Partial resolutions in monoid cohomology

Partial resolutions are constructed for the EilenbergMacLane cohomology of monoids, with applications to examples.

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Опубліковано в: :	Algebra and Discrete Mathematics
Дата:	2004
Автор:	Grillet, P.A.
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Мова:	English
Опубліковано:	Інститут прикладної математики і механіки НАН України 2004
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Цитувати:	Partial resolutions in monoid cohomology / P.A. Grillet // Algebra and Discrete Mathematics. — 2004. — Vol. 3, № 4. — С. 12–31. — Бібліогр.: 9 назв. — англ.

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spelling	Grillet, P.A. 2019-06-18T17:31:43Z 2019-06-18T17:31:43Z 2004 Partial resolutions in monoid cohomology / P.A. Grillet // Algebra and Discrete Mathematics. — 2004. — Vol. 3, № 4. — С. 12–31. — Бібліогр.: 9 назв. — англ. 1726-3255 2000 Mathematics Subject Classification: 20M50. https://nasplib.isofts.kiev.ua/handle/123456789/156593 Partial resolutions are constructed for the EilenbergMacLane cohomology of monoids, with applications to examples. en Інститут прикладної математики і механіки НАН України Algebra and Discrete Mathematics Partial resolutions in monoid cohomology Article published earlier
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
title	Partial resolutions in monoid cohomology
spellingShingle	Partial resolutions in monoid cohomology Grillet, P.A.
title_short	Partial resolutions in monoid cohomology
title_full	Partial resolutions in monoid cohomology
title_fullStr	Partial resolutions in monoid cohomology
title_full_unstemmed	Partial resolutions in monoid cohomology
title_sort	partial resolutions in monoid cohomology
author	Grillet, P.A.
author_facet	Grillet, P.A.
publishDate	2004
language	English
container_title	Algebra and Discrete Mathematics
publisher	Інститут прикладної математики і механіки НАН України
format	Article
description	Partial resolutions are constructed for the EilenbergMacLane cohomology of monoids, with applications to examples.
issn	1726-3255
url	https://nasplib.isofts.kiev.ua/handle/123456789/156593
citation_txt	Partial resolutions in monoid cohomology / P.A. Grillet // Algebra and Discrete Mathematics. — 2004. — Vol. 3, № 4. — С. 12–31. — Бібліогр.: 9 назв. — англ.
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fulltext	Algebra and Discrete Mathematics RESEARCH ARTICLE Number 4. (2004). pp. 12 – 31 c© Journal “Algebra and Discrete Mathematics” Partial resolutions in monoid cohomology P. A. Grillet Communicated by B. V. Novikov Abstract. Partial resolutions are constructed for the Eilenberg- MacLane cohomology of monoids, with applications to examples. Introduction The Eilenberg-MacLane cohomology groups of a monoid S are usually computed from projective resolutions of the trivial S-module Z, as com- putation from cocycles tends to be very unwieldy [6], [7], [8]. When S has a complete (Church-Rosser) and reduced presentation, Squier’s par- tial resolution [9] leads to simpler computations in dimensions n ≤ 2 (in some cases, n ≤ 3 [5]). When S is commutative, its commutative cohomology groups are like- wise difficult to compute, but the overpath method, introduced in [4], yields markedly simpler computations of H2 by cocycles, directly from presentations of S [2], [3]. It turns out that the two results are closely related. When the over- path method is adapted to Eilenberg-MacLane cohomology, forsaking commutativity and using cycles rather than cocycles, the result is a par- tial resolution, constructed in Section 1, which is very similar to Squier’s but applies to any presentation. Squier’s resolution is retrieved in Section 2, with minor modifications, if the presentation is complete and reduced, or almost reduced. Section 3 sets up the computation of H0, H1, H2 by cocycles. Section 4 computes H2 for five examples: finite cyclic groups and monoids; the bicyclic semigroup; the free commutative monoid on two generators; the 2000 Mathematics Subject Classification: 20M50. P. A. Grillet 13 monoid freely generated by two idempotents; and the free band with identity on two generators. The author is very thankful to Prof. B.V. Novikov for timely advice and suggestions. 