3-Crossed Modules of Commutative Algebras

On 3-crossed mo dules of algebras T.S. Kuzpınarı, A. O da ba¸ s and E. ¨ O. Uslu Abstract In this pap er we define 3-crossed mod u les for comm utative (Lie) algebras and inves tigate the re lation betw een this construction and the simplicial algebras. Also we define the pro- jectiv e 3 -crossed resolution fo r in v estigate a higher dimensional homological information and sho w the existence of this resolution for an arbitrary k -algebra. Keyw ords: crossed mo dule, 2-crossed mo dule, simplicial alg ebra, Mo ore complex. 1 In tro duction As an extension of cr o ssed modules (Whitehead) and 2-cr ossed modules (Co nduch ´ e); Arv asi, Kuzpınarı and Uslu in [4], defined 3-c rossed modules as a mo del for homo topy 4-types. Kan, in [17], also prov ed that simplicial groups are a lg ebraic mo dels for homo to py t ypes. It is known from [3, 14, 2 3] that simplicial algebra s with Mo o re complex o f length 1 , 2 lea d to cr ossed mo dule and 2-cros sed mo dules which ar e r elated to K oszul co mplex and Andre-Quillen homolo gy constructions for use in homoto pic a l and homolo gical algebra . PJ.L.Doncel, A.R. Grandjean and M.J.V ale in [10] exten t the 2-cros sed modules of gro ups to commutativ e alg ebras. As in indicated in [1 0], they defined a homolog y theory and obtain the relation with Andre-Quillen homolog y for n = 0 , 1 , 2 , . This homology theory includes the pro jective 2-cro ssed resolution and the homotopy op era to r given in [15]. Of course thes e results based on the work of T.Porter [23], which inv olves the relatio n betw een Koszul complex a nd the Andre-Quillen homology by means of free crossed mo dules of comm utativ e algebra. In this v ein, we hop e that it would b e p ossible to genera lise these results by using commutativ e algebra case of higher dimensiona l cr o ssed algebra ic gadgets. The present work inv olves the relatio n b etw een 3-crosse d mo dules and simplicial alg ebra with- out details sinc e the most calcula tion ar e sa me as gro up case given in [4]. F urther more the work inv olves the existence of pro jective 3-cross ed resolution of a k- a lgebra to obtain an higher di- mensional homolog ical information ab out c o mmut ative alg ebras. Here the constr uction is a bit different from the 2-cros sed re solution given in [10] beca use of the num b er of Peiffer liftings . At the end of the w ork we give the L ie algebra 3-c r ossed mo dules . The main r esults of this work are; 1. Intro duce the notion o f 3-c r ossed mo dules of commutativ e alg ebras a nd Lie alg ebras; 2. Cons truct the pass age from 3 -cros s ed modules of a lgebras to simplicial a lg ebras and the conv erse passag e as an a nalogue result given in [4]; 3. Define the pro jective 3-cro ssed resolution for investigate a higher dimensional homolog ic al information and show the existence of this resolution for a n ar bitrary k -a lgebra which was shown for tw o crossed mo dules in [10]. 1 2 Preliminaries In this work k will b e a fixed comm utative ring with iden tit y 1 not equal to zer o a nd all algebr as will b e comm utativ e k- algebras , we a ccept they a r e no t requir ed to hav e the identit y 1. 2.1 Simplicial Algebras See [20], [9] for most of the basic prop er ties o f simplicial structures. A simplicial algebra E consists of a family of algebra s { E n } together with face and degenera cy maps d n i : E n → E n − 1 , 0 ≤ i ≤ n , ( n 6 = 0) and s n i : E n − 1 → E n , 0 ≤ i ≤ n , satisfying the usual simplicial identities given in [1], [16]. The categor y of simplicial algebra s will b e deno ted by SimpAlg . Let ∆ denotes the catego ry of finite o rdinals. F or each k ≥ 0 we obtain a sub categ ory ∆ ≤ k determined by the o b jects [ i ] of ∆ with i ≤ k . A k -tr uncated simplicial algebra s is a functor fro m ∆ op ≤ k to Alg (the category of algbe ras). W e will denote the catego ry of k - truncated simplicial algebras b y T r k SimpAlg . By a k - tru ncation of a simpl i cial alg ebr a, we mean a k -trunca ted simplicial a lgebra tr k E obta ine d b y forg etting dimensions of order > k in a simplicia l algebra E . Then we have the adjoints situations SimpAlg tr k / / T r k SimpAlg st k o o cost k / / SimpAlg tr k o o where st k and cost k are called the k -skeleton and the k -coskeleton functors resp ectivily F o r de- tailed definitions s ee [11]. 