Exact sequences for mixed coproduct/tensor-product ring constructions

Publicacions Matemàtiques, Vol 38 (1994), 89-126 . EXACT SEQUENCES FOR MIXE D COPRODUCT/TENSOR-PRODUC T RING CONSTRUCTION S WARREN DICKS AND I . J . LEAR Y Abstr act To a commutative ring K, and a family of K -algebras indexed b y the vertex set of a graph, we associate a K -algebra obtained by a mixture of coproduct and tensor product constructions . For this , and related constructions, we give exact sequences and deduc e homological properties . 1 . Int roduct io n 1 .1 . Notation . Throughout, let K be a ring (associative, with 1) . The symbols ~ and ~ with no subscript are understood to mean ~ K and  respectively . Throughout, let X be a set, and A Ç X x X a symmetric antireflexiv e relation on X, that is, (x, y) E A implies (y, x) E A, and no (x, x) lies i n A . This corresponds to a (nonoriented) graph X A with vertex set X : w e define X A to have an edge connecting two elements x and y if and onl y if (x, y) E A . If Y is any subset of X, the full subgraph Y An (Y x Y ) of X A will b e abbreviated Y A . Recall that a graph is complete if each pair of vertices is connected b y an edge . Thus the complete subgraphs of X A correspond to the subset s Y of X such that any two elements of Y are connected by an edge i n XA . In this article we shall be studying a general situation described i n Hypothesis 3 .1, below ; it is wel l - illustrated by the following special case . 90  WARREN DICKS, 1 . J . LEAR Y 1 .2 . Construction . Suppose that K is commutative, and that w e are given, for each x E X, a K -algebra R(x) (associative with 1) . Form the coproduct II R(x) as K -algebras, and factor out the ideal generate d xE x by {ab — ba a E R(x), b E R(y),(x, y) E A} . Let R(X A ) denote th e resulting K -algebra. This construction is a mixture of the coproduct an d tensor -product constructions on K -algebras . For example, if X A is a complete graph then R(X A ) = ~ R(x) . sE X Our objective is to obtain homological information about R(X A ), fo r example, estimates for the left global dimension . From a homologica l point of view the formation of tensor products of K -algebras is not generally well - behaved, so if we are to obtain any information at all o n R(X A ), we should have prior knowledge of, say, the R(Y A ), for the complete subgraphs Y A of X A . Thus we take certain such information a s given, and try to extend it to all of R(X A ) . By constructing an exac t sequence motivated in section 2, described in section 3, and verified i n section 4, we are able to obtain a practical upper bound for the left globa l dimension of R(X A ), in terms of X A and the left global dimensions o f rings of the form R(Y A ), where Y A is a complete subgraph ofX A . I n section 5, we show how the left global dimension estimate can be refined . In section 6 we give G . M . Bergman's hitherto unpublished descriptio n of K-bases of R(X A ) constructed from K--bases of the R(x) . Consider the following group-theoretic situation . 1 .3 . Construction . Suppose that we are given a family of group s (G(x) x EX) . The X A -product of the family is the group G(X A ) formed by taking the free product * G(x) as groups, and factoring ou t xE X the normal subgroup generated b y {aba'b' i a E G(x), b E G(y), (x, y) E A} . This construction is a mixture of the coproduct and restricted direc t product constructions on groups . For example, if X A is a finite complete graph then G(X A ) = x G(x) ; x E X and, for infinite complete graphs, we get the restricted direct product , generated by the G(x) . For another example, if each G(x) has arder 2, then G(X A ) is a rightangled Coxeter group . Our results will give homological information on the group rin g R(X A ) = K[G(X A )] in terms of the group rings K[G(Y A )], for certai n complete subgraphs Y A of XA . MIXER COPRODUCTS/TENSOR-PRODUCTS  9 1 As was the case with [10], many of the results in this article ar e inspired by, and, in some cases, even copied from, the preprint [3] o f G . M . Bergman, and must be considered joint work of the three of us . Due to other commitments, Bergman reluctantly abandoned the project , and declined all offers to be listed as a co-author on either paper . Sinc e there now appears to be an incipient interest in Construction 1 .3, c f [1], [6], [11], we have decided to go ahead and publish, with Bergman' s approval . 2 . Motivating example s Throughout this section we consider the following . 2 .1 . Hypothesis . Suppose K is commutative, and that we hav e a family of faithfully flat K-algebras (R(x) x E X), indexed by th e vertex set of the graph X A . For any subset Y of X, we can appl y Construction 1 .2 and form the K -algebra R(Y A ) . We set R = R ( X A) . As we shall see in Proposition 3 .4, R is flat as right R(Y A )-module, fo r every subset Y of X . The following examples illustrate what we might hope to be able t o say about 1 . gl . dim . R . 2 .2 . Example . Let X A be the grap h a  b  c  d so R = (R(a)0R(b)) j (R(b)0R(c)) Ij (R(c)0R(d)) . By [9, Theorem 2 ] R(b)  R(c ) there is an exact sequence of R-bimodule s o —> ROR(b) R ROR ( c ) R --+ RO R(a)OR(b) R R OR(b)0R(c) R R OR(c}OR(d) R _} R ~ O , and, as in the proof of [9, Corollary 7], since R is flat as right modul e over R(b), R(c), R(a)0R(b) , R(b)0R(e) and R(c)0R(d) , we hav e 1 . gl . dim . R c max{1 + 1 . gl . dim . R(x), 1 . gl . dim . R(y)0R(z) x E Xo, yz E X I } , where Xo = {b, c} and X 1 = {ab, bc, cd } . WARREN DICKS, 1 . J . LEAR Y 2 .3 . Example . Let X A be the grap h a ,  b d  c Then R = (R(a) ~ R(c)) ~ (R(bM R(d)) . Consider the following subgraphs of X A : Y A  YÁ  Z A  Z Á a  b  c  a  b  b  b  c a  ■  •  .  •  a Since R O Y A } = R (YÁ) II R (Z'Á) , by [9, Theorem 2], there is a shor t R(Z A ) exact sequences of R(Y A )-bimodule s o —> R ( Y A )0 R(ZA) R(Y A ) ~ R(YA)® R ( Y) R(Y A ) R ( YA )® R(Z Á } R(Y A ) —> R(Y A ) —> O . This is a sequence of flat right R(Y A )-modules, and R is flat as righ t R(YA)-module, so the sequence remains exact under R O R ( YA }— O R ( YA }R . This gives us a short exact sequence of R -bimodules which, in an obviou s notation, can be abbreviated t o o—> b—> ab+bc~abc~o . In a similar manner we can obtain other short exact sequences o f R-bimodules which, in the saure notation, are expressed as follows : o —> ac --+ abc + acd --~ abcd --> o , o---}~—> a + c--- . -~ac—> p , 0 ---4 d —> da + cd -> acd --4 0 . 92 MIXED COPRODUCTS/TENSOR-PRODUCTS  9 3 These four exact sequences can be combined to give a new exact sequenc e of R -bimodule s o —+ - --} a+ b +c + d —> ab + bc + cd +da ----* abcd --> O . This sequence has the desirable property that the final term is R, and the other terms are induced from the rings associated with the complet e subgraphs of X A , which we have agreed to accept as our building blocks . As in the proof of [9, Corollary 7], we deduce tha t 1 . gl .dim . R Ç max{2 + 1 . gl. dim . K, 1 + 1 . gl .dim .R(x) , 1 . gl . dim . R(yz) x E Xo, yz E X 1 } , where Xo = {a,b,c,d} and X 1 = {ab, bc, cd, da} . To see that the term 2 + 1 . gl . dim . K cannot be omitted from ou r estimate, take the case where, for each x E X , R(x) K[e x i=e x ]ÑK xK . Then, for all yz E X 1 , R(yz) = R(y)0R(z) ~ KxKx K xK . But, from [2, Examples 12 .2-12 .3 (i)] , it is known that R(a)~ R(c) i s free of rank 4 as a module over its center, a polynomial ring K[s] in on e indeterminate . Thus R is free of rank 16 as a module over its center, a polynomial ring K [s, t] in two indeterminates . It foliows tha t 1 . gl . dim . R = 1 . gl. dim . K [s, t] = 2 + 1 . gl . dim . K . The next example illustrates one sort of complication that can aris e as we go about constructing our exact sequences . 2 .4 . Example . Let X A be the grap h a  f 94  WARREN DICKS, 1 . J . LEAR Y As in Example 2 .3, we can constructshort exact sequences of R -bimodules and combine them to construct an exact sequence of R -bimodule s whose finalterm is R, and whose other terms are induced from the ring s associated with the complete subgraphs of X A . For instance, we ca n forra the short exact sequence s o ---} ade -4 abcde + ade f — } abcde f —> O, . O~cd —> abcd+cde---~abcde--> o , o - --} ef —> def + ae f --> ade f ~ O , 0 ->  a + de ----~ ade --} 0 , O—> b—> abc+bd -+ abcd~O , o --}e ---} ce+ de —> cde—> o , 0—>~-> c +d --4 cd—> O , 0—> f—> a ,f + e f ~ ae f -4 o , and combine these in stages, finally arriving at an exact sequence o f R-bimodule s O ---} O+O—> a +b +c+d+e+ f +de+e f —> af +bd+ ce+de +ef +abc+def —> abcdef --} O . This exact sequence contains a redundancy, in that the componen t de + e f —> de + e f is an isomorphism which can be eliminated ; th e process is explained in detail in section 5 . We are then left with an exac t sequence of R -bimodule s 0 -4  a+ b+c+d+e+ f—> a f -}-bd +ce+ abc+ de f—> abcde f—> 0 , and we see that here the only subgraphs that arise are the interseccions o f maximal complete subgraphs! The associated global dimension formul a is l .gl .dim .R Ç max{2 + 1 . gl . dim . K, 1 +1 .gl .dim .R(x) , 1 .gl .dim .R(Y) ~ xEXo, YEX I } , where Xo = {a, b, c, d, e, f} and X ~ = {af, bd, ce, abc, den . MIXED COPRODUCTS/TENSOR-PRODUCTS  9 5 The concluding example illustrates the sensitivity to the choice of K . 2 .5 . Example . Let X A be the grap h 3 where h, i, j, k, and the associated edges, have been repeated, in orde r to achieve a planar representation . Thus X A arises as the 1 -skeleton ofa triangulation of the pro j ective plane, which has 11 vertices, 30 edges, an d 20 triangles . The triangulation is full, in the sense that every complet e subgraph of the 1 -skeleton, X A , is the vertex set of a simplex . To simplif y notation somewhat, let us writ e Xo = X = {a,b,c,d,e,f,g,h,i,j,k} , X~ = {ab,af,ag,ah,ak,aj,bc,bg,bh,bi,bk,cd,cg,ci,cj , de, dk, dj, dg, e f, eg, eh, ei, ek, f g, fi, f j, hi, hk, j k } , X 2 = {abg, abk, a f g, ah, ajk, bcg, bci, bhi, bhk, cdg , cdj, cij, deg, dek, djk, e f g, e f i, ehi, ehk, f j i } . By considering the circuit given by the full subgraph on {a, b, c, d, e, f } , and proceedingas in Example 2 .3, we get the following exact sequenc e of R -bimodules : 0 —> 0 —> a+b+c +d+ e+ f —> ab + bc + cd + de + e f + fa --} abcdef —> O . By considering R(— U {g}) in place of R(—), we get a similar exac t sequence : 0 —>g —>ag + bg + cg + dg + eg +f g --} abg + bcg + cdg + deg + e fg + fag ~ abcdefg ---} 0 . 96  WARREN DICKS, 1 . J . LEAR Y Continuing as before, we build up sequences : 0--> be —> abcde fg +beh- - }abcdefgh—> O , O—>b +e -- 4 be-- 4 O , 0—> h----> bh -feh —> óeh~o , 0 -4 fehbc ~ abcde f gh + fehbci —> abcde f ghi ~ O , 0--- 4 e+h+ b----~ fe + eh+ hb + bc~ f ehbc—> O , 0 —> ei+hi +bi ~ fei + ehi + hbi + bci —> fehbci —> 0 , 0 ~ dcif a - -} abcde f ghi + dci f aj —> abcde f ghi j--} O , o-> c+i+ f -4 dc+ci+if + fa —> dcifa —> O , 0--~ cj +i .7 +fj - - > dej + ci.7+ i f .7+f aj —> dcifaj —> 0 y 0 —> abhedj —> abcde f ghi j + abhedjk - --} abcde f ghi j k --> O , 0--›sçb—>b+ h+e+d+j+ a —) ab + bh + he + ed + dj + ja ~ abhedj —> O , --} k—>bk+hk+ek+dk+jk+a k —> abk + bhk + hek + edk + djk + jak —> abhedjk —> O . These then combine to give an exact sequenc e 0 -4-4 O+b+e+h+ E x xEX q —> b+e+h +bh+eh+ E x y xyEX 1 --> bh + eh + E xyz —> abcde f ghi j k —> 0 . xyzEX 2 Here we have three components b+ e+ h ~ b+ e + h, bh + eh —> bh + eh , and ~ --4 0, which we might hope to eliminate . It turns out that the firs t two are isomorphisms, and can be eliminated, but the third is given by multiplication by 2, which is an isomorphism if and only if 2 is invertibl e in R, which is equivalent to 2 being invertible in K, by the faithfu l flatness assumptions . Thus, in general, we have an exact sequenc e 0—>—)~+ E x---} E xy-- -}  xyz -4 abcdefghijk --}O . xEX Q  xyEX i  xyzEX 2 There is an underlying exact sequence obtained by taking all the R(x ) to be Z, and it is the augmented chain complex of a contractible threedimensional CW-complex with 20 vertices, 30 edges, 12 polygonal faces , o MIXED COPRODUCTS/TENSOR-PRODUCTS  97 and 1 