1. First resolution 1. In what follows, S is an arbitrary monoid, determined by a presentation as the the quotient of the free monoid F on a set X (often denoted by X∗) by the congruence generated by a binary relation R on F (often called a rewrite system). The elements of R are ordered pairs r = (r′, r′′) of elements of F . We denote the identity elements of S and F by 1, and the projection F −→ S by a 7−→ ā. In F , a connecting sequence from a ∈ F to b ∈ F consists of sequences a0, a1, . . . , an, u1, . . . , un, v1, . . . , vn of elements of F and a sequence r1, . . . , rn of elements of R, such that n ≥ 0, a = a0, an = b, and, for every 1 ≤ i ≤ n, either ai−1 = ui r ′ i vi and ai = ui r ′′ i vi, or ai−1 = ui r ′′ i vi and ai = ui r ′ i vi. This definition is justified by the description of the congruence ∼ generated by R: namely, a ∼ b (ā = b̄) if and only if there exists a connecting sequence from a to b. In this section we construct a partial projective resolution M3 ∂3−→ M2 ∂2−→ M1 ∂1−→ M0 ǫ −→ Z of the trivial §-module Z, which resembles Squier’s resolution [9] but has a different module M3 and requires no hypothesis on R. As in [9], M0 = Z[S]; M1 is the free §-module with one basis element [x] for each generator x ∈ X of F ; ǫ : Z[S] −→ Z is the augmentation homomorphism ǫ ( ∑ s∈S nss s ) = ∑ s∈S nss; and ∂1 : M1 −→ Z[S] is the module homomorphism such that ∂1 [x] = x̄− 1 for all x ∈ X; equivalently, the additive homomorphism such that ∂1 s[x] = sx̄− s. (1) Lemma 1.1. [9] Im ∂1 = Ker ǫ. Proof. First, ǫ ∂1 [x] = ǫ (x̄− 1) = 0 for all x ∈ X, so that Im ∂1 ⊆ Ker ǫ. For the converse we construct some maps which will be used later. 14 Partial resolutions in monoid cohomology The expansion mapping ξ : F −→M1 is defined by: ξ (x1 x2 . . . xn) = ∑ 1≤i≤n ¯x1 x2 . . . xi−1 [xi] (2) for all x1, x2, . . . , xn ∈ X (in particular, ξ x = [x] and ξ1 = 0). We see that ∂1 ξ (x1 x2 . . . xn) = x̄1−1+ ∑ 2≤i≤n ( ¯x1 x2 . . . xi− ¯x1 x2 . . . xi−1 ) = ¯x1 x2 . . . xn − 1, so that ∂1 ξ a = ā− 1 (3) for all a ∈ F . For each s ∈ S choose any representative word wss ∈ F such that w̄ss = s. Let σ0 : Z[S] −→M1 be the additive homomorphism such that σ0 s = ξwss (4) for all s ∈ S. Let ζ : Z −→ Z[S] be the additive homomorphism ζ(n) = n1. By the above, ∂1 σ0 s + ζ ǫ s = w̄ss− 1 + 1 = s for all s ∈ S, so that ∂1 σ0 + ζ ǫ is the identity on Z[S]. Hence Ker ǫ ⊆ Im ∂1. 2. As in [9], M2 is the free §-module with one basis element [r] for each r ∈ R; ∂2 : M2 −→M1 is the module homomorphism such that ∂2 [r] = ξr′ − ξr′′ (5) for all r ∈ R, where ξ is the expansion mapping above. To study M2 we construct another trace map τ . Let P be a connecting sequence, consisting of sequences a0, a1, . . . , an, u1, . . . , un, v1, . . . , vn of elements of F and a sequence r1, . . . , rn of elements of R, such that n ≥ 0, a = a0, an = b, and, for every 1 ≤ i ≤ n, either ai−1 = ui r ′ i vi and ai = ui r ′′ i vi, or ai−1 = ui r ′′ i vi and ai = ui r ′ i vi. Then τ P = ∑ 1≤i≤n ǫi ūi [ri], (6) where ǫi = +1 if ai−1 = uir ′ ivi, ai = uir ′′ i vi, ǫi = −1 if ai−1 = uir ′′ i vi, ai = uir ′ ivi. Since ξ(ab) = ξa + ā ξb for all a, b ∈ F , we have, in the above, ξai−1 = ξuir ′ ivi = ξui + ūi ξr ′ i + ¯uir′i ξvi, ξai = ξuir ′′ i vi = ξui + ūi ξr ′′ i + ¯uir′′i ξvi, P. A. Grillet 15 or vice versa; since ¯uir′i = ¯uir′′i this yields ξai−1 − ξai = ǫi ūi (ξr ′ i − ξr′′i ) = ǫi ūi ∂2[ri] and ξa− ξb = ∑ 1≤i≤n (ξai−1 − ξai) = ∂2τP, (7) for every connecting sequence P from a to b. Lemma 1.2. [9] Ker ∂1 = Im ∂2. Proof. By (3), (5), ∂1 ∂2[r] = ∂1 ξr′ − ∂1 ξr′′ = r̄′ − r̄′′ = 0 for all r ∈ R; hence Im ∂2 ⊆ Ker ∂1. To prove the converse we expand the partial contracting homotopy σ. For every s ∈ S and x ∈ X choose one arbitrary connecting sequence Ps,x from wss x to ws x̄. Let σ1 : M1 −→ M2 be the additive homomorphism such that σ1 s[x] = τPs,x (8) for all s ∈ S and x ∈ X. Then ∂2 σ1 s[x] + σ0 ∂1 s[x] = ∂2 τPs,x + σ0(sx̄− s) by (8), (1) = ξ(wss x)− ξws x̄ + ξws x̄ − ξwss by (7), (4) = ξ(wss x)− ξwss = w̄ss[x] = s[x], by (2), so that ∂2 σ1 + σ0 ∂1 is the identity on M1. Hence Ker ∂1 ⊆ Im ∂2. 