2.2 The Mo or e Complex. The Mo o re co mplex NE of a simplicia l a lgebra E is defined to b e the normal chain complex ( NE , ∂ ) with N E n = n − 1 \ i =0 k er d i and with differen tial ∂ n : N E n → N E n − 1 induced from d n by restrictio n. W e say that the Mo o re co mplex N E of a simplicial alg e br a E is of length k if NE n = 0 for all n ≥ k + 1. W e denote the categ ory of simplicia l a lgebras with Mo o re complex of length k by SimpAlg ≤ k . The Mo ore complex, NE , carr ies a hyper crossed complex structure (see Car rasco [5] ) from which E can b e rebuilt. Now we will ha v e a lo ok to this co nstruction slightly . The details ca n b e found in [5]. 2.3 The Poset of Surjectiv e Maps The following nota tion and ter minology is derived from [6]. F or the o rdered se t [ n ] = { 0 < 1 < · · · < n } , let α n i : [ n + 1] → [ n ] be the increasing surjectiv e map given by; α n i ( j ) =  j if j ≤ i, j − 1 if j > i. Let S ( n, n − r ) b e the set of all monotone increasing surjective maps from [ n ] to [ n − r ]. This ca n be genera ted from the v ar ious α n i by compos itio n. The compos ition of these gener ating maps is sub ject to the following rule: α j α i = α i − 1 α j , j < i . T his implies that every elemen t α ∈ S ( n, n − r ) has a unique expressio n as α = α i 1 ◦ α i 2 ◦ · · · ◦ α i r with 0 ≤ i 1 < i 2 < · · · < i r ≤ n − 1, wher e the indices i k are the elemen ts of [ n ] such that { i 1 , . . . , i r } = { i : α ( i ) = α ( i + 1) } . W e thus ca n ident ify S ( n, n − r ) with the set { ( i r , . . . , i 1 ) : 0 ≤ i 1 < i 2 < · · · < i r ≤ n − 1 } . In particular, the 2 single e le men t o f S ( n, n ), de fined by the iden tit y ma p on [ n ], corresp onds to the empt y 0 -tuple ( ) denoted b y ∅ n . Similarly the only element of S ( n, 0) is ( n − 1 , n − 2 , . . . , 0). F o r all n ≥ 0, le t S ( n ) = [ 0 ≤ r ≤ n S ( n, n − r ) . W e 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 o r dered set. F or example S (2) = { φ 2 < (1) < (0) < (1 , 0) } S (3) = { φ 3 < (2) < (1) < (2 , 1) < (0) < (2 , 0) < (1 , 0 ) < (2 , 1 , 0) } S (4) = { φ 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) < (2 , 1 , 0) < (3 , 2 , 1 , 0) } 2.4 The Semidirect Decomp osition of a Simplicial A lgebra The fundamen ta l idea behind this can b e found in Conduch ´ e [8 ]. A detailed investigation of this for the ca se of s implicial g roups is g iven in Ca rrasco and Cega rra [6].The alg ebra case of the structure is also g iven in [5]. Prop ositi o n 1 If E is a simplicial algebr a, then for any n ≥ 0 E n ∼ = ( . . . ( N E n ⋊ s n − 1 N E n − 1 ) ⋊ . . . ⋊ s n − 2 . . . s 0 N E 1 ) ⋊ ( . . . ( s n − 2 N E n − 1 ⋊ s n − 1 s n − 2 N E n − 2 ) ⋊ . . . ⋊ s n − 1 s n − 2 . . . s 0 N E 0 ) . Pro of. This is by rep eated use of the following lemma. Lemma 2 L et E b e a simplicial algebr a. Then E n c an b e de c omp ose d as a semidir e ct pr o duct: E n ∼ = ker d n n ⋊ s n − 1 n − 1 ( E n − 1 ) . The bracket ing and the order o f terms in this multiple semidirect product are g e nerated by the sequence: E 1 ∼ = N E 1 ⋊ s 0 N E 0 E 2 ∼ = ( N E 2 ⋊ s 1 N E 1 ) ⋊ ( s 0 N E 1 ⋊ s 1 s 0 N E 0 ) E 3 ∼ = (( N E 3 ⋊ s 2 N E 2 ) ⋊ ( s 1 N E 2 ⋊ s 2 s 1 N E 1 )) ⋊ (( s 0 N E 2 ⋊ s 2 s 0 N E 1 ) ⋊ ( s 1 s 0 N E 1 ⋊ s 2 s 1 s 0 N E 0 )) . and E 4 ∼ = ((( N E 4 ⋊ s 3 N E 3 ) ⋊ ( s 2 N E 3 ⋊ s 3 s 2 N E 2 )) ⋊ (( s 1 N E 3 ⋊ s 3 s 1 N E 2 ) ⋊ ( s 2 s 1 N E 2 ⋊ s 3 s 2 s 1 N E 1 ))) ⋊ s 0 (decomp osition of E 3 ) . Note that the term corr esp onding to α = ( i r , . . . , i 1 ) ∈ S ( n ) is s α ( N E n − # α ) = s i r ...i 1 ( N E n − # α ) = s i r ...s i 1 ( N E n − # α ) , where # α = r. Hence an y element x ∈ E n can b e wr itten in the fo r m x = y + X α ∈ S ( n ) \{∅ n } s α ( x α ) with y ∈ N E n and x α ∈ N E n − # α . 