 ) o (s 0 NE 1 o s 1 s 0 NE 0 ) E 3 = ((NE3 o s 2 NE 2 ) o (s 1 NE 2 o s 2 s 1 NE 1 ))o ((s 0 NE 2 o s 2 s 0 NE 1 ) o (s 1 s 0 NE 1 o s 2 s 1 s 0 NE 0 )): E 4 = (((NE4 o s 3 NE 3 ) o (s 2 NE 3 o s 3 s 2 NE 2 ))o ((s 1 NE 3 o s 3 s 1 NE 2 ) o (s 2 s 1 NE 2 o s 3 s 2 s 1 NE 1 )))o s 0 (decomposition of E 3 ); and so correspond to an ordered indexing by the elements of S(n) where the term corresponding to = (i r ; : : :; i 1 ) 2 S(n) is s (NE n?# ) = s ir :::i1 (NE n?#) = s ir :::s i 1 (NE n?#); where # = r: Hence any element x 2 E n can be written in the form x = y + X 2S(n)nf; ng s (x ) with y 2 NE n and x 2 NE n?# : Hypercrossed complex pairings We recall from Carrasco [?] the construction of a family of k-linear morphisms. We dene a set P (n) consisting of pairs of elements (; ) from S(n) with \ = ;; where = (i r ; : : :; i 1 ); = (j s ; :::; j 1 ) 2 S(n): The k-linear morphisms that we will need, fc ; : NE n?# NE n?#?! NE n : (; ) 2 P (n); n 0g are given as composites where C ; (x y ) = p(s s )(x y ) = p(s (x )s (y )) = (1? s n?1 d n?1 ) : : :(1? s 0 d 0 )(s (x )s (y )): s = s ir : : :s i 1 : NE n?#?! E n ; s = s js : : : s j 1 : NE n?#?! E n ; p : E n! NE n is dened by composite projections, p = p n?1 : : : p 0, with p j = 1? s j d j for j = 0; 1; : : :n? 1

6 6 Z. Arvasi and T. Porter and where : E n E n! E n denotes multiplication. We now dene the ideal I n in E n to be that generated by all elements of the form C ; (x y ) where x 2 NE n?# and y 2 NE n?# and for all (; ) 2 P (n). The image of I n by the dierential of the Moore complex will be denote P n and will be called the n th -order Peier ideal. We will be primarily concerned with the case n = Crossed Modules of Algebras 2.1 From simplicial algebras to 2-crossed modules Crossed modules of groups were initially dened by Whitehead as models for (homotopy) 2-types. Conduche, [?], in 1984 described the notion of 2-crossed module as a model for 3-types. Both crossed modules and 2-crossed modules have been adapted for use in the context of commutative algebras (cf. [?] and [?]). Throughout this paper we denote an action of an element r 2 R on c 2 C by r c or simply rc if no confusion will arise. A crossed module is an algebra : C! R with an action of R on C c) = r@c c 0 = cc 0 for all c; c 0 2 M; r 2 R: In this section, we describe a 2-crossed module (cf. [?]) and a free 2-crossed module of algebras by using the second order Peier elements of [?]. Grandjean and Vale [?] have given a denition of a 2-crossed module of algebras. The following is an equivalent formulation of that concept. A 2-crossed module of k-algebras consists of a complex of C 0 -algebras C 2 - C 1 - C 0 2 1, morphisms of C 0 -algebras, where the algebra C 0 acts on itself by multiplication, such that C 2 - C 1 is a crossed module in which C 1 acts on C 2, (we require that for all x 2 C 0 ; y 2 C 1 and z 2 C 2 that (xy)z = x(yz)), further, there is a C 0 -bilinear function giving f g : C 1 C 0 C 1?! C 2 ; called a Peier lifting, which satises the following axioms: P L1 2 fy 0 y 1 g = y 0 y 1 (y 1 ) y 0 ; P L2 : f@ 2 (x 1 2 (x 2 )g = x 1 x 2 ; P L3 fy 0 y 1 y 2 g = fy 0 y 1 y 2 g 1 y 2 fy 0 y 1 g; P L4 : a) f@ 2 (x) yg = y 1 (y) x; b) 2 (x)g = y x; P L5 : fy 0 y 1 g z = fy 0 z y 1 g = fy 0 y 1 zg;