3. Coterminal connecting sequences are connecting sequences P , Q from the same a ∈ F to the same b ∈ F . M3 is the free S-module generated by all ordered pairs [P, Q] of coterminal connecting sequences; ∂3 is the module homomorphism such that ∂3[P, Q] = τQ− τP, (9) where τ is the trace map above. (One may further assume that the generators of M3 satisfy [P, P ] = 0 and [Q, P ] = −[P, Q].) Lemma 1.3. Ker ∂2 = Im ∂3. Proof. By (7), ∂2 τP = ξa − ξb when P is a connecting sequence from a to b. If now P and Q are connecting sequences from a to b, then ∂2 ∂3[P, Q] = ∂2(τQ− τP ) = 0; hence Im ∂3 ⊆ Ker ∂2. For the converse inclusion we use some properties of the trace map τ . Combining a connecting sequence P from a to b with a connecting sequence Q from b to c yields a connecting sequence P + Q from a to c; 16 Partial resolutions in monoid cohomology if P consists of sequences a0, a1, . . . , an, u1, . . . , un, v1, . . . , vn ∈ F and a sequence r1, . . . , rn ∈ R, and Q consists of sequences b0, b1, . . . , bm, t1, . . . , tm, w1, . . . , wm ∈ F and a sequence s1, . . . , sn ∈ R, then P + Q consists of the sequences a0, a1, . . . , an = b0, b1, . . . , bm; u1, . . . , un, t1, . . . , tm; v1, . . . , vn, w1, . . . , wm; and r1, . . . , rn, s1, . . . , sn. We see that τ (P + Q) = τP + τQ. (10) Reversing a connecting sequence P from a to b yields a connecting se- quence −P from b to a; if P consists of sequences a0, a1, . . . , an, u1, . . . , un, v1, . . . , vn ∈ F and r1, . . . , rn ∈ R, then −P consists of an, an−1, . . . , a1, a0, un, . . . , u1, vn, . . . , v1 and rn, . . . , r1. We see that τ (−P ) = − τP. (11) When c, d ∈ F and P is a connecting sequence from a to b, consisting of sequences a0, a1, . . . , an, u1, . . . , un, v1, . . . , vn ∈ F and a sequence r1, . . . , rn ∈ R, then cPd is a connecting sequence from cad to cbd, consisting of sequences ca0d, ca1d, . . . , cand, cu1, . . . , cun, v1d, . . . , vnd, and the same r1, . . . , rn. We see that τ (cPd) = c̄ τP. (12) For every a = x1 x2 . . . xn ∈ F , where x1, x2, . . . , xn ∈ X, we have a connecting sequence Psa = P1,x1 x2 . . . xn + Px̄1,x2 x3 . . . xn + · · ·+ P ¯x1...xn−1,xn (13) from a to wā obtained by combining the connecting sequences P1,x1 from x1 to wx̄1 , Px̄1,x2 from wx̄1 x2 to w ¯x1x2 , P ¯x1x2,x3 from w ¯x1x2 x3 to w ¯x1x2x3 , . . . , P ¯x1...xn−1,xn from w ¯x1...xn−1 xn to w ¯x1...xn . By the above, τPsa = τP1,x1 + τPx̄1,x2 + · · ·+ τP ¯x1...xn−1,xn = σ1 ξa. (14) For every r ∈ R there is also a connecting sequence, which may be denoted by r, from r′ to r′′, with trace τr = [r]. This yields two con- necting sequences Pr′ and r + Pr′′ from r′ to wr̄′ = wr̄′′ . We can now extend our partial contracting homotopy with the module homomorphism σ2 : M2 −→M3 such that σ2[r] = [Pr′ , r + Pr′′ ]. (15) For every r ∈ R, ∂3 σ2[r] + σ1 ∂2[r] = τr + τPr′′ − τPr′ + σ1 ξr′ − σ1 ξr′′ = [r], by (9), (5), and (14). Hence Ker ∂2 ⊆ Im ∂3. P. A. Grillet 17 Since M0, M1, M2, M3 are free Z[S]-modules, Lemmas 1.1, 1.2, 1.3 yield Theorem 1.4. M3 −→M2 −→M1 −→M0 −→ Z is a partial projective resolution of the trivial §-module Z. 2. Squier’s resolution 1. In F , a path P from a ∈ F to b ∈ F consists of sequences a0, a1, . . . , an, u1, . . . , un, v1, . . . , vn of elements of F and a sequence r1, . . . , rn of elements of R, such that n ≥ 0, a = a0, an = b, and, for every 1 ≤ i ≤ n, ai−1 = ui r ′ i vi and ai = ui r ′′ i vi (always going from r′ to r′′); P is trivial if n = 0, simple if n = 1. We write a P −→ b when P is a path from a to b, a −→ b when there is a path from a to b (the latter is often