3 2.5 Hyp ercrossed Complex P airings In the following we recall from [3] hypercro ssed complex pairings for commutativ e algebr as. The fundamen tal idea b e hind this can b e found in Carras co and Cegar ra (cf. [6]). The construction depe nds on a v a riety of sourc e s, mainly Conduch ´ e [8], Z. Arv asi and T. Porter, [3]. Define a set P ( n ) cons isting of pa irs of elemen ts ( α, β ) from S ( n ) with α ∩ β = ∅ and β < α , with resp ect to lexicogra phic ordering in S ( n ) where α = ( i r , . . . , i 1 ) , β = ( j s , . . . , j 1 ) ∈ S ( n ). The pair ings that we will need, { C α,β : N E n − ♯α ⊗ N E n − ♯β → N E n : ( α, β ) ∈ P ( n ) , n ≥ 0 } are given as composites b y the diag ram N E n − # α ⊗ N E n − # β s α ⊗ s β   C α,β / / N E n E n ⊗ E n µ / / E n p O O where s α = s i r , . . . , s i 1 : N E n − ♯α → E n , s β = s j s , . . . , s j 1 : N E n − ♯β → E n , p : E n → N E n is defined b y co mpo site pro jections p ( x ) = p n − 1 . . . p 0 ( x ) , where p j ( z ) = z s j d j ( z ) − 1 with j = 0 , 1 , . . . , n − 1 . µ : E n ⊗ E n → E n is given b y mult iplication ma p and ♯α is the num ber of the elements in the s et of α, similarly for ♯β . 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 )) Let I n be the ideal in E n generated by elements of the form C α,β ( x α ⊗ y β ) where x α ∈ N E n − ♯α and y β ∈ N E n − ♯β . W e illustrate this for n = 3 a nd n = 4 a s follows: F or n = 3, the p o ssible Peiffer pa irings a r e the following C (1 , 0)(2) , C (2 , 0)(1) , C (0)(2 , 1) , C (2)(0) , C (2)(1) , C (1)(0) F or all x 1 ∈ N E 1 , y 2 ∈ N E 2 , the corresp onding g e ne r ators of I 3 are: C (1 , 0)(2) ( x 1 ⊗ y 2 ) = ( s 1 s 0 x 1 − s 2 s 0 x 1 ) s 2 y 2 , C (2 , 0)(1) ( x 1 ⊗ y 2 ) = ( s 2 s 0 x 1 − s 2 s 1 x 1 )( s 1 y 2 − s 2 y 2 ) C (0)(2 , 1) ( x 2 ⊗ y 1 ) = s 2 s 1 x 2 ( s 0 y 1 − s 1 y 1 + s 2 y 1 ) C (1)(0) ( x 2 ⊗ y 2 ) = [ s 1 x 2 ( s 0 y 2 − s 1 y 2 ) + s 2 ( x 2 y 2 ) , C (2)(0) ( x 2 ⊗ y 2 ) = ( s 2 x 2 )( s 0 y 2 ) , C (2)(1) ( x 2 ⊗ y 2 ) = s 2 x 2 ( s 1 y 2 − s 2 y 2 ) . F or n = 4, the key pairing s are thus the following C (3 , 2 , 1)(0) , C (3 , 2 , 0)(1) , C (3 , 1 , 0)(2) , C (2 , 1 , 0)(3) , C (3 , 0)(2 , 1) , C (3 , 1)(2 , 0) , C (3 , 2)(1 , 0) , C (3 , 2)(1) , C (3 , 2)(0) , C (3 , 1)(0) , C (0)(2 , 1) , C (3 , 1)(2) , C (2 , 1)(3) , C (3 , 0)(2) , C (3 , 0)(1) , 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) , 4 Theorem 3 ([3]) L et E b e a simplici al algeb r a with Mo or e c omplex NE in which E n = D n , is an ide al of E n gener ate d by the de gener ate elements in dimensio n n, then ∂ n ( N E n ) = P I ,J [ K I , K J ] for I , J ⊆ [ n − 1] with I ∪ J = [ n − 1] , I = [ n − 1] − { α } J = [ n − 1] − { β } wher e ( α, β ) ∈ P ( n ) for n = 2 , 3 and 4 , Remark 4 Shortly in [21] they define d t he n ormal s ub gr oup ∂ n ( N G n ∩ D n ) by F α,β elements which wer e define d first by Carr asc o in [5]. Castiglioni and L adr a gener alise d this inclusion in [7]. F ollowing [3 ] we have Lemma 5 L et E b e a simplici al algebr a with Mo or e c omplex N E of length 3 . Then for n = 4 the images of C α,β elements under ∂ 4 given in T able 1 ar e trivial. Pro of. Since N G 4 = 1 b y the re s ults in [3] r esult is tr ivial. 5 1 d 4 [ C (3 , 2 , 1)(0) ( x 1 ⊗ y 3 )] = s 2 s 1 x 1 ( s 0 d 3 y 3 − s 1 d 3 y 3 + s 2 d 3 y 3 − y 3 ) 2 d 4 [ C (3 , 2 , 0)(1) ( x 1 ⊗ y 3 )] = ( s 2 s 0 x 1 − s 2 s 1 x 1 )( s 1 d 3 y 3 − s 2 d 3 y 3 + y 3 ) 3 d 4 [ C (3 , 1 , 0)(2) ( x 1 ⊗ y 3 )] = ( s 1 s 0 x 1 − s 2 s 0 x 1 )( s 2 d 3 y 3 − y 3 ) 4 d 4 [ C (2 , 1 , 0)(3) ( x 1 ⊗ y 3 )] = ( s 2 s 1 s 0 d 1 x 1 − s 1 s 0 x 1 ) y 3 5 d 4 [ C (3 , 2)(1 , 0) ( x 2 ⊗ y 2 )] = ( s 1 s 0 d 2 x 2 − s 2 s 0 d 2 x 2 − s 0 x 2 ) s 2 y 2 6 d 4 [ C (3 , 1)(2 , 0) ( x 2 ⊗ y 2 )] = ( s 1 x 2 − s 0 x 2 + s 2 s 0 d 2 x 2 − s 2 s 1 d 2 x 2 )( s 1 y 2 − s 2 y 2 ) 7 d 4 [ C (3 , 0)(2 , 1) ( x 2 ⊗ y 2 )] = ( s 2 s 1 d 2 x 2 − s 1 x 2 )( s 0 y 2 − s 1 y 2 + s 2 y 2 ) 8 d 4 [ C (3 , 2)(1) ( x 2 ⊗ y 3 )] = s 2 x 2 ( s 1 d 3 y 3 − s 2 d 3 y 3 + y 3 ) 9 d 4 [ C (3 , 2)(0) ( x 2 ⊗ y 3 )] = s 2 x 2 ( s 2 d 3 y 3 − s 1 d 3 y 3 + s 0 d 3 y 3 − y 3 ) 10 d 4 [ C (3 , 1)(2) ( x 2 ⊗ y 3 )] = ( s 1 x 2 − s 2 x 2 )( s 2 d 3 y 3 − y 3 ) 11 d 4 [ C (3 , 1)(0) ( x 2 ⊗ y 3 )] = ( s 1 x 2 − s 2 x 2 )( s 2 d 3 y 3 − s 1 d 3 y 3 + s 0 d 3 y 3 − y 3 ) 12 d 4 [ C (3 , 0)(2) ( x 2 ⊗ y 3 )] = ( s 0 x 2 − s 1 x 2 + s 2 x 2 )( s 2 d 3 y 3 − y 3 ) 13 d 4 [ C (3 , 0)(1) ( x 2 ⊗ y 3 )] = ( s 0 x 2 − s 1 x 2 + s 2 x 2 )( s 1 d 3 y 3 − s 2 d 3 y 3 + y 3 ) 14 d 4 [ C (2 , 1)(3) ( x 2 ⊗ y 3 )] = ( s 2 s 1 d 2 x 2 − s 1 x 2 ) y 3 15 d 4 [ C (0)(2 , 1) ( x 2 ⊗ y 3 )] = ( s 2 s 1 d 2 x 2 − s 1 x 2 )( s 2 d 3 y 3 − s 1 d 3 y 3 + s 0 d 3 y 3 − y 3 ) 16 d 4 [ C (2 , 0)(3) ( x 2 ⊗ y 3 )] = ( s 2 s 0 d 2 x 2 − s 0 x 2 + s 1 x 2 − s 1 s 1 d 2 x 2 ) y 