7 Free 2-Crossed Modules 7 for all x; x 1 ; x 2 2 C 2 ; y; y 0 ; y 1 ; y 2 2 C 1 and z 2 C 0 : A morphism of 2-crossed modules of algebras may be pictured by the diagram C 2 - C 1 - C 0 f 2? f 1? f 0? C C C where f f 1; f f 2, f 1 (c 0 c 1 ) = f 0 (c 0 ) f 1 (c 1 ); f 2 (c 0 c 2 ) = f 0 (c 0 ) f 2 (c 2 ); for all c 2 2 C 2 ; c 1 2 C 1 ; c 0 2 C 0 ; and f gf 1 f 1 = f 2 f g: We thus can dene the category of 2-crossed modules, denoting it by X 2 Mod. The following theorems, in some sense, are well known in other algebraic settings such as those of groups, and Lie algebras. We do not give all details of the proofs as analogous proofs can be found in the literature [?], [?] and the adaptation to the case of commutative algebras is routine. The rst although well known is included for completeness and because it indicates the pattern of later results. In the following we denote the category of simplicial algebras with Moore complex of length n by SimpAlg n. Theorem 2.1 The category of crossed modules is equivalent to the category of simplicial algebras with Moore complex of length 1. Proof: Let E be a simplicial algebra with Moore complex of length 1: Put M = NE 1 ; N = NE 0 1 = d 1 (restricted to M). Then NE 0 acts on NE 1 by multiplication via s 0 : Since the Moore complex is of length 1; we 2 NE 2 = Kerd 0 Kerd 1 = 0 and the generators of this ideal are of the form x(s 0 d 1 y? y) with x; y 2 NE 1 (see [?]). It then follows that for all x; x (x) x 0 = d 1 (x) x 0 = s 0 d 1 (x)x 0 by the action, = xx 0 2 NE 2 = 0: 1 : M! N is a crossed module. This yields a functor N 1 : SimpAlg 1 - XMod:

8 8 Z. Arvasi and T. Porter Conversely, 1 : M! N be a crossed module. By using the action of N on M; one forms the semidirect product M o N together with homomorphisms d 0 (m; n) = n; d 1 (m; n) 1 m + n; s 0 (n) = (0; n): Dene E 0 = N and E 1 = M o N: Then we have a 1-truncated simplicial algebra E 1: There is a functor t 1] from the category of 1-truncated simplicial algebras to that of simplicial algebras (see section 1). This enables us to dene a functor XMod - SimpAlg 1 ; given by sending fm; to E = t 1] E 1: Using lemma 1.1, E is a simplicial algebra whose Moore complex is of length 1: The correspondence gives rise to an equivalence of categories. 2 Some indication of the proof has been given above as it can be adapted to dimension 2. First we recall from [?] the following result. Proposition 2.2 Let E be a simplicial commutative algebra with the Moore complex NE. Then the complex of algebras NE 2 =@ 3 (NE 3 \ D NE 1 - NE 0 is a 2-crossed module of algebras, where the Peier lifting map is dened as follows: f g : NE 1 NE 1?! NE 2 =@ 3 (NE 3 \ D 3 ) (y 0 y 1 ) 7?! s 1 y 0 (s 1 y 1? s 0 y 1 ): (Here the right hand side denotes a coset in NE 2 =@ 3 (NE 3 \ D 3 ) represented by the corresponding element in NE 2 ). 2 This proposition leads to the generalisation of theorem 1.2 as follows. The methods we use for proving it are based on ideas of Ellis, [?]. A dierent proof of this result is given in [?]. Theorem 2.3 The category of 2-crossed modules is equivalent to the category of simplicial algebras with Moore complex of length 2. Proof: Let E be a simplicial algebra with Moore complex of length 2: In the previous proposition, a 2-crossed module NE 2 - NE 1 - NE 0