three-cell . The CW-complex arises by taking the dual of the give n tessellation of the pro j ective plane, adding on a polygonal disk to kil l the fundamental group, which here coincides with the homology grou p in dimension 1, and adding on a three-cell to kill the resulting homolog y in dimension 2 . It was not obvious at the outset that the relativel y unsophisticated algebraic process of forming exact sequences, and the n refining them, had to yield such a relatively sophisticated topologica l ob ject, although it was encouraged in that direction by the original choic e of a full triangulation of the pro jective plano . One consequence is tha t 1 .gl .dim .R Ç max{3+ 1 . gl .dim .K , 2 + 1 . gl .dim .R(x) , 1 + 1 . gl .dim . R(xy) , 1 . gl . dim . R(xyz) ~ x E Xo,xy E X 1 , xyz E X 2 } . If 2 E K, then we also have an exact sequenc e o-> E x E xy-->  xyz---+ abcdefghzjk —}o , xEXo  xyEX 1  xy .zEX 2 so that here the term 3 + 1 . gl . dim . K can be omitted from our previou s estimate . To see that the term 3 + 1 . gl . dim . K cannot be omittedin general , take K to be a field, and, for each x E X, tak e R(x) =K[e x ~ el =ex]KXK . Let 1 = E Rex, so 1 is a two-sided ideal of R such that R= K E B I . Let K e xE X denote the R -bimodule R/1 . Then it is not difficult to use the resolutio n to show that ExtR (K E , R) K E /2K E , as R-bimodules . Hence if K has characteristic 2 then1 . gl . dim . R > 3, while if K has characteristi c different from 2 then 1 . gl . dim . R Ç 2 . In fact equality holds in bot h cases, the reverse inequalities coming from our estimate, and from a surjective map ExtR(K E , R) --3 K~, respectively . This example is closely related to one of the right - angled Coxete r groups which have virtual cohomological dimension 2 over Q, and 3 ove r Z, described by M . Bestvina [5, Remark (3)I • What general pattern emerges from these examples? It is clear tha t for any graphX A (even infinite) this procedure, of combining exact sequences and eliminating redundancies, will always give an exact sequenc e that gives information about R(X A ) in terms of the rings associated with 104  WARREN DICKS, I . J . LEAR Y If Y A is complete then there is a well-defined, degree +1, R -bimodul e endomorphism s * of R (F * Y ) which is given on generators b y (X0, . . .,Xm)8m(_1)m+l(XO, . . .,Xm,y) . It is readily verified that s * is well-defined, and that s * o ~ * + a * s * acts a s the identity on R(F * Y) . Thu s Ker o~ * = (Ker a * )(s * a * + c7 * s * ) = (Ker O * )(s * a * ) Ç Im c7 * , so H * (R(,F * Y ) } = 0, in this case . Thus we may assume that Y A is not complete, so there exist tw o vertices v, w of YA which are not joined by an edge . Let Z = link y A (w) , let W = ZU {w}, and let U = Y—{w} . Since Z c W Ç Y - - {v}, we see that Z, W, and U are proper subset s of Y . So, by the induction hypothesis, H * (R( .f ' * Z)), H, k (R( F * W)}, an d H * (R( .F * U)} are all o . Clearly U A n W A = Z A and U A U W A = Y A . Notice that we have .F * U A n .F * V [T A = .F * Z A in ,F * Y A . We claim that .F * U A U .F * WA = F * Y A . Consider any element z E Y ` * Y A so z = {Xo, . . . , X n } C F D Y A , where X D c ••• c Xn . If X n does no t contain w, then X n Ç U, so z E On the other hand, if Xn doe s contain w, then, since X n is complete, X n Ç link y,(w) U {w} = W . Thus z E .F * V [ l A . This proves .F * Y A Ç .~* U A U .F * WA , and the revers e inclusion is obvious . It now follows easily that the natural sequence of differential grade d Rbimodule s (4) 0 —> R(F*Z) (+ +) R( .F * W) # R(F*U) (+) R( .T*Y) 0 is exact in all non-negative degrees . The degree -1 part of (4) has th e for m (5) 0 —> R OR(z) R (~~ R ORw) R ROR(w) R(+) R® R(Y) R —+ 0 , and we now show this is exact . By 3 .1 (iv), R(Y) = R(U)  R(W), so , R(Z ) by [9, Theorem 2], there is a short exact sequence of R(Y)-bimodule s 0 —' R(Y)OR(Z)R(Y ) (~) R(Y)O R( u)R(Y) R E Y ) •-•R(w)R(Y) R(Y) — . 0 . MIXED COPRODUCTS/TENSOR-PRODUCTS  10 5 By Proposition 3 .4, this is a resolution of o by (faithfully) flat righ t R(Y)-modules, so remains exact under —O R(Y) R, since the resulting homology is TorR ( Y ) (o, R) = O . Again by Proposition 3 .4, R is flat as righ t R(Y)-module, so the resulting sequence remains exact under RO R(Y) — , giving the exact sequence (5) . Thus (4) is a short exact sequence of differential graded R -bimodules . Applying the homology functor, we get a natural exact triangle H),(R( .F,,W))ED .H*(R(F .U) ) 11 .(R(,F*Z)) <  II .(R( .F*Y)) , where S has degree -1 . Since we have verified that the groups at tw o vertices of the triangle are zero, it follows immediately that the thir d group,  is also 0, as desired . Thus, we have proved by induction that H * ~R~ .~` * Y }} = 0, if Y i s finite . Hence, for arbitrary Y, we have prove d (6)  H * (R (,F . Y ' } } = o for all finite subsets Y ' of Y . If Y ' Ç Y" are finite subsets of Y, then there is a natural ma p RGF * Y ' ) — } R( .F * Y " ) of differential graded R -bimodules, which is actually an embedding in al l non -negative degrees . Hence we get a directed syste m (R(FY ' ) ~ Y ' a finite subset of Y ) of differential graded R -bimodules . We claim tha t ( 7 )  lim  R(,T' * Y ' ) — R( .F * Y ) . finite Y' C Y By 3 .1 (iii), R(Y) is generated by the R(y), y E Y, s o l  ROR(Y,) R -= = R OR(Y) R , finite Y' C Y so the degree -1 part of (7) holds . WARREN DICKS, 1 . J . LEAR Y Since every element of . F * YA lies in . F . YÁ for some finite subset Y ' o f Y, we have  lim  = .F' * Y A , and (7) follows . No w finite Y' C Y H * (R(,F * Y ) } = H * (  lim  R(.F * Y ' ) }, by (7) , finite Y' C Y - -  lim  H * (R( .T . *Y' ) } = o, by (s) . finte Y'C Y This completes the proof . ■ We now want to show that all the other complexes defined aboye ar e acyclic . We shall make use of the following well-known fact . 