written a ∗ −→ b, with a −→ b denoting a simple path). A connecting sequence from a to b (as in Section 1) can be decomposed into paths running in alternating directions: a ←− . −→ . ←− . −→ . ←− . −→ b, with the first and last paths possibly trivial. A presentation, or its set R of defining relations, is terminating or Noetherian if there is no infinite sequence . −→ . −→ . . . −→ . −→ . . . of simple paths; equivalently, no infinite sequence . −→ . −→ . . . −→ . −→ . . . of nontrivial paths. In a terminating presentation, there is for every a ∈ F a path a −→ w in which w is irreducible, that is, there is no simple path w −→ c; equivalently, there is no equality w = ur′v with r ∈ R. A wedge is a pair of paths a −→ b and a −→ c. A diamond is a quadruple of four paths a −→ b −→ d, a −→ c −→ d. a �� >> >> >> >> b �� == == == == c �� d R is confluent if every wedge can be completed to a diamond: if for every paths a −→ b and a −→ c there exist paths b −→ d and c −→ d. R is complete if it is terminating and confluent. R is Church-Rosser if for every a, b ∈ F such that ā = b̄ there exist paths a −→ c, b −→ c. A terminating presentation is Church-Rosser if 18 Partial resolutions in monoid cohomology and only if it is complete, if and only if for every a ∈ F there is a unique irreducible w ∈ F with a path a −→ w. We call R semi-reduced if r′ = us′v implies r′ = s′ when r, s ∈ R; R is reduced if R is semi-reduced and r′′ is irreducible for every r ∈ R [9]. Every terminating Church-Rosser presentation is equivalent to a reduced terminating Church-Rosser presentation [9]. Squier’s resolution assumes a reduced terminating Church-Rosser presentation, and can therefore be applied, after reduction, to any terminating Church-Rosser presentation. 2. Presentations with some of these properties are readily constructed from existing presentations by means of order relations on F . When < is a strict order relation on F , a presentation is ordered (by <) if r′ > r′′ for every r ∈ R. If < is compatible (if a < b implies uav < ubv for all u, v ∈ F ), then paths are descending (a −→ b implies a ≥ b). Proposition 2.1. Every free monoid F has a compatible well order. Rel- ative to any such: (a) every monoid presentation R ⊆ F ×F is equivalent to an ordered presentation; every ordered presentation is terminating; (b) every ordered monoid presentation is equivalent to a complete, ordered presentation; (c) every complete, ordered monoid presentation is equivalent to a semi-reduced complete, ordered presentation; (d) every semi-reduced, complete, ordered monoid presentation is equiv- alent to a reduced complete, ordered presentation. Parts (c) and (d) are due to [9]. Part (a) has a converse of sorts: any terminating presentation is ordered for the corresponding order relation a > b if and only if there exists a nontrivial path a −→ b, which is a compatible strict partial order relation on F ; moreover, the descending chain condition holds in F . Proof. The generating set X can be well ordered; words of fixed length can be well ordered lexicographically; then a > b if and only if either \|a\| > \|b\|, or \|a\| = \|b\| and a > b (where \|a\| denotes the length of a) is a compatible well order on F . (a) Given R ⊆ F × F , we can delete from R all trivial pairs (w, w), and replace (r′, r′′) ∈ R by (r′′, r′) whenever r′ < r′′. This yields a presentation which is ordered and equivalent to R. Ordered presentations are terminating since the descending chain condition holds in F . P. A. Grillet 19 (b) Given an ordered presentation R, let R̄ be the union of R and the set of all r = (r′, r′′) ∈ F × F such that r′ > r′′ and there exists a wedge x −→ r′, x −→ r′′ which cannot be completed to a diamond. Since x −→ r′, x −→ r′′ implies x̄ = r̄′ = r̄′′, R and R̄ generate the same congruence and are equivalent. Moreover, R̄ is ordered by definition, and every wedge a −→ b, a −→ c which cannot be completed to a diamond in R yields some r ∈ R̄ and can be completed to a