9 Free 2-Crossed Modules 9 has already been constructed by adding the relevant structure to the Moore complex. Thus there exists an obvious functor N 2 : SimpAlg 2 - X 2 Mod: Conversely suppose given a 2-crossed module 2?! 1?! N: Dene E 0 = N. We can create the semidirect product E 1 = M o N by using the action of N on M and we get homomorphisms, d 0 ; d 1 ; and s 0, dened by d 0 (m; n) = n; d 1 (m; n) 1 m + n; s 0 (n) = (0; n): By using axioms a) and b) of P L3, there is an action of m 2 M on l 1 2 L given by m l 1 1 m l 1? f@ 2 l 1 mg: Using this action we form the semidirect product L o M: An action of (m; n) 2 M o N on (l 1 ; m 1 ) 2 L o M is given by (m; n) (l 1 ; m 1 ) = (m l 1 + n l 1 ; mm 1 + n m): Using this action we get the semidirect product E 2 = (LoM)o(M on): (The bilinearity of f g together with axioms P L3 and P L5 ensure that these last two actions are indeed commutative actions.) There are homomorphisms, d i, s j given by d 0 (l 1 ; m 1 ; m 2 ; n) = (m 2 ; n) s 0 (m 2 ; n) = (0; 0; m 2 ; n); d 1 (l 1 ; m 1 ; m 2 ; n) = (m 1 + m 2 ; n) s 1 (m 2 ; n) = (0; m 2 ; 0; n): d 2 (l 1 ; m 1 ; m 2 ; n) = (m 1 1 m 2 + n); We thus have a 2-truncated simplicial algebra E 2 = fe 0 ; E 1 ; E 2 g: There is a functor t 2] from the category of 2-truncated simplicial algebras to that of simplicial algebras, and this enables us to dene a functor X 2 Mod - SimpAlg 2 given by sending fl; M; 2 1 g to E = t 2] E 2. Using lemma 1.1, E is a simplicial algebra whose Moore complex is of length 2: This correspondence gives rise to an equivalence of categories. The remaining details are left to the reader Free 2-Crossed Modules The denition of a free 2-crossed module is similar in some ways to the corresponding denition of a free crossed module. However, the construction of a free 2-crossed module is naturally a bit more complicated.

10 10 Z. Arvasi and T. Porter It will be helpful to have the notion of a pre-crossed module: this is just a : C! R with an action c) = r@c for c 2 C; r 2 R: Let (C; be a pre-crossed module, let Y be a set and let : Y! C be a function, then (C; is said to be a free pre-crossed R-module with basis or, alternatively, on the : Y! R if for any pre-crossed R-module (C 0 ; 0 ) and function 0 : Y! C 0 such 0 0 there is a unique morphism : (C; (C 0 ; 0 ) such that = 0. The pre-crossed module (C; is totally free if R is a free algebra. Let fc 2 ; C 1 ; C g be a 2-crossed module, let Y be a set and let # : Y! C 2 ; then fc 2 ; C 1 ; C g is said to be a free 2-crossed module with basis # or, alternatively, on the 2 # : Y! C 1 ; if for any 2-crossed module fc2 0 ; C 1 ; C 0 2 1g and function # 0 : Y! C 0 such that 2# 0 2 #0 ; there is a unique morphism : C 2! C 0 2 such 2 0 2: Remark. `Freeness' in any setting corresponds to a left adjoint to a forgetful functor, so what are the categories involved here? Let 2CD=P CM be the category whose objects consist of a precrossed module (C; and a pair (Y; ) where : Y! C is simply a function to the underlying set of the algebra C such = 0. Morphisms of such objects consist of a pair (; 0 ), where : (C; (C 0 ; D 0 0 ) is a morphism of precrossed modules and 0 : Y! Y 0, is a function such that 0 0 =. Forgetting the algebra structure of the top algebra, C 2, of a 2-crossed module provides one with a functor from 2-crossed modules to this category. The objects of 2CD=P CM are thought of as 2-(dimensional) construction data on a given precrossed module. We will show that free 2-crossed modules always exist on such `gadgets' and hence that the forgetful functor described above has a left adjoint. The description of this category 2CD=P CM may seem a bit articial, but given an algebra presented as a quotient of, say, a polynomial algebra by an ideal, the free crossed module of that presentation has been found to contain valuable information on the algebra. Given a crossed module, for example, an arbitrary algebra R together with an ideal I in R, what should one mean by a presentation of (R; I) or more generally of a crossed module on R? The `yoga' of crossed algebraic methods suggests several possible replies. For the sake of exposition, we will describe only the ideal-pair case, (R; I). Picking a set of generators X for I gives a function 1 : X! R and hence a free precrossed module (R + [X]; R; ] ) on 1 1. This gives a morphism of precrossed modules from (R + [X]; R; ] 1 ) to (I; R; inc) measuring in part the freeness of I on X. Taking the kernel of this morphism K! R, we pick a set of generators of K, 2 : Y! K as a precrossed module and we have an object of 2CD=P CM. Thus to analyse an ideal pair homologically, one natural method to use is to compare it via a free 2-crossed module, with a free precrossed module. This process is based to some