4 .5 . Lemma . Suppose . . . —> C~ 2 —> C * a 1 ~ C *,O ._-> C *, _ 1 —+ 0 i s an exact sequence of differential (N — 1)-graded R-bimodules, such that , for all j E N, C *, ~ is acyclic . Then C *, _1 is acyclic . Proof : Let j E N — 1 . Let Z *, ~ denote the image of C *,i + 1 in C *,j ; notice that Z *, _ 1 = By the exactness of the given sequence, there is a short exact sequenc e of chain complexes 0 —> —> C *,i +1 —> Z *,i —> 0, and, sinc e C *,j + 1 is acyclic, the resulting exact triangle for homology gives identifications Hz ( Z *, i ) = Hi_ 1 (Z *,i +1 ), for all i E Z . Let n E N — 1, and apply the foregoing with (i,j) = (n, --1), (n — 1, 0), . . . , (—1,n), to obtain H n (Z *, _1) = Hn_1(Z *, o) = . . = H_2(Z*,n +1 ) . The final term , H__ 2 ( Z *, n + 1 ) , is 0, since the complexes are N — 1 graded . Thus the firs t term, H n (Z *, _ 1 ) , is 0, for all n E N — 1 . Hence Z *, _ 1 is acyclic, that is , C,_ 1 is acyclic . ■ We can now prove our main result . 4 .6 . Theorem . If Hypothesis 3 .1 holds, and Y is a subset of X, the n R(JV[ * Y}, R(z * Y), and R(C * Y), are all acyclic . Proof : Choose a total _weZl ordering ~ of .IW[oYA . There is a ma p C o X A —> M o X A , X o 1—> Xo , where Xo is the least element of M o X A , with respect to -<, which contains Xo . Consider any m, n E N — 1 . Let I(m, n) denote the set of all rn+ n -tuples (Xo , . . . , Yo, ... , Y n ) such that the Y~ lie in M O Y A with Yo ~ • • • -~ Yn, and the X i lie in C O Y A with XoC . . CXm CYo rl . . r1 Yn . 106 MIXED COPRODUCTS/TENSOR-PRODUCTS  10 7 Let C m,n denote the R -bimodule ROR(Xo, . . .,X,n,yo, . . .,Yr,,) R , (X 0 , . . ., X r , ., , ,Yo, . . .,Y n )E I(m,n )  R(Xo)  if m E N , R(Yon•••nY n ) ifm=—1,nEi `N ,  1 R(Y)  if m =—1,n =—1 . w e ábbreviate aOR(xo, . . .,xm,Yo, . . .,Y,) b to a(Xo, . . . , X m ; Y o, . . . , Yn)b . There is then a bi-graded R -bimodule C *,* = e) EB1 C, . TZ,n . W e m>—In>— 1 think of the indices (m, n) aslying in the m-n plane, largely in the firs t quadrant, and will speak of m indexing the columns, and n indexing th e rows . There are two commuting Rbimodule differentials  a y : C *,* —} C *, * given b y wher e R(Xo, .,Xm,Ya, . . .,Yn) = (a(Xo, ., X m ; m =  (—1)a(Xo,  X ; Y 0 , . . . , Yn)b , iT a (a(Xo, .. . , X m ; Yo, . . . , Yn)b)a y n „ = E(-- 1 )ja(Xa, . . ., Xmy Yo, . . . Yj . . ., Y n)b . j= o It is straightforward to check that O s 2 = o, and = a y a s . W e think of a s as acting on the m -co-ordinate, horizontally to the left, an d a y as acting on the n-co-ordinate, vertically downwards . In summary , we have a large commuting diagram . Notice that the row with index n = -1 agrees with R(C * Y) . Defin e C*,+ =  El) C, . n} n, and let s s : C *,+ ~ C *,+ be given b y m7--1 n> 0 (a(Xo, . . . ,X m ;Yo, .. . , Yn»)s s = (_1)m+la(Xo, . . . , X, Yo n . . . nYn ;Yo, . . .,Yn)b , where the result is understood to be o whenever X m = Yo n • - - n Yn . I t is readily verified that s s is well-defined, and that s~r~~ + o~~sx acts as 108  WARREN DICKS, I . J . LEAR Y the identity on C,, ,+ . It follows that H * (C *,+ ) = O, which means that , for all n E N, the nth row is acyclic . Notice t hat the column wit h index m = -1 agrees with R (M * Y ) . Define C +, * = E) EE1 C m,n , and let s y :C +, . ~ C +, * be given b y m>Q rz> - 1 (a(Xo, . . .,Xm ;Yo, . . .,Yn)b)sy = a(XO, . . .,Xm ;Xm,Yo, . . . ,Yn)b , where the result is understood to be o whenever X m = Yo . It is readil y verified that s s is well - defined, and that s y a y + ~ y s y acts as the identit y on C +, * . It follows that, far all m E N, the mth columnis acyclic . We can interchange m and n, and apply Lemma 4 .5, to deduce that , if R(M * Y ) is acyclic, then R(C * Y) is acyclic also . If we consider the commuting subdiagram obtained by reducing C O Y A to ZoYA (or, indeed, to any set between Z O .Y Aand C O Y A ) throughout , then the maps s s and s y act on the subdiagram, and we again deduc e that if R (M * Y ) is acyclic, then R(Z * Y) is acyclic . Now consider the commuting subdiagram obtained by reducing C O Y A to ,J 'o Y A throughou t . Here sy acts on t he sub -diagram, so t he column s with non -negative index are acyclic . If any element of NIoY A is infinite , then sx does not act on the subdiagram . However, the rows in th e subdiagram are easily seen to consist of direct sums of copies of th e complexes R( .F * W), where W ranges over the family consisting of Y and sets of the form Yo n • • • n Y n , with Yo -< • • • -< Y n in M O Y A . W e proved, in Theorem 4 .4, that all such complexes R(F * W) are acyclic . Thus all the rows of the subdiagram are acyclic, so, by Lemma 4 .5 , R(M * Y) is acyclic. Now, by the two preceding paragraphs, we see tha t R(C * Y) and R(Z * Y ) are acyclic also . ■ 4 .7 . Remarks . There is a certain topological flavour to the abov e proof, and it is interesting to identify the sources . (i) The essence of the argument used in proving Theorem 4 .6 may b e viewed as an extension to non-constant coefficient systems of a theore m in simplicial homology provedby A . Weil [12], who also attributes a similar result to Leray . If an augmented simplicialcomplex S * is a union of subcomplexe s the nerve of this covering is the simplicial complex with vertices the S ,I z } , and simplices the finite collections with non -empty intersection . The main result of section 3 of [12] is that, if each non -empty intersec - tion of the is acyclic, then the homology (with constant coefficients ) of the nerve of the covering is isomorphic to the homology of S * . MIXED COPRODUCTS/TENSOR-PRODUCTS  10 9 To state the generalization of this to nonconstant coefficients, it i s more convenient to use (non -augmented) simplicial complexes, becaus e it is no longer clear what coefficient ob j ect should be assigned to th e simplex of dimension -1 . Given a poset (Po, -<), and a functor F from Po to an abelian category , one may define a chain complex with F-coefficients for the complex P *+ , where the simplex {po , . . . , pn } , with p o ~ - - • ~ p n , is assigned coefficien t ob j ect F ( p o ) . The homology of this chain complex,denoted H * (P *+ , F) , is called the homology of 7~ *+ with coefficients in F . If F is a constan t functor, then