diamond in R̄: to a −→ b 1,(b,c),1 −−−−−→ c, a −→ c −→ c if b > c, to a −→ b −→ b, a −→ c 1,(z,y),1 −−−−−→ b if b < c. Starting with R0 = R, construct Rn by induction as Rn+1 = R̄n, and let Rω = ⋃ Rn. The congruence generated by R contains R1, R2, . . . , and Rω; hence R and Rω are equivalent. Moreover, Rω is ordered, hence terminating, and a wedge of Rω contains only finitely many edges, is a wedge of some Rn, and can be completed to a diamond of Rn+1; hence Rω is complete. (c),(d) Given a complete presentation R, the proof of Theorem 2.4 of [9] first constructs a semi-reduced complete presentation R′ which is equivalent to R, by deleting some pairs from R (namely, all s such that s′ = ur′v for some r ∈ R and u, v ∈ F not both equal to 1). If R is ordered, then so is R′. From R′ the second part of the proof then constructs a reduced complete presentation R′′ which is equivalent to R′, and hence to R, which consists of pairs (u, v) with a nontrivial path u −→ v in R′ (namely, all pairs (r′, w) such that r ∈ R′, there is a nontrivial path r′ −→ w, and w is irreducible); if R′ is ordered then so is R′′. Parts (a), (c), and (d) hold verbatim for finite presentations [9], but not part (b): it is not true that every finite ordered presentation is equiv- alent to a finite complete ordered presentation; [9] provides a counterex- ample. 3. We now let R be a semi-reduced complete presentation and obtain Squier’s resolution M ′ 3 ∂3−→ M2 ∂2−→ M1 ∂1−→ M0 ǫ −→ Z from a suitable submodule M ′ 3 of M3. An essential wedge is a wedge a 1,r,v −−−→ b, a u,s,1 −−−→ c, with r, s ∈ R, u, v ∈ F , r′ = ut, s′ = tv for some t ∈ F , t 6= 1, and u, v 6= 1 in case r = s; then a = r′v = us′ = utv is an essential word. For each essential wedge a −→ b, a −→ c we choose any one diamond a −→ b −→ d, a −→ c −→ d (for instance, with d irreducible, as in [9]). An essential diamond is one of these chosen diamonds. Thus every essential wedge can be completed 20 Partial resolutions in monoid cohomology to an essential diamond. The two paths a −→ b −→ d, a −→ c −→ d in an essential diamond are an essential pair of paths. M ′ 3 is the submodule of M3 (freely) generated by all essential pairs [P, Q] of paths. (As before one may further assume that [P, P ] = 0 and [Q, P ] = −[P, Q].) Lemma 2.2. When R is a semi-reduced complete presentation, ∂3(M ′ 3) = Ker ∂2. Proof. By Lemma 1.3 it suffices to show that () τQ− τP ∈ ∂3(M ′ 3) for every coterminal paths P , Q. We note the following: (a) () holds for every essential diamond, by definition of M ′ 3. (b) if () holds for P and Q, then () holds for T + P and T + Q, for P + U and Q + U , and for uPv and uQv; (c) c if in the diagram of paths a P �� Q �� ?? ?? ?? ?? b T �� == == == == c U �� V �� == == == == d W �� == == == == e Y �� f () holds for the two inner diamonds, then () holds for the outer dia- mond. Claim 1. Every wedge a p,r,q −−−→ b, a u,s,v −−−→ c of simple paths can be com- pleted to a diamond for which () holds. We have a = pr′q = us′v, so that r′ and s′ are subwords of a. If r = s, then we may assume that p 6= u and q 6= v, otherwise b = c and adding trivial paths a p,r,q −−−→ b −→ b, a u,s,v −−−→ b −→ b yields a diamond for which () holds. If the subwords r′ and s′ do not overlap in a, say, a = pr′ts′v for some t ∈ F , there is a diamond a p,r,ts′v −−−−→ b pr′′t,s,v −−−−−→ d, a pr′t,s,v −−−−→ c p,r,ts′′v −−−−−→ d, where d = pr′′ts′′v, for which () holds: with P = p r ts′v + pr′′t s v, Q = pr′t s v + p r ts′′v we have τQ− τP = ¯pr′t[s] + p̄[r]− p̄[r]− ¯pr′′t[s] = 0 P. A. Grillet 21 since ¯pr′t = ¯pr′′t in S. Now assume that the subwords r′ and s′ overlap in a. Since R is semi- reduced, neither of r′, s′ is a proper subword of the other. Hence the end of one overlaps the beginning of the other. Let t 6= 1 be the common part. If the end of r′ overlaps the beginning of s′, then r′ = ht, s′ = tk, u = ph, q = kv, a = phtkv; if r = s, then h, k 6= 1, since in that case we may assume that p 6= u and q 6= v. Then htk 