11 Free 2-Crossed Modules 11 extent on the intuition from related CW-complex constructions from topology. Andre's use, [?], of simplicial resolutions provides the bridge between the two settings. The sort of 2-construction data one obtains from a simplicial resolution corresponds to a special type of 2-crossed module: A free 2-crossed module fc 2 ; C 1 ; C g is totally free 1 : C 1! C 0 is a totally free pre-crossed module. At the moment, totally free 2-crossed modules seem to be of more immediate use and interest than those that are merely free. We will therefore concentrate on their construction, but later on will indicate how to adapt that construction to the more general case. We thus start by giving an explicit description of the construction of a totally free 2-crossed module on 2-construction data. For this, we need to recall the 2-skeleton of a free simplicial algebra given as d 0 ; d 1 ; d - 2 d 0 ; d 1 E (2) : ::: (R[s 0 (X); s 1 (X)])[Y ] s 0 ; s 1 - R[X] s 0 - R; with the simplicial structure dened as in section 3 of [?]. Analysis of this 2-dimensional construction data, (cf. [?]), shows that it consists of some 1-dimensional data, namely the function ' : X! R that is used to induce d 1 : R[X]! R, together with strictly 2-dimensional data consisting of the function : Y! R + [X], where R + [X] is the positively degree part of R[X] and which is used to induce d 2 from R[s 0 (X); s 1 (X)][Y ] to R[X]. We will denote this 2-dimensional construction data by (Y; X; ; '; R): Theorem 2.4 A totally free 2-crossed module fl; E; R; 0 1 g exists on the 2-dimensional construction data (Y; X; ; '; R): Proof: Suppose given the 2-dimensional construction data as above, i.e., given a function ' : X! R and : Y! R + [X]: Set E = R + [X]. With the obvious action of R on E, the function ' gives a free pre-crossed 1 : E?! R: Now take D = R[s 0 (X)] + [s 1 (X); Y ] \ ((s 0? s 1 )(X)); so that E acts on D by multiplication via s 1. The function induces a morphism of E-algebras, : R[s 0 (X)] + [s 1 (X); Y ] \ ((s 0? s 1 )(X))?! R + [X] given by (y) = (y): (Of course D = NE (2) 2, the Moore complex of the 2-skeleton of the free simplicial algebra on the data.) Let fa; E; R; ; g be any 2-crossed module and let # 0 : Y! A. Recall from [?], and as summarised above, the second order Peier ideal P 2 in D. It is easily checked that (P 2 ) = 0

12 12 Z. Arvasi and T. Porter as all generator elements of P 2 are in Kerd 2 : By taking the factor algebra L = D=P 2 ; there exists a morphism 0 : L! E such that the diagram, D @@R????? 0? commutes, where q is the quotient morphism of algebras. Also 0 is a crossed module. Indeed, given the elements y + P 2 ; y 0 + P 2 2 L; 0 (y + P 2 ) (y 0 + P 2 ) = (y) y 0 + P 2 L = s 1 d 2 (y)y 0 + P 2 yy 0 + P 2 = (y + P 2 )(y 0 + P 2 ): Hence there exists a unique morphism : L! A, given by (y + P 2 ) = # 0 (y), such that = 0 : Therefore fl; E; R; 0 ; 'g; i.e. the complex R[s 0 (X)] + [s 1 (X); Y ] \ ((s 0? s 1 )(X)) P 2 is the required free 2-crossed module on (Y; X; ; '; R): 2 0?! R + [X] '?! R Note: In the group case, a closely related structure to that of 2-crossed module is that of a quadratic module, dened by Baues [?]. Although it seems intuitively clear that the results above should extend to an algebra version of quadratic modules, we have not checked all the details and so have omitted a study of this idea from this paper. Having given the construction above, we will briey turn to the more general case of a free 2-crossed module generated by 2-construction data over a given precrossed module, i.e. by an object ((C; (Y; )) of 2CD=P CM. The precrossed module (C; gives us a simplicial algebra E with E 0 = R, E 1 = C o R, E 2 = C o C o R and so on. The 2-construction data for gluing in the new 2-generators : Y! C allows one to form a new simplicial algebra F with F 0 = E 0, F 1 = E 1, and F 2 = E 2 [Y ], etc., as in the step-by-step construction of a simplicial resolution, cf. Andre, [?]. The 2-crossed module associated to F will be the desired free 2-crossed module on the construction data. The proof is essentially the same as that of Theorem 2.4 above. 3 The n-type of the k-skeleton The key invariant studied in [?] was the commutative algebra version of the module of identities amongst the relations from combinatorial group theory. Given a way of `present-