this is just the ordinary homology of P . If (Po,  is a union of subposets (Pr, ~ ) , such that every chain in Po is containe d in one of the Po i} , then the simplicial complex ~*+ is equal to the unio n of the subcomplexes PZ? . If each non -empty intersection of the poset s is F-acyclic (in the sense that if (Qo, is such an intersection , then H i (Q . + , F) = o for i ~ O), then an argument similar to that give n by Weil shows that, even in this generality, the homology of P*+ wit h coefficients in F may be calculated as the homology of a chain comple x associated to the nerve of the covering . This chain complexassociate s to each non -empty intersection (Qo,  of the (Pr, ~}, the coefficien t object Ho(Q *+ , F) . - We shall not prove the aboye result in its full generality here, becaus e we do not use it, and also because three special cases of it appear in ou r proof of Theorem 4 .6 . We may define a functor from Co Y A (resp . F 0 Y A , ZoY A ) to R -bimodules, which sends the subset Z to R OR(Z) R . The corresponding chain complex is then R(C *+ Y) (resp . R(F *+ Y ) , R(Z, k + Y ) } . Any chain in C O Y A (resp . .F O Y A , Z O Y A ) is contained in CO ZA (resp . .F O Z A , Z O Z A ) for some Z E .Jt / tgYA . Thus we may view ,1V1 *+ Y A as the nerve o f t he covering of C * + Y A (resp . .F *+ Y A , Z *+ Y A ) by its sub complexes whic h come from maximal complete subgraphs . Since an intersection of complete graphs is a complete graph, we may apply the generalization o f Weil ' s theorem, once we have proved that for any complete graph Y the chain complex R(C * + Y ) (resp . R( .1 ' *+ Y ), R(Z *+ Y ) } is acyclic . Fo r the case of R(,F *+ Y) this was done in the proof of Theorem 4 .4 . Th e proof of the claim for R(C *+ Y) and R(Z *+ Y ) is contained in the proo f of Theorem 4 .6 . The latter proof generalizes to arbitrary functors F as follows . If P o is any poset with a greatest element p, and F is any functor from Po t o an abelian category, then it is easily verified tha t Hn(p* +,F )  0 F(p) if r~ = o , ifrt~o . WARREN DICKS, 1 . J . LEAR Y On the other hand, the proof of acyclicity of R( .F * Y ) refies on the fac i that R( . T * —) preserves direct limits, so does not generalize to arbitrar y functors F . (ji) M .W . Davis used finitely generated Coxeter groups in [8] to con - struct interesting contractible CW - complexes . The barycentric subdivisions of his complexes give acyclic augmented chain complexes whic h are closely related to the acyclic complexes of the form R( C * X ) ; the two complexes are basically the same in the case of integral group rings o f finitely generated right-angled Coxeter groups . We now deduce consequences about pro jective dimensions . 4 .8 . Corollary . Suppose that Hypothesis 3 .1 holds . Then, for an y left R . -module M, there is an exact sequence of left R-module s (8) . . . ---+  ROR(xo) M ~ . . XoC•••CXn in Zo(XA) . . . -4  e RoR(xo) m m —> a , Xa ETo(XA ) so prof . dim . R M G sup{d(Y, X A ) + proj . dim . R(Y) M iY E ZoX A } . Thu s 1 . gl . dim . R < sup{d(Y, X A ) + 1 . gl . dim . R(Y) 1 Y E ZoX A } . Here pro j . dim . R( y ) M denotes the minimum of the lengths of projective R(Y)-resolutions of M, and 1 . gl . dim . R(Y) denotes the supremu m of the projective dimensions of left R(Y) - modules . Proof : We have the resolution (1), by Theorem 4 .6 . All the term s of (1) are flat as right R - modules, so (1) remains exact under — OR M , giving us the exact sequence (8) . Consider any Xo E Z O X A . Since R is flat as right R(Xo)- module, an y projective R(Xo)-resolution of M lifts, under R*R(xo) —, to a projectiv e R -resolution of R ►+ . R(Xo ) M . Thus we get certain pro j ective R -resolutions of all the terms of th e sequence (8), and these can be used to construct a double complex . The corresponding total complex is then a pro jective Rresolution of M , whose length i s sup {n + proj . dim . R( x 4) M ~ Xo c • • • c X~ in Z OXA } , and this gives the desired bounds . ■ The main motivation for our work was the result of Bergman [3], that , if Hypothesis 3 .1 holds and X A is finite, the n 1 . gl . dim . R Ç sup{IY ' — Y i +1 . gl . dim . R(Y) 1 Y C Y ' in C O X A } ; 110 MIXED COPRODUCTS/TENSOR-PRODUCTS  11 1 this implication follows from Corollary 4 .8, or even from Theorem 4 . 4 together with the argument used in the proof of Corollary 4 .8 . Let us record some of the consequences for the special case of grap h products of groups . 4 .9 . Corollary . Let (G(x) x E X) be a family of groups indexe d by X, and let G = G(X A ) . Then there there is an exact sequence of lef t KG-module s . . ~  . .  K [G/G(Xo}] -- ~ .. . XpC ... CX, 7 , in Za(X A ) . . —>  K[G/G(Xo) ] --> K ----> o . XaETo ()C A } Hence cd K G(XA) sup{d(Y, X A ) + cd K G(Y A ) Y E ZoX A } . Proo f : This is the case of Corollary 4 .8 where R(x) = K[G(x)], for al l x E X, and M = K~, where K E denotes the left K - module K made int o a left K[G(X A )]- module with trivial G(X A )-action . ■ 5 . Refining resolution s This section is devoted to describing how the resolution (1) given b y Theorem 4 .6 can be refined, by choosing a useful subresolution wit h split exact quotient . Throughout this section, let us suppose that Hypothesis 3 .1 holds, s o k is the principal ideal domain chosen in 3 .1 (vi) . 5 .1 . Definition . Let D . be a differential graded R-bimodule wit h differential a * . Suppose we have a decomposition of differential graded R-bimodule s = D *,1 D *,2 , and that D *,1 is split, which means that there exist s a graded R -bimodule endomorphism s * of D *,1 of degree -}-1, such tha t s * a * + a * s * = 1 on D *,1 ; we have already used Ç such maps to goo d advantage, in the proofs of Theorems 4 .4 and 4 .6 . Here s * a * s * a * = s*a* ( 1 — a *s *) = S * a * --- s *a*a *s* = s * a * + o = s * a * . Thus s * a * is idempotent, and so is a * s * = 1 — * 9 .0 . . In particular, s * a * and a * s * commute, and have product o, so (s8s) 2 = O . Now 1 = s * a * +a .S . = s *a*S*(~*+a*s*a*S* = ( .9 * a .s . ) a . + (s * 0 . s .) . Thus we can replace s * with s * a * s * , and so assume that s * a * s * = s * , and that s* = o . 