3 17 d 4 [ C (2 , 0)(1) ( x 2 ⊗ y 3 )] = ( s 2 s 0 d 2 x 2 − s 0 x 2 + s 1 x 2 − s 2 s 1 d 2 x 2 ) ( s 1 d 3 y 3 − s 2 d 3 y 3 + y 3 ) 18 d 4 [ C (1 , 0)(3) ( x 2 ⊗ y 3 )] = ( s 2 s 0 d 2 x 2 − s 0 x 2 − s 1 s 0 d 0 x 2 ) y 3 19 d 4 [ C (1 , 0)(2) ( x 2 ⊗ y 3 )] = ( s 1 s 0 d 2 x 2 − s 2 s 0 d 2 x 2 + s 0 x 2 )( s 2 d 3 y 3 − y 3 ) 20 d 4 [ C (3)(2) ( x 3 ⊗ y 3 )] = x 3 ( s 2 d 3 y 3 − y 3 ) 21 d 4 [ C (3)(1) ( x 3 ⊗ y 3 )] = x 3 ( s 1 d 3 y 3 − s 2 d 3 y 3 + y 3 ) 22 d 4 [ C (3)(0) ( x 3 ⊗ y 3 )] = x 3 ( s 2 d 3 y 3 − s 1 d 3 y 3 + s 0 d 3 y 3 − y 3 ) 23 d 4 [ C (2)(1) ( x 3 ⊗ y 3 )] = ( s 2 d 3 x 3 − x 3 )( s 1 d 3 y 3 − s 2 d 3 y 3 + y 3 ) 24 d 4 [ C (2)(0) ( x 3 ⊗ y 3 )] = ( s 2 d 3 x 3 − x 3 )( s 2 d 3 y 3 − s 1 d 3 y 3 + s 0 d 3 y 3 − y 3 ) 25 d 4 [ C (1)(0) ( x 3 ⊗ y 3 )] = ( s 1 d 3 x 3 − s 2 d 3 x 3 + x 3 )( s 2 d 3 y 3 − s 1 d 3 y 3 + s 0 d 3 y 3 − y 3 ) T able 1 where x 3 , y 3 ∈ N G 3 , x 2 , y 2 ∈ N G 2 , x 1 ∈ N G 1 . 6 2.6 Crossed Mo dules Here we will recall the notio n o f cro ssed mo dules of commutativ e algebra s given in [23] and [12] Let R b e a k -algebra with identit y . A cr ossed modul e of c o mmu tative algebr a is an R - algebra C , tog ether with a comm utative action of R on C and R -a lg ebra mor phism ∂ : C → R to gether with an ac tio n of R on C , written r · c for r ∈ R a nd c ∈ C , satisfying the co nditions. CM1) for a ll r ∈ R , c ∈ C ∂ ( r · c ) = r∂ c CM2) (Peiffer Identit y) for a ll c, c ′ ∈ C ∂ c · c ′ = cc ′ W e will denote such a cros sed mo dule b y ( C, R , ∂ ). A m orphism of cr osse d mo dule from ( C, R, ∂ ) to ( C ′ , R ′ , ∂ ′ ) is a pa ir of k -algebr a morphisms φ : C − → C ′ , ψ : R − → R ′ such that φ ( r · c ) = ψ ( r ) · φ ( r ) a nd ∂ ′ φ ( c ) = ψ ∂ ( c ). W e th us get a categ ory XMo d of cr ossed mo dules. Examples of Cr osse d Mo dules (i) An y idea l, I , in R gives a n inclusion ma p I − → R, which is a cro ssed module then w e will s ay ( I , R, i ) is an idea l pair. In this case, of co urse, R acts on I by multiplication and the inclusion homomor phism i makes ( I , R , i ) int o a cro ssed mo dule, an “ inclusion crossed mo dules”. Conv e r sely , Lemma 6 If ( C, R, ∂ ) is a cr osse d mo dule, ∂ ( C ) is an ide al of R. (ii) Any R -mo dule M can b e cons idered as an R -alg ebra with zer o multip lication a nd hence the zero morphism 0 : M → R sending everything in M to the z ero element of R is a crossed mo dule. Again co nversely: Lemma 7 If ( C, R, ∂ ) is a cr osse d mo dule, ker ∂ is an ide al in C and inherits a natur al R -mo dule structur e fr om R -action on C. Mor e over, ∂ ( C ) acts trivial ly on ker ∂ , henc e ker ∂ has a natur al R/∂ ( C ) -mo dule structu r e. As these tw o exa mples sug g est, g eneral cross ed mo dules lie betw e e n the t wo extr emes o f ide a l and mo dules. Bo th asp ects a re imp ortant. (iii) In the category o f alg ebras, the appropr iate replac e ment for a utomorphism g roups is the m ultiplication a lgebra defined by Mac Lane [19]. Then a utomorphism cr o ssed mo dule corr esp ond to the multiplication cros sed mo dule ( R, M ( R ) , µ ). T o see this crossed mo dule, we need to a ssume Ann ( R ) = 0 or R 2 = R a nd let M ( R ) b e the set of a ll m ultiplier s δ : R → R such tha t for a ll c, c ′ ∈ C , δ ( r r ′ ) = δ ( r ) r ′ . M ( R ) ac ts on R by M ( R ) × R − → R ( δ, r ) 7− → δ ( r ) and there is a mo rphism µ : R → M ( R ) defined b y µ ( r ) = δ r with δ r ( r ′ ) = rr ′ for all r, r ′ ∈ R . 2.7 2 -Crossed Mo dules Now we recall the commut ative alg e br a cas e of 2 -crosse d mo dules due to A.R.Gra ndjean and V ale, [14]. A 2 -cross ed mo dule of k -algebra s is a complex C 2 ∂ 2 − → C 1 ∂ 1 − → C 0 7 of C 0 -algebra s with ∂ 2 , ∂ 1 morphisms of C 0 -algebra s, wher e C 0 acts on C 0 by multiplication, with a bilinear function { ⊗ } : C 1 ⊗ C 0 C 1 − → C 2 called as Peiffer lifting which sa tisfies the following axio ms: 2CM1 ) ∂ 2 { y 0 ⊗ y 1 } = y 0 y 1 − ∂ 1 ( y 1 ) y 0 2CM2 ) { ∂ 2 ( x 1 ) ⊗ ∂ 2 ( x 2 ) } = x 1 x 2 2CM3 ) { y 0 ⊗ y 1 y 2 } = { y 0 y 1 ⊗ y 2 } + ∂ 1 y 2 { y 0 ⊗ y 1 } 2CM4 ) ( i ) { ∂ 2 ( x ) ⊗ y } = y · x − ∂ 1 ( y ) x ( ii ) { y ⊗ ∂ 2 ( x ) } = y · x 2CM5 ) z { y 0 ⊗ y 1 } = { z y 0 ⊗ y 1 } = { y 0 ⊗ z y 1 } for all x, x 1 , x 2 ∈ C 2 , y , y 0 , y 1 , y 2 ∈ C 1 and z ∈ C 0 . A morphism of 2-crossed mo dules can be defined in an obvious wa y . W e th us define the categ ory o f 2- crossed mo dules denoting it by X 2 Mo d . The pro of of the following theorem ca n b e found in [3 ]. Theorem 8 The c ate gory of 2 -cr osse d mo dules is e quivalent to t he c ate gory of simplicial algebr as with Mo or e c omplex of lengt h 2 . Now we will g ive some remar k s on 2-c r ossed mo dules where the g roup ca se can b e found in [22]. 