1,r,k −−−→ r′′k, htk h,s,1 −−−→ hs′′ is an essential wedge, and can be completed to an essential diamond htk 1,r,k −−−→ r′′k P ′′ −−→ d, htk h,s,1 −−−→ hs′′ Q′′ −−→ d. Hence the given wedge a p,r,q −−−→ b, a u,s,v −−−→ c, in which q = kv, b = pr′′kv, u = ph, c = phs′′v, and a = phtkv, can be completed to a diamond phtkv p,r,kv −−−→ pr′′kv pP ′′v −−−→ pdv, phtkv ph,s,v −−−→ phs′′v pQ′′v −−−→ pdv for which () holds by (b), since it holds for the essential diamond. The case where the end of s′ overlaps the beginning of r′ is similar. Claim 2. () holds for all paths a P −→ w and a Q −→ w where w is irre- ducible. This is proved by artinian induction on a, the induction hypoth- esis being that () holds for all paths b −→ w and b −→ w where w is irreducible and b is lower than a (= there is a nontrivial path a −→ b). If P is trivial, then a = w, Q is trivial (otherwise there is an infinite sequence a Q −→ a Q −→ a . . . of nontrivial paths), and τQ− τP = 0− 0 = 0. Similarly () holds if Q is trivial. We may now assume that P and Q are nontrivial, and are the paths in a diamond a p,r,q −−−→ b P ′ −→ w, a u,s,v −−−→ c Q′ −→ w in which a −→ b, a −→ c are simple paths and P ′, Q′ are shorter that P and Q. By the above there is a diamond a p,r,q −−−→ b P ′′ −−→ d, a u,s,v −−−→ c Q′′ −−→ d for which () holds. Since R is complete there is a path d W −→ w: indeed there is a path d −→ w′ where w′ is irreducible; this yields two paths b P ′ −→ w and b P ′′ −−→ d W −→ w′, and implies w′ = w. In the diagram a r ��~~ ~~ ~~ ~~ s �� @@ @@ @@ @@ b P ′ �� P ′′ �� ?? ?? ?? ?? c Q′′ �� Q′ �� ω �� @@ @@ @@ @@ d W �� ω ��~~ ~~ ~~ ~~ ω in which the two w −→ w paths are trivial, () holds for every inner 22 Partial resolutions in monoid cohomology diamond, by the induction hypothesis; by (c), () holds for the given outer diamond. With Claim 2 we now show that () holds for all coterminal connecting sequences a P −→ b and a Q −→ b. Any connecting sequence a P −→ b can be analyzed into a sequence c1 P1 {{ww ww ww ww w Q1 AA AA AA AA c2 P2 ~~}} }} }} }} Q2 AA AA AA AA cn Pn \|\|yyyyyyyy Qn ## GGGGGGGG a = a0 a1 a2 · · · an−1 an = b of possibly trivial paths whose directions alternate; that is, P = − P1 + Q1 − P2 + Q2 · · · − Pn + Qn. As above there are paths ai Wi−−→ w to the same irreducible element w. By Claim 2, () holds for ci Pi−→ ai−1 Wi−1 −−−→ w and ci Qi −→ ai Wi−−→ w. Hence τQi + τWi − τPi − τWi−1 ∈ ∂3(M ′ 3). Adding from i = 1 to i = n yields τP−τW0+τWn = −τP1+τQ1 · · ·−τPn+τQn−τW0+τWn ∈ ∂3(M ′ 3). If now Q is another path from a to b, then τQ− τW0 + τWn ∈ ∂3(M ′ 3); hence τQ− τP ∈ ∂3(M ′ 3) and (*) holds for P and Q. Since M0, M1, M2, and M ′ 3 are free Z[S]-modules, Lemmas 1.1, 1.2, 2.2 yield Theorem 2.3. When R is a semi-reduced complete presentation, M ′ 3 −→ M2 −→ M1 −→ M0 −→ Z is a partial projective resolution of the trivial §-module Z. 3. Cochains 1. Let R ⊆ F × F be a presentation of a monoid S and A be an §- module. For n ≤ 2 a minimal n-cochain with values in A is an §-ho- momorphism U : Mn −→ A. (Minimal 2-cochains are functions of one variable and are called minimal 1-cochains in [2], [3], [4].) The cobound- ary of U : Mn −→ A is δU = U ◦ ∂n+1. Under pointwise addition, min- imal n-cochains, cocycles, and coboundaries constitute abelian groups MCn(R, A) ⊇MZn(R, A) ⊇MBn(R, A). By Theorem 1.4, Hn(S, A) ∼= Extn Z[S](Z, A) ∼= MZn(R, A)/ MBn(R, A), P. A. Grillet 23 by isomorphisms which are natural in A. Since Mn is free, there is an isomorphism U 7−→ Û which is natural in A and sends a mapping U of the set of generators of Mn to the minimal n-cochain Û with the same value on every generator. This isomorphism MC0(R, A) ∼= A takes MZ0(R, A) to B = { a ∈ A ∣∣ sa = a for all s ∈ S }, so that H0(S, A) ∼= A/B, as with the bar resolution. For 1-cochains, recall that a crossed homomorphism is a mapping U : S −→ A such that U(st) = U(s) + sU(t) for all s, t ∈ S. For every a ∈ A, a′ : s 7−→ sa− a is a crossed homomorphism. Proposition 3.1. Up to