112  WARREN DICKS,I . J . LEAR Y Let us extend s * to all of D * by specifying thatD *,2 s * = O . Then s * is an R -bimodule endomorphism ofD * , of degree +1, such that s* = 0 , s * a * s * = s * , D, k,1 = D * (s * a * + a * s * ), and D *,2 = D * ( 1 An endomorphism s * of degree +1 of a differential graded R -bimodul e such that s* = 0 and s * a * s * = s * will be called a component-contractin g homo top y . In this event, e = s * o ~ *+ c7 * s * is idempotent, and, as differen - tial graded R-bimodule, D * decomposes as D . e ~ D * (1 — e), such tha t D * e is split, with contracting homotopy s * . 5 .2 . Definition . Let M be a k-module . The rank of M over k , denoted r k k M, is the minimum of the cardinals of the k-generating set s of M . The relation rank of M over k, denotedrel -rk k M, is the minimu m of the ranks of kernels of surjective klinear maps from free kmodule s to M . By a minimal presentation of M we mean a short exact sequenc e of kmodules, 0 —> G ~ F —> M -+ 0, with F, G free, such tha t rel -rk k M = rk k G . In this event it is easy to see that rk k M = rk k F . The foliowing is well-known . 5 .3 . Lemma . Let k be a principal ideal domain . For any exac t sequence of kmodules 0—> G---} F--> M-4 0, such that G Ç F ar e free kmodules, there exist decompositions F-- H ~ F ' , G = H® G ' such that rk k F ' = rk k M and rk k G ' = rel -rk k M, and thus the quotien t presentation 0 —> G ' --+ F ' - --} M —> 0 is minimal . Proof : Let 0 -4 A --4 B --} M ---} 0 be any minimal presentation of M , with A C B free . Since F is free, there is a map F --~ B such tha t F / ~ B—>M—> o commutes . Let B' be the image of F —> B, so B ' is free, and B ' ~ M i s surjective . Let A ' be the kernel of B ' - -} M, so A ' C A, so the rank of A ' is at most the rank of A, so 0 —> A ' —> B ' —> M --} 0 is also a minima l presentation . Now 0  0  0 H' —> H --4 0 - ---} 0 G —> F --4 M -4 0 A' —> B' —> M ---} 0 1 0  0  0 0 ~ 0 ~ 0 . ~ MIXED COPRODUCTS/TENSOR-PRODUCTS  11 3 is a commuting diagram with exact columns and rows . Hence H' = H . Since the middle column splits, H has the desired properties . ■ 5 .4 . Definition . Let D . be a differential graded kmodule that i s free as kmodule . We now define a noncanonical component -contracting homotopy s * which reduces rk k D * as much as possible ; in particular, s * reduces D n to o whenever H(D) = o and H_ 1 (D) is free, as kmodule . Let n be an integer . cae have a presentation o --> Im a n+1 —> Ker On —> H(D) --} O . By Lemma 5 .3 we can choose decomposition s Ker an = An EE) Xn f Im an+1 = A n e Yn, .Y n Ç X n , such that rk k X n= rk k .FI n (D, k ) , rk k Yn = rel -rk k Hn (D * ) , and we the n have a minimal presentatio n o~Y n ~ n } X nHn(D * ) --> o , where i n is the inclusion map . Since D_ 1 is a free k-module, the image of a n : D n —+ 14_ 1 is a fre e kmodule, so we may choose a klinear isomorphis m D n Ker a n El) Im a n = An ~ X n ~ An -1~ Yn _ 1 . Thus we have a commuting diagra m r , D n +1  } An +l EE) Xn +1 ~ A .n ~ Y n 0 1 (0° 000 l i000 1 oin oo / D n ----~ A n EE) X n e) An — 1 E) Yn— • Let s n : D n —> D n + 1 be the map in the reverse direction that corresponds to ooi o 000 0 000 0 000 0 It is clear that sn s n + l = o and snan+1 sn = sn, so s * is a component -con - tracting homotopy, and H * (D * ) ,' :--, H * (D * (l — s * ~* — a * s * ) } , as grade d kmodules . Although s * depends on the choice of the decompositions, there is a t least one invariant, namely the graded k - module isomorphism class o f a n+ 120  WARREN DICICS, I . J . LEAR Y 5 .9 . Corollary . Let X A be finite-dimensional, and let (G(x) ~ x E X ) be a family of groups indexed by X, and let G = G(X A ) be their grap h product . Then there there is a finite exact sequence of left KG-module s • —> cn , Y K [ G I G ( Y )] —> . . . YEZo(XA) ▪  ▪ —> ED co ,Y K[GIG(Y)] ~ K —} o , YEZ4(XA ) where cn ,Y K[G/G(Y)] denotes a direct sum of c n (Y, X A , k) copies o f K[G/G(Y)] . Hence cd K G(X A ) c sup{hd k (Y, X A ) + cd K G(Y A ) ~ Y E ZoX A } . Bestvina [5], refining techniques of Davis [8], gives a similar result fo r finitely generated Coxeter groups, and there is some overlap with th e aboye result in the case of right-angled Coxeter groups . 6 . Normal forrns and modulefreenes s adapted from G . M . Bergman [3, Section 2 ] The results of this section shed some light on the structure of the ring s with which we have been dealing . Here we examine the following special case of Hypothesis 3 .1 . 6 .1 . Hypothesis . For each x E X, let R(x) be a K-ring given wit h a K -centralizing (right) K -basis B(x) U {1}, where 1 « B ( x ) . For eac h (x, y) E A, let B(x, y) = {ab — ba i a E B(x), b E B(y)} . For each subse t Y of X, let R(Y) be the quotient of II R(y) by the ideal generated b y yE Y the images of the B (x, y), (x, y) E A n (Y x Y) . The object of this section is to prove that, for any Y Ç X, R = R(X ) is free as right R(Y)-module on a basis containing 1 ; this is stronger tha n the conclusion of Proposition 3 .4 . The form taken by the basis of R R( y ) will show in particular that if, for each x E X, we are given a subse t C(x) of B(x) U {1}, such that 1 E C(x), and C(x) is the K -basis of a subring S(x) of R ( x ) , then 8(X), which is defined in the obvious way, i s naturally embedded in R(X ) . 