13 Free 2-Crossed Modules 13 ing' an algebra A as a quotient R[X]=I where I = (a 1 ; : : :; a n ); there may be equations or identities amongst the a i so that I is not `free' on the specied generators. The invariant measured that lack of freeness. Of course its study is linked to that of syzygies. The data (X : a 1 ; : : :; a n ) allows a 1-skeleton of a free simplicial resolution of A to be constructed and the free crossed module used in [?] was that constructed from that 1-skeleton. The various stages of the construction of a resolution, i.e. the k-skeleta, can be observed as they change with k by looking at the way the invariants of the n-type of these k-skeleta change with increasing k. We examine a slightly more general situation namely that of a free simplicial algebra homotopy equivalent to a given simplicial algebra. The case of a resolution corresponds to the given simplicial algebra being a K(A; 0); i.e. constant with value K(A; 0) n = A and all face and degeneracies the identity isomorphism. In the lowest dimensional cases this process corresponds to going from a `presentation' to nding a description of the identities between the given `relations', and then, on being given generators for this module of identities, observing the change in the various invariants that result from adding these into dimension 2. Recall from [?] that a morphism f : E! F of simplicial algebras is called an n- equivalence if i (f) : i (E)?! i (F); is an isomorphism for all i; 0 i n: Two simplicial algebras E and F are said to have the same n-type if there is a chain of n-equivalences linking them. A simplicial algebra F is an n-type if i (F) = 0 for i > n: Suppose given any simplicial algebra A and n 1, then we can use the step-by-step construction (see [?]) to produce a free simplicial algebra E and an epimorphism ' : E?! A which is an n-equivalence: Take a free algebra R and an epimorphism ' : R?! A 0 set E (0) = K(R; 0), the constant simplicial algebra with value R: We have a morphism ' (0) : E (0)?! A: Now take a free R-algebra on construction data to enable the kernel of ' to be killed and A 1 to be \covered" '1 R[X]?! A 1 "## ##" R '0?! A 0 ;

14 14 Z. Arvasi and T. Porter to make 0 (' 0 ) an isomorphism. The construction then follows the obvious routine, mirroring the construction of a resolution. You add new generators in dimension k + 1 to adjust the kernel at level k and to produce an epimorphism onto A k+1. We need only continue until E (n+1) has been reached if we are only interested in invariants of the n-type. In fact we can represent any n-type of a simplicial algebra by an n-equivalent free simplicial algebra constructed using construction data and a step-by-step construction of skeleta, E (0),! E (1),!,! E (n),! E (n+1),! : : :E: As an n-type does not retain homotopy information in dimensions greater than n, we have that any algebraic n-type can be represented by a simplicial algebra having NE k = 0 if k > n; and such that E k is a free algebra if k < n, for instance by taking cosk n E (n+1) as given above Types Assume given an R-algebra, B = R=I, then 0 (K(R; 0)) = R and i (K(R; 0)) = 0 for i > 0: Thus algebras are algebraic models of the 0-types of the 0-skeleton E (0) of the `step-by-step' construction Types Given data for the 1-step of the construction of a free simplicial algebra, we get E (1) : d 0 ; d 1 ; d 2 d 0 ; d 1 ::: R[s 0 (X); s 1 (X)] - R[X] - R s 0 ; s s 0 1 f - R=I; with X = fx i : i 2 Ig and d 1 0(X i ) = 0; d 1 1(X i ) = x i 2 Kerf; s 0 (r) = r 2 R: From the denition, there is an isomorphism 0 (E (1) ) = E (1) 0 =d1 1 (Kerd1 0 ): Considering the morphism d 1 : 1 Kerd1 0! R; one readily obtains Imd1 1 = I and E(1) 0 = R: Thus 0 (E (1) ) = R=I: Take a 1-truncation of a free simplicial algebra E (1) as follows: d 0 ; d 1 R[X]=P 1 - R s 0 f - R=I; where P 1 is the Peier ideal of the pre-crossed module R + 1! R.