isomorphisms which are natural in A, a minimal 1-cochain is a mapping of X into A; a minimal 1-cocycle is a crossed homomorphism of S into A; and a minimal 1-coboundary is a crossed homomorphism of the form s 7−→ sa− a for some a ∈ A. Proof. A minimal 1-cochain is a homomorphism Û : M1 −→ A induced by a mapping U of X into A, so that Û [x] = U(x). Then δÛ [r] = Û(∂2[r]) = Û(ξr′ − ξr′′). If Û is a minimal 1-cocycle, then Û(ξr′) = Û(ξr′′) for every r ∈ R. Since ξ(pq) = ξ(p)+p̄ξ(q) for all p, q ∈ F , it follows that Û(p) = Û(q) whenever p ∼ q. Hence Û induces a mapping U ′ : S −→ A, which is well defined by: U ′(p̄) = Û(ξp) for all a ∈ F . Then U ′ is a crossed homomorphism, since ξ(pq) = ξ(p) + p̄ξ(q) for all p, q ∈ F , and uniquely determines Û , by Û [x] = U ′(x̄). If Û is a coboundary, Û = δâ for some a ∈ A, then Ûs[x] = â∂1s[x] = â(sx̄− s) = sx̄a− sa; hence Ûξ (x1 . . . xn) = Û ( ∑ 1≤i≤n ¯x1 . . . xi−1[xi] ) = ∑ 1≤i≤n ( ¯x1 . . . xi−1 x̄i a− ¯x1 . . . xi−1 a ) = ¯x1 . . . xn a− a and U ′(s) = sa− a for all s = ¯x1 . . . xn ∈ S. A minimal 2-cochain is a homomorphism V̂ : M2 −→ A induced by a mapping V of R into A, so that V̂ [r] = V (r). If P is a connecting sequence in F , consisting of sequences a0, a1, . . . , an, u1, . . . , un, v1, . . . , vn ∈ F and r1, . . . , rn ∈ R, let V (P ) = V̂ (τP ) = ∑ 1≤i≤n ǫi ūi V (ri), (16) 24 Partial resolutions in monoid cohomology where ǫi = +1 if ai−1 = uir ′ ivi, ai = uir ′′ i vi, ǫi = −1 if ai−1 = uir ′′ i vi, ai = uir ′ ivi (ǫi = +1 for a path). V is independent of path if V (P ) = V (Q) whenever P and Q are coterminal. Proposition 3.2. Up to isomorphisms which are natural in A, a minimal 2-cochain is a mapping of R into A; a minimal 2-cocycle is a mapping of R into A which is independent of path; a minimal 2-coboundary is a mapping of R into A of the form V (r) = Û(ξr′) − Û(ξr′′) for some mapping U : X −→ A. By (2), Ûξ(x1 . . . xn) = ∑ 1≤i≤n ¯x1 x2 . . . xi−1 U(xi). (17) Very similar characterizations of minimal cocycles and coboundaries are given in [2] and [3]. Proof. When V is a minimal 2-cochain and P , Q are coterminal chains, δV̂ [P, Q] = V̂ ∂3[P, Q] = V̂ (τQ− τP ) = V (Q)− V (P ), by (16); therefore V̂ is a minimal 2-cocycle if and only if V is independent of path. If furthermore Û is a minimal 2-cochain, then δÛ [r] = Û∂2[r] = Û(ξr′ − ξr′′) = Û(ξr′)− Û(ξr′′). Proposition 3.3. If R is semi-reduced and complete, then V is indepen- dent of path if and only if V (P ) = V (Q) whenever P , Q is an essential pair of paths. Proof. By Lemma 2.2, V̂ ∂3[P, Q] = 0 for all coterminal pairs [P, Q] (gen- erators of M3) if and only if V̂ ∂3[P, Q] = 0 for all essential pairs [P, Q] (generators of M ′ 3). If R is finite in Proposition 3.3, then there are only finitely many essential words, essential diamonds, and essential pairs, so that minimal 2-cocycles are characterized by finitely many conditions. It is easy to devise an algorithm which yields all these conditions. P. A. Grillet 25 4. Examples 1. Example 1 is the finite cyclic monoid C = 〈c \| cm = cm+n〉 of index m ≥ 0 and period d > 0; for instance, a cyclic group of order d. (Infinite cyclic monoids and groups have H2(C, A) = 0 for all A.) In this example F is free on X = {c}, with c̄ = c, and R = {r}, where r = (cm+d, cm). A minimal 2-cochain V̂ ∈MC1(F, A) may be identified with V (r); this identifies MC2(R, A) with A. If p −→ q and p −→ t are simple paths, then p = cn, where n ≥ m + d, and q = t = cn−d. Therefore we have a complete and reduced presentation. Let V = V (r) ∈ A. An essential word p = utv has u = ci, t = cj 6= 1, v = ck, r′ = ut, s′ = tv, and u, v 6= 1 if r = s; hence i, j, k > 0, i + j = j + k = m + d, and i = k = m + d− j, where 0 < i, j, k < m + d. An essential diamond consists of an essential wedge ci+m+d 1, r, ci −−−→ ci+m and ci+m+d ci, r, 1 −−−→ ci+m, followed by paths ci+m R −→ d and ci+m S −→ d. We may choose d = ci+m and trivial paths ci+m R −→ d and ci+m S −→ d; then the paths in the diamond are rci and cir. Hence