6 .2 . Examples . (i) If K is a field contained in the center of eac h R(x), x E X, then R(x) obviously has a K -centralizing K -basis containing 1 . Thus the situation considered here includes the case of Construction 1 .2 where K is a field . MIXED COPRODUCTS/TENSOR-PRODUCTS  12 1 (u) In the situation of Construction 1 .3, set R(x) = K[G(x)] and B(x) = G(x) — {1} . Then B(x) U {1} is a K -centralizing K -basis of R(x ) and R(X) = K[G(X A )] . Further, if, far each x E X, C(x) is a submonoid of the group G(x) , then C(x) is the K -basis of a subring S(x) = K[C(x)] of R(x), and S ( X ) = K[C(X A )], where C(X A ) is defined in the obvious way, so th e aboye statement implies that C(X A ) Ç G(X A ) . Let B denote the disjoint union of the B(x) . Each element b E B wil l be said to be "associated to" the index x E X such that b E B(x) . Le t B * denote the free monoid on the set B . Since R is generated by the images of the R(x), it will be spanne d as right K - module by the products of the images of the elements o f B (counting the empty product, 1), i .e . by the natural image of B* . We shall call these products monomials, and denote them by the sam e symbols as the elements of B* of which they are images, though the ma p B*  R is generally not oneto -one . But we will be careful to distinguish between speaking of two mono - mials as being "equal in R " , and being "equal " , which will mean "equa l in B* " Note that if a monomial b1 • • • b n has two succesive terms bi, bi+1 bot h associated with the same index x E X, then, by writing the produc t bib i + l E R(x) as a K-linear combination of elements of B ( x ) U{ 1}, w e can reduce b 1 • • • b 7z in R to a K-linear combination of monomials o f shorter length . More generally, if b1 - • - b 7z has two terms b i and b i ( i C j ) associated with the same index x E X, and if . all terms b k occurin g between these ( i .e ., i < k < j) are associated with indices y such tha t (x, y) E A, then, in R, we can commute b i past these terms until it i s adjacent to and then reduce our monomial as aboye to a K-linea r combination of shorter monomials . We deduce that R will be spanned as a right K-module by those mono - mials b 1 • • - b , z with the property that any two terms bi and b ; therein , that are associated with the same index x E X, are separated by at leas t one intermediate term b k associated with an index y such that (x, y) 1 A . W e shall call such b 1 - - • bn " acceptable monomials " , and denote the se t of acceptable monomials S Ç B* . An acceptable monomial can still have adjacent terms bi b i+1 associate d with indices x and y such that (x, y) E A, and in this case it will b e equal in R to the (also acceptable) monomial obtained by transposin g these terms . To obtain invariants of acceptable monomials under suc h transpositions, let us associate to any acceptable monomial b 1 . - - bn a partial ordering of its terms, setting 122  WARREN DI C KS, I . J . LEAR Y (10) b i ~ b i if i C j and there exists a sequenc e i =m 1 C . . .CmgC . . . CrïmT= j such that, writing ind(q) for the index associated with b ryn q , w e have (ind(q), ind(q + 1) ) 1 A for all q C r . We are being sloppy in our notation, since a monomial may repea t terms of B, so that it is not really the terms b i that are being partiall y ordered, but, if you will, their subscripts i ; or, if you prefer, the pair s (i, bi ) . In any case, the point is that we obtain from our monomial a finit e partially ordered set, with its vertices labeled with certain elements o f B, possibly with repetitions . This partially ordered set will (by (10) and the definition of acceptable monomial) have the properties that any tw o vertices labelled with elements of B associated to indices x, y such tha t (x, y) 1 A must be related under our ordering (one -< the other ; note tha t this includes the case x = y) ; and when one vertex covers another (is a minimal vertex >- than it ) , the associated indices in X must be distinct . 6 .3 . Lemma . Let w — b 1 • • • b, and w' = ¿I I - • - b ; n be acceptabl e rnonomials of the same length . Then the following conditions are equivalent : (a) w ' can be obtained from w by a series o f transpositions o f adjacen t terms bi, bi + 1 associated to indices x, y such that (x, y) E A . (b) There is an isomorphism between the partially ordered sets associated with these two monomials, which preserves the B -labels o n the vertices . Equivalently : there exists a permutation rr E Sym n such that for all i, bi = b, (i) , and far all i, j if bi ~ b ' .i in w ' the n b„(i) --< b 7 ( .i) in w . Further, when these condicions hotd, the isomorphism of (b ) (equivalently, rr ) is unique . Broof . ( ~ (b) : We easily see that each transposition leaves th e isomorphism class of the B -labeled partially ordered set unchanged . (b) ~ (a) : If rr is not the identity, there will be some i such tha t 7r(i) ~ 7r (i + 1) . We see that b i and bi+ 1 must be unrelated under - < (otherwise rr would not respect the partial ordering) , so they must b e associated with a pair of indices (x, y) E A . Hence we may transpos e them, transforming w to a monomial the order of whose terms is " closer " to that of w ' (fewer pairs of terms b i , b i occuring in different orders) . Iterating this procedure, we see that w will be transformed in a finit e number of steps into w ' . MIXED COPRODUCTS/TENSOR-PRODUCTS  12 3 To see the last assertion of the lemma, note that in our partially ordered sets, any two vertices bearing the same label in B must be relate d under --< . Since the sets are finite, there cannot therefore be more tha n one order-preserving and label-preserving bijection . ■ Let us write w w ' if the equivalent conditions of the aboye lemm a hold . This gives an equivalence relation on the set S of acceptable mono - mials . We shall write 5~- - ► = S ' , and represent the equivalence class o f w by [w] E S ' . Clearly, the map S —> R factors through S' . We shal l soon see that this map is one -to -one, and its image is a K -centralizin g K -basis of R . But first we need a result on the structure of S ' . For any subset Y of X , let us define SY to be the set of al l [b 1 • • • bn] E S ' such that all b i E U B (x) . Let us also define S ; Y t o xE Y be the set of all [w] E S ' such that in the partially ordered set associate d with [w], no maximal vertex is labeled with a member of any B(y) wit h y E Y . We note that the maximal vertices of the partially ordered se t associated with [w] correspond to those terms that can be transpose d to the rightmost position . (E .g ., if (x, y), (x, z) E A, y z, a E B(x) , b