15 Free 2-Crossed Modules 15 Let Tr(E (1) ; 1) denote this 1-truncated simplicial algebra. The corresponding free crossed module 1 : R + [X]=P 1! R: By proposition 1.6 in [?], this becomes R n =Imd! R; where d is the rst Koszul dierential. From the routine calculation above and corollary 3.4 of [?] and using results from [?], there are the following isomorphisms 0 (Tr(E (1) ; 1)) = B; 1 (Tr(E (1) ; 1)) = Ker(R n =Imd?! R) = H 2 (B; B) the second Andre-Quillen homology group, whilst i (Tr(E (1) ; 1)) = 0 for i > 1: We see that free crossed modules are algebraic models of 1-types of the 1-skeleton of a free simplicial algebra Types Suppose given the 2-skeleton E (2) of the construction of a free simplicial algebra d 0 ; d 1 ; d 2 d 0 ; d 1 E (2) : ::: (R[s 0 (X); s 1 (X)])[Y ] - R[X] - R s 0 ; s s 0 1 f - R=I: As above, one gets the same homotopy modules, for E (2) ; in dimension 0: 0 (E (2) ) = B but 1 (E (2) ) = Ker(R + [X]=P 1?! R)=(Image of Y ): and there is an isomorphism 2 (E (2) ) = Ker(NE (2) 2 =@ 3(NE (2) 3 )?! E(2) Since NE (2) 2 = R[s 0 (X)] + [s 1 (X); Y ] \ ((s 0? s 1 )(X)); the second homotopy module of the 2-skeleton looks like 2 (E (2) ) R[s0 (X)] + [s = Ker 1 (X); Y ] \ ((s 0? s 1 )(X))?! R[X] where P 2 is the second order Peier ideal. Take a 2-truncation of a free simplicial algebra P 2 1 ) d 0 ; d 1 ; d 2 d 0 ; d 1 (R[s 0 (X); s 1 (X)])[Y ]=P 2 - R[X] - R s 0 ; s s 0 1 f - R=I:

16 16 Z. Arvasi and T. Porter Let Tr(E (2) ; 2) denote this 2-truncated simplicial algebra. Using theorem 2.4, the corresponding free 2-crossed module is R[s 0 (X)] + [s 1 (X); Y ] \ ((s 0? s 1 R [X]?! R: P 2 From the above calculation, one gets the isomorphisms: and and nally 0 (Tr(E (2) ; 2)) = B; 2 (Tr(E (2) ; 2)) R[s0 (X)] = Ker + [s 1 (X); Y ] \ ((s 0? s 1 )(X))?! R[X] P 2 i (Tr(E (2) ; 2)) = 0 for i > 2: Again free 2-crossed modules of algebras are algebraic models of 2-types of the 2-skeleton of the free simplicial algebra. References [1] M. Andre, Homologie des algebres commutatives, Springer-Verlag, Die Grundlehren der mathhematicschen Wisssenschaften in Einzeldarstellungen Band 206 (1974). [2] Z.Arvasi and T.Porter, Simplicial and crossed resolutions of commutative algebras, J. Algebra, 181, (1996), [3] Z. Arvasi and T. Porter, Higher order Peier elements in Simplicial Commutative Algebras, Theory and Applications of Categories, 3, 1997, [4] H.J. Baues, Combinatorial homotopy and 4-dimensional complexes, Walter de Gruyter, (1991). [5] P. Carrasco, Complejos hipercruzados cohomologia y extensiones, Ph.D. Thesis, Universidad de Granada, (1987). [6] P. Carrasco and A.M. Cegarra, Group-theoretic algebraic models for homotopy types, J. Pure Applied Algebra, 75 (1991), [7] D. Conduche, Modules croises generalises de longueur 2, J. Pure Applied Algebra, 34 (1984), [8] J. Duskin, Simplicial methods and the interpretation of triple cohomology, Memoirs A.M.S. Vol (1975). [9] G.J. Ellis, Higher dimensional crossed modules of algebras, J. Pure Applied Algebra, 52 (1988), [10] G.J. Ellis, Homotopical aspects of Lie algebras, J. Austral. Math. Soc. (Series A) 54 (1993),

17 Free 2-Crossed Modules 17 [11] A.R. Grandjean and M.J. Vale, 2-Modulos cruzados en la cohomologia de Andre- Quillen, Memorias de la Real Academia de Ciencias, 22 (1986), [12] T. Porter, Homology of commutative algebras and an invariant of Simis and Vasconcelos, J. Algebra, 99 (1986), Z.Arvasi Department of Mathematics, Faculty of Science, Osmangazi University, Eskisehir, Turkey T.Porter School of Mathematics University of Wales, Bangor Gwynedd LL57 1UT, UK.