1) Let C 1 ∂ 1 − → C 0 be a cr o ssed mo dule. If we take C 2 trivial then C 2 ∂ 2 − → C 1 ∂ 1 − → C 0 is a 2 -cros sed mo dule with the Peiffer lifting defined by { x ⊗ y } = 0 for x, y ∈ C 1 . 2) If C 2 ∂ 2 − → C 1 ∂ 1 − → C 0 is a 2 -cros sed mo dule then C 1 I m∂ 2 ∂ 1 − → C 0 is a c rossed mo dule. 3) Let C 2 ∂ 2 − → C 1 ∂ 1 − → C 0 be a 2-cr ossed module with triv ial Peiffer lifting then C 1 ∂ 1 − → C 0 will b e a cros sed module. Also in this situation we hav e the tr ivial action o f C 0 on C 2 . 3 Three Crossed Mo dules As a conseque nce of [4], her e we will define 3- crossed mo dules o f co mmut ative algebras . The wa y is similar but so me of the conditions are different. Let E be a simplicial algebra with Mo o re complex of length 3 and N E 0 = C 0 , N E 1 = C 1 , N E 2 = C 2 , N E 3 = C 3 . Thus we have a k -alg ebra complex C 3 ∂ 3 − → C 2 ∂ 2 − → C 1 ∂ 1 − → C 0 Let the actions of C 0 on C 3 , C 2 , C 1 , C 1 on C 2 , C 3 and C 2 on C 3 be as follows; x 0 x 1 = s 0 x 0 x 1 x 0 x 2 = s 1 s 0 x 0 x 2 x 0 x 3 = s 2 s 1 s 0 x 0 x 3 x 1 x 2 = s 1 x 1 x 2 x 1 x 3 = s 2 s 1 x 1 x 3 x 2 · x 3 = s 2 x 2 x 3 8 Then, since ( s 2 s 1 s 0 ∂ 1 x 1 − s 1 s 0 x 1 ) y 3 = 0 ( s 2 s 1 ∂ 2 x 2 − s 1 x 2 ) y 3 = 0 x 3 ( s 2 ∂ 3 y 3 − y 3 ) = 0 we get ∂ 1 x 1 y 3 = s 1 s 0 x 1 y 3 ∂ 2 x 2 y 3 = s 1 x 2 y 3 ∂ 3 x 3 · y 3 = x 3 y 3 and using the s implicial identities we get, ∂ 3 ( x 2 · x 3 ) = ∂ 3 ( s 2 x 2 x 3 ) = ∂ 3 ( s 2 x 2 ) ∂ 3 ( x 3 ) = x 2 ∂ 3 ( x 3 ) Thu s ∂ 3 : C 3 → C 2 is a cro ssed mo dule. Definition 9 L et C 3 ∂ 3 − → C 2 ∂ 2 − → C 1 ∂ 1 − → C 0 b e a c omplex of k -algebr as defin e d ab ove. We define Peiffer liftings as fol lows; { ⊗ } : C 1 ⊗ C 1 → C 2 { x 1 ⊗ y 1 } = s 1 x 1 ( s 0 y 1 − s 1 y 1 ) { ⊗ } (1 , 0)(2) : C 1 ⊗ C 2 → C 3 { x 1 ⊗ y 2 } (1 , 0)(2) = ( s 1 s 0 x 1 − s 2 s 0 x 1 ) s 2 y 2 { ⊗ } (2 , 0)(1) : C 1 ⊗ C 2 → C 3 { x 1 ⊗ y 2 } (2 , 0)(1) = ( s 2 s 0 x 1 − s 2 s 1 x 1 )( s 1 y 2 − s 2 y 2 ) { ⊗ } (0)(2 , 1) : C 1 ⊗ C 2 → C 3 { x 1 ⊗ y 2 } (0)(2 , 1) = s 2 s 1 x 1 ( s 0 y 2 − s 1 y 2 + s 2 y 2 ) { ⊗ } (1)(0) : C 2 ⊗ C 2 → C 3 { x 2 ⊗ y 2 } (1)(0) = ( s 0 x 2 − s 1 x 2 ) s 1 y 2 + s 2 ( x 2 y 2 ) { ⊗ } (2)(0) : C 2 ⊗ C 2 → C 3 { x 2 ⊗ y 2 } (2)(0) = s 2 x 2 s 0 y 2 { ⊗ } (2)(1) : C 2 ⊗ C 2 → C 3 { x 2 ⊗ y 2 } (2)(1) = s 2 x 2 ( s 1 y 2 − s 2 y 2 ) wher e x 1 , y 1 ∈ C 1 , x 2 y 2 ∈ C 2 . Then using T able 1 we g et the follo wing identities. 9 { x 2 ⊗ ∂ 2 y 2 } (0)(2 , 1) = { x 2 ⊗ y 2 } (1)(0) + { x 2 ⊗ y 2 } (2)(1) { x 1 ⊗ ∂ 3 y 3 } (2 , 0)(1) = { x 1 ⊗ ∂ 3 y 3 } (0)(2 , 1) + { x 1 ⊗ ∂ 3 y 3 } (1 , 0)(2) − ∂ 1 x 1 y 3 { ∂ 2 x 2 ⊗ y 2 } (1 , 0)(2) = − { x 2 ⊗ y 2 } (0)(2) { ∂ 3 x 3 ⊗ ∂ 3 y 3 } (1)(0) = x 3 y 3 { ∂ 2 x 2 ⊗ ∂ 3 y 3 } (0)(2 , 1) = ∂ 2 x 2 y 3 { ∂ 2 x 2 ⊗ ∂ 3 y 3 } (1 , 0)(2) = − { x 2 ⊗ ∂ 3 y 3 } (0)(2) { ∂ 2 x 2 ⊗ ∂ 3 y 3 } (2 , 0)(1) = ∂ 2 x 2 · y 3 − { x 2 ⊗ ∂ 3 y 3 } (0)(2) { x 2 ⊗ y 2 y ′ 2 } (1)(0) = { x 2 ⊗ ∂ 2 ( y 2 y ′ 2 ) } (0)(2 , 1) − { x 2 ⊗ ( y 2 y ′ 2 ) } (2)(1) { x ′ 2 x 2 ⊗ y 2 } (1)(0) = { x ′ 2 x 2 ⊗ ∂ 2 y 2 } (0)(2 , 1) − { x ′ 2 x 2 ⊗ y 2 } (2)(1) { x 2 ⊗ y 2 y ′ 2 } (2)(1) = { x 2 ⊗ ∂ 2 ( y 2 y ′ 2 ) } (1)(2 , 0) + { x 2 ⊗ y 2 y ′ 2 } (2)(0) − { x 2 ⊗ y 2 y ′ 2 } (1)(0) { x 2 x ′ 2 ⊗ y 2 } (2)(1) = { x 2 x ′ 2 ⊗ ∂ 2 y 2 } (1)(2 , 0) + { x 2 x ′ 2 ⊗ y 2 } (2)(0) − { x 2 x ′ 2 ⊗ y 2 } (1)(0) { x 2 ⊗ y 2 y ′ 2 } (2)(0) = − { x 2 ⊗ ∂ 2 ( y 2 y ′ 2 ) } (2)(1 , 0) { x 2 ⊗ ∂ 3 y 3 } (2)(1) = x 2 · y 3 { ∂ 3 x 3 ⊗ y 2 } (2)(1) = x ∂ 2 y 2 3 + x 3 · y 2 { ∂ 3 x 3 ⊗ y 2 } (1)(0) = y 2 · x 3 { ∂ 3 x 3 ⊗ y 2 } (2)(0) = 0 ∂ 3 { x 2 ⊗ y 2 } (2)(0) = − ∂ 3 { ∂ 2 x 2 ⊗ y 2 } (1 , 0)(2) ∂ 3 { x 2 ⊗ y 2 } (1)(0) = { ∂ 2 x 2 ⊗ ∂ 2 y 2 } (1)(0) + x 2 y 2 ∂ 3 { x 2 ⊗ y 2 } (2)(1) = ∂ 2 y 2 x 2 − x 2 y 2 ∂ 3 { x 1 ⊗ y 2 } (2 , 0)(1) = ∂ 3 { x 1 ⊗ y 2 } (1 , 0)(2) + { x 1 ⊗ ∂ 2 y 2 } − ∂ 1 x 1 y 2 + x 1 y 2 ∂ 3 { x 1 ⊗ y 2 } (0)(2 , 1) = { x 1 ⊗ ∂ 2 y 2 } + x 1 y 2 T able 2 x 0 { x 1 ⊗ y 1 } = { x 0 x 1 ⊗ y 1 } = { x 1 ⊗ x 0 y 1 } x 0 { x 1 ⊗ y 2 } (1 , 0)(2) = { x 0 x 1 ⊗ y 2 } (1 , 0)(2) = { x 1 ⊗ x 0 y 2 } (1 , 0)(2) x 0 { x 1 ⊗ y 2 } (0)(2 , 1) = { x 0 x 1 ⊗ y 2 } (0)(2 , 1) = { x 1 ⊗ x 0 y 2 } (0)(2 , 1) x 0 { x 1 ⊗ y 2 } (2 , 0)(1) = { x 0 x 1 ⊗ y 2 } (2 , 0)(1) = { x 1 ⊗ x 0 y 2 } (2 , 0)(1) x 0 { x 2 ⊗ y 2 } (1)(0) = { x 0 x 2 ⊗ y 2 } (1)(0) = { x 2 ⊗ x 0 y 2 } (1)(0) x 0 { x 2 ⊗ y 2 } (2)(0) = { x 0 x 2 ⊗ y 2 } (2)(0) = { x 2 ⊗ x 0 y 2 } (2)(0) x 0 { x 2 ⊗ y 2 } (2)(1) = { x 0 x 2 ⊗ y 2 } (2)(1) = { x 2 ⊗ x 0 y 2 } (2)(1) T able 3 z 1 { x 1 ⊗ y 1 } = { z 1 x 1 ⊗ y 1 } = { x 1 ⊗ z 1 y 1 } z 1 { x 1 ⊗ y 2 } (1 , 0)(2) = { z 1 x 1 ⊗ y 2 } (1 , 0)(2) = { x 1 ⊗ z 1 y 2 } (1 , 0)(2) z 1 { x 1 ⊗ y 2 } (0)(2 , 1) = { z 1 x 1 ⊗ y 2 } (0)(2 , 1) = { x 1 ⊗ z 1 y 2 } (0)(2 , 1) z 1 { x 1 ⊗ y 2 } (2 , 0)(1) = { z 1 x 1 ⊗ y 2 } (2 , 0)(1) = { x 1 ⊗ z 1 y 2 } (2 , 0)(1) z 1 { x 2 ⊗ y 2 } (1)(0) = { z 1 x 2 ⊗ y 2 } (1)(0) = { x 2 ⊗ z 1 y 2 } (1)(0) z 1 { x 2 ⊗ y 2 } (2)(0) = { z 1 x 2 ⊗ y 2 } (2)(0) = { x 2 ⊗ z 1 y 2 } (2)(0) z 1 { x 2 ⊗ y 2 } (2)(1) = { z 1 x 2 ⊗ y 2 } (2)(1) = { x 2 ⊗ z 1 y 2 } (2)(1) T able 4 where x 0 ∈ C 0 , x 1 , y 1 ∈ C 1 , x 2 , y 2 ∈ C 2 , x 3 , y 3 ∈ C 3 . F rom these results all liftings given in definition 1 a re C 0 , C 1 -bilinear maps. Definition 10 A 3-cr osse d mo dule c onsist of a c omplex C 3 ∂ 3 − → C 2 ∂ 2 − → C 1 ∂ 1 − → C 0 to gether with ∂ 3 , ∂ 2 , ∂ 1 which ar e C 0 , C 1 -algebr a morphisms, an action of C 0 on C 3 , C 2 , C 1 , an 10 action of C 1 on C 2 , C 3 and an action of C 2 on C 3 , further C 0 , C 1 -biline ar maps { ⊗ } (1)(0) : C 2 ⊗ C 2 − → C 3 , { ⊗ } (0)(2) : C 2 ⊗ C 2 − → C 3 , { ⊗ } (2)(1) : C 2 ⊗ C 2 − → C 3 , { ⊗ } (1 , 0)(2) : C 1 ⊗ C 2 − → C 3 , { ⊗ } (2 , 0)(1) : C 1 ⊗ C 2 − → C 3 , { ⊗ } (0)(2 , 1) : C 2 ⊗ C 1 − → C 3 , { ⊗ } : C 1 ⊗ C 1 − → C 2 c al le d P eiffer liftings which satisfy the fol lowing axioms for al l x 1 ∈ C 1 , x 2 , y 2 ∈ C 2 , and x 3 , y 3 ∈ C 3 : 3CM1 ) C 3 ∂ 3 − → C 2 ∂ 2 − → C 1 is a 2 -cr osse d mo dule with the Peiffer lifting { ⊗ } (2) , (1) 3CM2 ) ∂ 2 { x 1 ⊗ y 1 } = ∂ 1 y 1 x 1 − x 1 y 1 3CM3 ) { x 2 ⊗ ∂ 2 y 2 } (0)(2 , 1) = { x 2 ⊗ y 2 } (2)(1) − { x 2 ⊗ y 2 } (1)(0) 3CM4 ) ∂ 3 { x 2 ⊗ y 2 } (1)(0) = { ∂ 2 x 2 ⊗ ∂ 2 y 2 } + x 2 y 2 3CM5 ) { x 1 ⊗ ∂ 3 y 3 } (2 , 0)(1) = { x 1 ⊗ ∂ 3 y 3 } (0)(2 , 1) + { x 1 ⊗ ∂ 3 y 3 } (1 , 0)(2) − ∂ 1 x 1 y 3 3CM6 ) { ∂ 2 x 2 ⊗ y 2 } (2 , 0)(1) = − { x 2 ⊗ y 2 } (0)(2) + ( x 2 y 2 ) · { x 2 ⊗ y 2 } (2)(1) + { x 2 ⊗ y 2 } (1)(0) 3CM7 ) { ∂ 3 x 3 ⊗ ∂ 3 y 3 } (1)(0) = y 3 x 3 3CM8 ) { ∂ 3 y 3 ⊗ ∂ 2 x 2 } (0)(2 , 1) = − ∂ 2 x 2 y 3 3CM9 ) { ∂ 2 x 2 ⊗ ∂ 3 y 3 } (1 , 0)(2) = − { x 2 ⊗ ∂ 3 y 3 } (0)(2) 3CM10 ) { ∂ 2 x 2 ⊗ ∂ 3 y 3 } (2 , 0)(1) = ∂ 2 x 2 y 3 − { x 2 ⊗ ∂ 3 y 3 } (0)(2) 3CM11 ) { ∂ 3 y 3 ⊗ x 1 } (0)(2 , 1) = − x 1 y 3 3CM12 ) { y 2 ⊗ ∂ 3 x 3 } (1)(0) = − y 2 · x 3 3CM13 ) { ∂ 3 x 3 ⊗ y 2 } (1)(0) = y 2 · x 3 3CM14 ) { ∂ 3 x 3 ⊗ y 2 } (2)(0) = 0 3CM15 ) ∂ 3 { x 1 ⊗ y 2 } (2 , 0)(1) = ∂ 3 { x 1 ⊗ y 2 } (1 , 0)(2) + { x 1 ⊗ ∂ 2 y 2 } − ∂ 1 x 1 y 2 + x 1 y 2 3CM16 ) ∂ 3 { x 1 ⊗ y 2 } (0)(2 , 1) = { x 1 ⊗ ∂ 2 y 2 } − x 1 y 2 W e denote such a 3-cr o ssed mo dule by ( C 3 , C 2 , C 1 , C 0 , ∂ 3 , ∂ 2 , ∂ 1 ) . A m orphism of 3 -cr osse d mo dules of gro ups ma y b e pictur ed by the diagram C 3 f 3   ∂ 3 / / C 2 f 2   ∂ 2 / / C 1 f 1   ∂ 1 / / C 0 f 0   C ′ 3 ∂ ′ 3 / / C ′ 2 ∂ ′ 2 / / C ′ 1 ∂ ′ 1 / / C ′ 0 where 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 ) , f 3 ( c 0 c 3 ) = ( f 0 ( c 0 )) f 3 ( c 3 ) for { ⊗ } (0)(2) , { ⊗ } (2)(1) , { ⊗ } (1)(0) { ⊗ } f 2 ⊗ f 2 = f 3 { ⊗ } for { ⊗ } (1 , 0)(2) , { ⊗ } (2 , 0)(1) , { ⊗ } (0)(2 , 1) { ⊗ } f 1 ⊗ f 2 = f 3 { ⊗ } for { ⊗ } { ⊗ } f 1 ⊗ f 1 = f 2 { ⊗ } for all c 3 ∈ C 3 , c 2 ∈ C 3 , c 1 ∈ C 3 , c 0 ∈ C 3 . These compos e in an obvious wa y . So we can define the category of 3-cr ossed modules of co mmu tative algebr as,which we will b e denoted by X 3 Mo dAlg . 