E B(y), c E B(z), then, in the partially ordered set associated wit h [abc], a and c are both maximal . ) Note that if [w1] E S ;y and [w2] E Sy then w1 w2 will be an acceptabl e monomial. Further, [w i w 2 ] will be determined by the equivalence classe s [w l ] and [w 2 ], since any transposition of terms that can be performed i n the latter elements can certainly be duplicated in the product . In fact , we have : 6 .4 . Lemma . Let Y C X . Then for any element [w] E S ' there exis t unique elements [w 1 ] E  [w 2 ] E SY such that [w] = [w 1 w 2 ] . Proof : To get the existence of such a decomposition, look for a maximal term of w associated with an index in Y ; if there is one, transpose i t to the last position . Then treat the remaining string of terms (shorter b y one) the same way . It may contain maximal terms that were not maximal in the original element, because they were covered by the first ter m extracted . Iterate until we are left with a string w 1 (possibly empty ) with no maximal terms associated with an index in Y, followed by a string w 2 with all terms associated to indices in Y . To get uniqueness, note that w2 must consist precisely of those term s of w which are associated to indices in Y, and are not ~ any terms wit h indices in X --- Y . ■ We can now prove : 124  WARREN DICKS, I . J . LEAR Y 6 .5 . Proposition . l f Hypothesis 6 .1 holds, then the images in R of the distinct elements of S ' are distinct, and form a K -centralizin g K -basis of R . Proof : Let M be a free right K - module on the basis S ' . We shal l show that M may be made a right R-module in a natural way, and tha t the actions on this module of distinct elements of S ' are right K -linearl y independent . (The idea of this trick goes back to van der Waerden, cf . [4, Section 11 .2, (28")]) . For each x E X, t ake Y = {x} in Lemma 6 .4 . Every member of S { s } i s of the form [b] (b E B(x)), or [1], so the lemma says that we get a bijectio n S~{x} x (B(x)U{1}) —> S ' , given by ([w i ],b) i—> [w 1 b] . But B(x)U{1} is a right K -basis for R(x), so this decomposition allowsus to give the fre e right K-module M on S ' a structure of free right R(x)-module on th e basis S~ { z } , extending the given right K - module structure . Doing thi s for all x E X, we get a structure of right II R(x)-module . ac E X Now t ake any (x, y) E A . We see that S {~ ,y} will consist of element s [ww] = [ww], w x E B(x)U{1}, w y E B(y)U{1} . (Because element s of B(x) and B(y) are transposable with one another, we can form fro m them no acceptable monomial of length greater than two, by the definition of acceptable monomial . } This gives a bi j ect io n S ,' { x ,y} x (B(x)U{1}) x (B(y)U{1}) -~ S ' . Since (B(x) U {1}} x (B(y) U {1}} commutes with K, we see that S, {s,y } is a right K -basis of R( {x, y } ) . Thus we may define a structure of (free ) right R( {x, y})-module on M which, clearly, extends the structures o f right R(x)- and R(y)-module already defined . This means that th e actions of B(x) and the actions of B(y) on M must commute wit h one another . Since this is so for all pairs (x, y) E A, our righ t ~ R(x)-module structure must in fact give a right R(X)-module strucx E X ture, by the definition of R(X ) . Now note that for any [b ] • • • bn ] E S ' , the product b i • • • b n E R , acting on the element [1] E M, will give [b l • • - bn ] E M . It follovcrs tha t any non-trivial K-linear combination in R of the images of elements o f S' are K -Iinearly independent . Since we already know they span R, thi s completes theproof of the proposition . ■ We can now get our desired freeness result from Lemma 6 .4 . 6 .6 . Proposition . lf Hypothesis 6.1 holds then, for anysubset Y o f X, R(X) is free as a right module over R(Y), with basis the (faithful ) image o f S,Y . MIXED COPRODUCTS/TENSOR-PRODUCTS  12 5 Acknowledgements . W e thank George Bergman for the ideas and extensive contributions which were fundamental to this work, and fo r permission to include section 6, which closely follows [3, Section 21 . We also thank Michael Barr for suggesting, in 1976, that an exampl e based on the pro jective plane should yield 2 -torsion ; we took a 16 vertex, full, triangulation of the projective plane, took the dual polygona l tessellation, completed each n -gon to an (n - - 1)-simplex, and took th e 1 -skeleton, a graph with 32 vertices . The O - component of the correspond - ing refined resolution was worked out by the Bedford College Compute r Centre between March 1977 and June 1978 ; it was found to be the desired multiplication -by-2 map, as in Example 2 .5- . We are grateful t o the programmers Tom Lake and Phil Taylor for their intense efforts . Bestvina's recent, elegant, approach [5, Remark (3)], shows that one ca n simply take the 1 -skeleton of a full triangulation ; this led us to Exampie 2 .5, which does not require a computer . We thank P . H . Kropholle r for illuminating conversations concerning Bestvina's article . The authors gratefully acknowledge that this research was generousl y funded by the DGICYT, through grant PB90 -0719 for the first-name d author, and a post-doctoral fellowship held at the Centre de Recerc a Matemática for the second-named author . Reference s 1. Y .-G . BAIK, J . HoWIE, S . J . PRIDE, The identity problem fo r graph products of groups, J . Algebra 162 (1993), 168-177 . 2. G . M . BERGMAN, Modules over coproducts of rings, Trans . Amer . Math . Soc . 200 (1974), 1-32 . 3. G . M . BERGMAN, The global dimension of mixed coproduct/tenso r product algebras, (Unpublished note, 1976, 19 pages) . 4. G . M . 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WEIL, Sur les théorémes de de Rham, Comment . . Math . Helv . 26 (1952), 119-145 . Warren Dicks :  I . J . Leary : Departament de Matemàtiques  Centre de Recerca Matemátic a Universitat Autònoma de Barcelona  Apartat 5 0 08193 Bellaterra (Barcelona)  08193 Bellaterra (Barcelona ) SPAIN  SPAI N Primera versis rebuda el 7 de Juliol de 1993 , darrera versis rebuda el 10 de Gener de 1994