11 4 Applications 4.1 Simplicial Algebras As an applica tion we co nsider the rela tion b etw een simplicial a lgebras and 3-cro ssed mo dules which w ere g iven in [4] for group ca se. So pr o ofs in this sectio n ar e omitted, since can b e check ed easily by using the pro ofs given in [4]. Prop ositi o n 11 L et E b e a simplicial algebr a with Mo or e c omplex NE . Then the c omplex N E 3 /∂ 4 ( N E 4 ∩ D 4 ) ∂ 3 − → N E 2 ∂ 2 − → N E 1 ∂ 1 − → N E 0 is a 3 -cr osse d mo dule with the Peiffer liftings define d b elow: { ⊗ } : N E 1 ⊗ N E 1 − → N E 2 { x 1 ⊗ y 1 } (1)(0) 7− → s 1 x 1 ( s 1 y 1 − s 0 y 1 ) { ⊗ } (1 , 0)(2) : N E 1 ⊗ N E 2 − → N E 3 /∂ 4 ( N E 4 ∩ D 4 ) { x 1 ⊗ y 2 } (1 , 0)(2) 7− → ( s 2 s 0 x 1 − s 1 s 0 x 1 ) s 2 y 2 { ⊗ } (2 , 0)(1) : N E 1 ⊗ N E 2 − → N E 3 /∂ 4 ( N E 4 ∩ D 4 ) C 3 { x 1 ⊗ y 2 } (2 , 0)(1) 7− → ( s 2 s 1 x 1 − s 2 s 0 x 1 )( s 1 y 2 − s 2 y 2 ) { ⊗ } (0)(2 , 1) : N E 1 ⊗ N E 2 − → C 3 N E 3 /∂ 4 ( N E 4 ∩ D 4 ) { x 1 ⊗ y 2 } (0)(2 , 1) 7− → s 2 s 1 x 1 ( s 1 y 2 − s 0 y 2 − s 2 y 2 ) { ⊗ } (1)(0) : N E 2 ⊗ N E 2 − → N E 3 /∂ 4 ( N E 4 ∩ D 4 ) { x 2 ⊗ y 2 } (1)(0) 7− → ( s 1 x 2 − s 2 x 2 ) s 1 y 2 + s 2 ( x 2 y 2 ) { ⊗ } (2)(0) : N E 2 ⊗ N E 2 − → N E 3 /∂ 4 ( N E 4 ∩ D 4 ) { x 2 ⊗ y 2 } (2)(0) 7− → − s 2 x 2 s 0 y 2 { ⊗ } (2)(1) : N E 2 ⊗ N E 2 − → N E 3 /∂ 4 ( N E 4 ∩ D 4 ) { x 2 ⊗ y 2 } (2)(1) 7− → s 2 x 2 ( s 2 y 2 − s 1 y 2 ) (The elements denote d by ( , ) ar e c osets in N E 3 /∂ 4 ( N E 4 ∩ D 4 ) and given by the elements in N E 3 . ) Pro of. Here w e will c heck some of the conditions.The others c a n b e chec ked easily 3CM9) Since ∂ 4  C (3 , 2)(1 , 0) ( x 2 ⊗ y 3  ) = ( s 2 s 0 d 2 x 2 − s 1 s 0 d 2 x 2 − s 0 x 2 ) s 2 d 3 y 3 iwe find  ∂ 2 x 2 ⊗ ∂ 3 y 3  3 (1 , 0)(2) = ( s 2 s 0 d 2 x 2 − s 1 s 0 d 2 x 2 − s 0 x 2 ) s 2 d 3 y 3 s 0 x 2 s 2 d 3 y 3 ∈ mod∂ 4 ( N E 4 ∩ D 4 ) = − { x 2 ⊗ ∂ 3 y 3 } 3 (0)(2) 3CM13) Since ∂ 4  C (3 , 2)(1) ( x 2 ⊗ y 3  = s 2 x 2 ( s 2 y 3 − s 1 y 3 − y 3 ) = we find { x 2 ⊗ ∂ 3 y 3 } 3 (2)(1) = s 2 x 2 ( s 1 d 3 y 3 − s 2 d 3 y 3 ) ≡ s 2 x 2 y 3 ∈ mod∂ 4 ( N E 4 ∩ D 4 ) = x 2 · y 3 (3.13) Theorem 12 The c ate gory of 3 -cr osse d mo dules is e quivalent to the c ate gory of simplicial algebr as with Mo or e c omplex of lengt h 3 . 12 4.2 Pro ject iv e 3-crossed R esolution Here as an application we will define pro jectiv e 3-cr ossed reso lution of commutativ e alge bras. This construction was de fined by PJ.L.Doncel, A.R. Gr andjean and M.J.V ale in [10] for 2 -cross e d mo dules. Definition 13 A pr oje ctive 3 -cr osse d r esolution of an k -algebr a E is an exact se quenc e ... − → C k +1 ∂ k − → C k − → ... − → C 3 ∂ 3 − → C 2 ∂ 2 − → C 1 ∂ 1 − → C 0 ∂ 0 − → E − → 0 of k -mo dules such that 1) C 0 is pr oje ctive in the c ate gory of k -algebr as 2) C i is a C i − 1 -algebr as and pr oje ctive in the c ate gory of C i − 1 algebr as for i = 1 , 2 3) F or any epimorphism F = ( f , i d, id, id ) : ( C ′ 3 , C 2 , C 1 , C 0 , ∂ ′ 3 , ∂ 2 , ∂ 1 ) − → ( C ′′ 3 , C 2 , C 1 , C 0 , ∂ ′′ 3 , ∂ 2 , ∂ 1 ) and morphism H = ( h, id, id, i d ) : ( C 3 , C 2 , C 1 , C 0 , ∂ 3 , ∂ 2 , ∂ 1 ) − → ( C ′′ 3 , C 2 , C 1 , C 0 , ∂ ′ 3 , ∂ 2 , ∂ 1 ) ther e exist a morphism Q = ( q , id, id, i d ) : ( C 3 , C 2 , C 1 , C 0 , ∂ 3 , ∂ 2 , ∂ 1 ) − → ( C ′ 3 , C 2 , C 1 , C 0 , ∂ ′ 3 , ∂ 2 , ∂ 1 ) su ch that F Q = H 4) for k ≥ 4 , C k is a pr oje ctive k -mo dule 5) ∂ 4 is a homo morphism of C 0 -mo dule wher e t he action of C 0 on C 4 is define d by ∂ 0 6) F or k ≥ 5 , ∂ k is a homo morphism of k -mo dules Prop ositi o n 14 Any c ommu tative k -algebr a with a unit has a pr oje ctive 3 - cr osse d r esolution. Pro of. Let E be a k -algebra and C 0 = k [ X i ] a polyno mial ring suc h that there exist an epimor- phism ∂ 0 : C 0 → B . Now let define C 1 as C 0 = [ k er∂ 0 ] the po sitively g raded par t of polynomial ring on ke r∂ 0 and define ∂ 1 : C 1 → C 0 by inducing fr o m the inclusion i : k er∂ 0 → C 0 . Now let define K 2 = C 0 ( C 1 × C 1 S k er ∂ 1 ), the fr e e C 0 -mo dule o n the disjo in t union ( C 1 × C 1 ) S k er ∂ 1 and define ∂ ′ 2 : K 2 → k e r∂ 1 as ∂ ′ 2 ( x 1 y 1 ) = x 1 y 1 − ∂ 2 ( y 1 ) x 1 , x 1 y 1 ∈ C 1 ∂ ′ 2 ( x ) = x, x ∈ k er ∂ 1 Let R ′ be the C 0 -mo dule genera ted by the re lations ( αx 1 + β y 1 , z 1 ) − α ( x 1 , z 1 ) − β ( y 1 , z 1 ) ( x 1 , αy 1 + β z 1 ) − α ( x 1 , y 1 ) − β ( x 1 , z 1 ) when α, β ∈ C 0 and x 1 , y 1 , z 1 ∈ C 1 . Now define C 2 = K 2 /R ′ . Now define ∂ 2 : C 2 → k er ∂ 1 with ∂ 2 π = ∂ ′ 2 where π : K 2 → ( C 2 = K 2 /R ′ ) is pro jection. Now we will define C 3 . Let K 3 is the C 0 -mo dule defined o n the disjoin t union C 0 ( A (1 , 0) ∪ A (0 , 2) ∪ A (2 , 1) ∪ A (1 , 0)(2) ∪ A (2 , 0)(1) ∪ A (0)(2 , 1) ∪ ke r∂ 2 ) where A (1 , 0) = A (0 , 2) = A (2 , 1) = C 2 × C 2 , A (1 , 0)(2) = A (2 , 0)(1) = C 1 × C 2 and A (0)(2 , 1) = C 2 × C 1 . W e ha v e ∂ ′ 3 : K 3 → ( C 1 × C 1 ∪ ke r∂ 1 ) ∂ ′ 3 ( x, y ) = xy − ∂ 2 y x, ( x, y ) ∈ A (2)(1) ∂ ′ 3 ( x, y ) = ( ∂ 2 x, ∂ 2 y ) + xy, ( x, y ) ∈ A (1)(0) ∂ ′ 3 ( x, y ) = ( ∂ 2 x, y ) + y x, ( x, y ) ∈ A (0)(2 , 1) ∂ ′ 3 ( x, y ) = ( x, ∂ 2 y ) − ∂ 1 xy + xy, ( x, y ) ∈ A (2 , 0)(1) ∂ ′ 3 ( x, y ) = 0 , ( x, y ) ∈ A (0)(2) ∂ ′ 3 ( x, y ) = 0 , ( x, y ) ∈ A (1 , 0)(2) ∂ ′ 3 ( x, y ) = x, x ∈ k e r∂ 2 13 Now define a C 0 -mo dule R , generated b y the re lations,where α, β ∈ C 0 . Now define C 3 = K 3 /R we have ∂ 3 : C 3 → k e r∂ 2 with ∂ 3 π : ∂ ′ 3 where π : K 3 → ( C 3 = K 3 /R ) is pro jection. With these constructions C 3 ∂ 3 − → C 2 ∂ 2 − → C 1 ∂ 1 − → C 0 is a pro jective 3-cr ossed mo dule. If we define C 4 as the pr o jection re s olution of the E -module k er ∂ 3 then we hav e the pro jective crossed resolutio n · · · C k − → C k − 1 · · · C 4 q − → C 3 ∂ 3 − → C 2 ∂ 2 − → C 1 ∂ 1 − → C 0 ∂ 0 − → E − → 0 where q is the pr o jection. 4.3 Lie Algebra Case Lie algebraic version of cro s sed mo dules were introduced by K assel and Loday in, [18], and that of 2-cros sed mo dules were introduced by G. J. Ellis in, [13]. Also higher dimensional Peiffer elements in simplicial Lie algebras intro duced in [2]. As a conse q uence of the co mm utative algebr a version of 3- c rossed mo dules in this section w e will define the 3- crossed mo dules of Lie a lgebras by using the r e sults g iven in [2] with a similar wa y used for defining the comm uta tive algebra case. The relations for commu tative alge br a case given in the prev ious section can be applied to the Lie algebra cas e with s ome slight differences in the pro o fs up to the definition. Definition 15 A 3 -cr osse d mo dule over Lie algebr as c onsists of a c omplex of Lie algebr as L 3 ∂ 3 − → L 2 ∂ 2 − → L 1 ∂ 1 − → L 0 to gether with an action of L 0 on L 3 , L 2 , L 1 and an action of L 1 on L 3 , L 2 and an action of L 2 on L 3 so that ∂ 3 , ∂ 2 , ∂ 1 ar e morphisms of L 0 , L 1 -gr oups and the L 1 , L 0 -e quivariant liftings { , } (1)(0) : L 2 × L 2 − → L 3 , { , } (0)(2) : L 2 × L 2 − → L 3 , { , } (2)(1) : L 2 × L 2 − → L 3 , { , } (1 , 0)(2) : L 1 × L 2 − → L 3 , { , } (2 , 0)(1) : L 1 × L 2 − → L 3 , { , } (0)(2 , 1) : L 2 × L 1 − → L 3 , { , } : L 1 × L 1 − → L 2 c al le d 3 -dimensional Peiffer liftings. This data must satisfy the fol lowing axioms: 3CM1 ) C 3 ∂ 3 − → C 2 ∂ 2 − → C 1 is a 2 -cr osse d mo dule with the Peiffer lifting { ⊗ } (2) , (1) 3CM2 ) ∂ 2 { l 1 ⊗ m 1 } = ∂ 1 m 1 l 1 − [ l 1 , m 1 ] 3CM3 ) { l 2 ⊗ ∂ 2 m 2 } (0)(2 , 1) = { l 2 ⊗ m 2 } (2)(1) − { l 2 ⊗ m 2 } (1)(0) 3CM4 ) ∂ 3 { l 2 ⊗ m 2 } (1)(0) = { ∂ 2 l 2 ⊗ ∂ 2 m 2 } + [ l 2 , m 2 ] 3CM5 ) { l 1 ⊗ ∂ 3 l 3 } (2 , 0)(1) = { l 1 ⊗ ∂ 3 l 3 } (0)(2 , 1) + { l 1 ⊗ ∂ 3 l 3 } (1 , 0)(2) − ∂ 1 l 1 l 3 3CM6 ) { ∂ 2 l 2 ⊗ m 2 } (2 , 0)(1) = − { l 2 ⊗ m 2 } (0)(2) + [ l 2 , m 2 ] · { l 2 ⊗ m 2 } (2)(1) + { l 2 ⊗ m 2 } (1)(0) 3CM7 ) { ∂ 3 l 3 ⊗ ∂ 3 m 3 } (1)(0) = [ m 3 , l 3 ] 3CM8 ) { ∂ 3 l 3 ⊗ ∂ 2 l 2 } (0)(2 , 1) = − ∂ 2 l 2 l 3 3CM9 ) { ∂ 2 l 2 ⊗ ∂ 3 l 3 } (1 , 0)(2) = − { l 2 ⊗ ∂ 3 l 3 } (0)(2) 3CM10 ) { ∂ 2 l 2 ⊗ ∂ 3 l 3 } (2 , 0)(1) = ∂ 2 l 2 l 3 − { l 2 ⊗ ∂ 3 l 3 } (0)(2) 3CM11 ) { ∂ 3 l 3 ⊗ l 1 } (0)(2 , 1) = − l 1 l 3 3CM12 ) { l 2 ⊗ ∂ 3 l 3 } (1)(0) = − l 2 · l 3 3CM13 ) { ∂ 3 l 3 ⊗ l 2 } (1)(0) = l 2 · l 3 3CM14 ) { ∂ 3 l 3 ⊗ l 2 } (2)(0) = 0 3CM15 ) ∂ 3 { l 1 ⊗ l 2 } (2 , 0)(1) = ∂ 3 { l 1 ⊗ l 2 } (1 , 0)(2) + { l 1 ⊗ ∂ 2 l 2 } − ∂ 1 l 1 l 2 + l 1 l 2 3CM16 ) ∂ 3 { l 1 ⊗ l 2 } (0)(2 , 1) = { l 1 ⊗ ∂ 2 l 2 } − l 1 l 2 14 References [1] M.Andr ´ e . 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[23] T. Por ter . Homology of commutative alg ebras and and inv ariant o f Simis and V asconcelos . J. A lgebr a 99 , 45 8-46 5 , (1986). [24] J.H. C. Whitehead . Combinatorial Homotopy I a nd I I. Bul l. Amer. Math. So c. 55 , 213-24 5 and 496- 543, (1 9 49). T.S. Kuzpınarı stufan@aksa r ay .edu.tr Department of Ma thematics Aksaray University Aksaray/T ur key A. Odaba¸ s ao dabas@ ogu.edu.tr Mathematics and Computer Scie nces Department Eski¸ sehir Osma ng azi Universit y Eski¸ sehir/T ur key E. ¨ O. Uslu euslu@aku.edu.tr Department of Ma thematics Afy on Ko catep e University Afy onk ara hisar/T urkey 16