V is a cocycle if and only if V (r) = c̄iV (r) = ci V (r) for all 0 < i < m + d, if and only if V = cV . Thus MZ2(R, A) = Ac = {a ∈ A \| ca = a}. If C acts trivially on A (if ga = a for all g ∈ G and a ∈ A), then MZ2(R, A) = Ac = A. Minimal 1-cochains may also be identified with elements of A. If a = U(c) ∈ A, then Û(ξck) = ∑ 1≤i≤k c̄i−1 a = ∑ 1≤i≤k ci−1 a by (17) and (δa)(r) = ā(r′)− ā(r′′) = ∑ 1≤i≤m+d ci−1 a− ∑ 1≤i≤m ci−1 a = ∑ m+1≤i≤m+d ci−1 a = cm ∑ 1≤i≤d ci−1 a. Hence MB2(R, A) = B = {cm ∑ 1≤i≤d ci−1 a \| a ∈ A}. If C acts trivially on A, then B = dA = {da \| a ∈ A}. This yields the well known result: Proposition 4.1. When C = 〈c \| cm = cm+d〉 is a finite cyclic monoid, or C is a cyclic group of order d, then H2(G, A) ∼= Ac/B, 26 Partial resolutions in monoid cohomology where Ac = {a ∈ A \| ca = a} and B = {cm ∑ 1≤i≤d ci−1 a \| a ∈ A}. If C acts trivially on A, then H2(C, A) ∼= A/dA. When C is a cyclic group, comparable results hold in higher dimen- sions [6], [1]. 2. Example 2 is the bicyclic semigroup, which has the monoid pre- sentation G = 〈p, q \| pq = 1〉. F is free on X = {p, q}, with p̄ = p, q̄ = q, and R = {r}, where r = (pq, 1); MC2(R, A) may be identified with A. Since pq cannot overlap itself, there are no essential words and no essential diamonds. R is a complete, reduced presentation, and every minimal 2-cochain is a minimal 2-cocycle. When U : {p, q} −→ A, δÛ [r] = Ûξ(pq)− Ûξ(1) = U(p) + p U(q) by (18) and (17); if we identify MC2(R, A) with A, then MB2(R, A) = A + pA = A. Hence: Proposition 4.2. When S is the bicyclic semigroup, H2(S, A) = 0 for all A. 3. Example 3 is the free commutative monoid S = 〈c, d \| cd = dc〉. F is free on X = {c, d}, with c̄ = c, d̄ = d; we let R = {r}, where r = (dc, cd); MC2(R, A) may be identified with A. R is complete, with every path ending when all c’s precede all d’s. Since dc cannot overlap itself, there are no essential words and no essential diamonds; R is reduced, and every minimal 2-cochain is a minimal 2- cocycle. When U : {c, d} −→ A, δÛ [r] = Ûξ(dc)− Ûξ(cd) = U(d) + d U(c)− U(c)− c U(d); if we identify MC2(R, A) with A, then MB2(R, A) = A(c)+A(d), where A(c) = {a− ca \| a ∈ A} and A(d) = {a− da \| a ∈ A}. If S acts trivially on A, then A(c) = A(d) = 0. We have proved: Proposition 4.3. When S is the free commutative monoid S = 〈c, d \| cd = dc〉, then H2(S, A) ∼= A / ( A(c) + A(d) ) , where A(c) = {a − ca \| a ∈ A} and A(d) = {a − da \| a ∈ A}. If S acts trivially on A, then H2(S, A) ∼= A. P. A. Grillet 27 On the other hand, in the commutative cohomology, Hn(S, A) = 0 for every abelian group valued functor A and n ≥ 2 [2]. 4. Example 4 is the monoid freely generated by two idempotents, S = 〈e, f \| e2 = e, f2 = f〉. F is free on S = {e, f}, with ē = e, f̄ = f , and R = {r, s}, where r = (ee, e) and s = (ff, f). Minimal 2-cochains have two values M(r) and M(s), and MC2(F, A) ∼= A⊕A. Since ee and ff can only overlap themselves, there are two essen- tial words eee and fff . This yields two essential wedges, four essential diamonds, and four essential pairs of paths: eee 1,r,e −−−→ ee 1,r,1 −−−→ e, eee e,r,1 −−−→ ee 1,r,1 −−−→ e; eee 1,r,e −−−→ ee −→ ee, eee e,r,1 −−−→ ee −→ ee; fff 1,s,f −−−→ ff 1,s,1 −−−→ f, fff f,s,1 −−−→ ff 1,r,1 −−−→ f ; fff 1,s,f −−−→ ff −→ ff, fff f,s,1 −−−→ ff −→ ff. In particular, R is complete. By 3.3, a minimal 2-cochain V is a minimal 2-cocycle if and only if it V (P ) = V (Q) for every essential pair [P, Q]. This yields four conditions: V (r) + V (r) = e V (r) + V (r), V (r) = e V (r), V (s) + V (s) = f V (s) + V (s), V (s) = f V (s); equivalently, V (r) = e V (r) and V (s) = f V (s). If we identify , C1(F, A) with A⊕ A, then MZ1(F, A) = Ae ⊕ Af , where Ae = {a ∈ A \| ea = a}, Af = {a ∈ A \| fa = a}. When U : {e, f} −→ A, δÛ [r] = Ûξ(ee)− Ûξ(e) = U(e) + eU(e)− U(e) = eU(e) and δÛ [s] = f U(f). If we identify MC2(R, A) with A⊕A, then MZ1(R, A) = eA⊕ fA. Since e2 = e holds in S, eea = ea for all a ∈ A, and Ae = eA; similarly Af = fA. Hence MZ2(R, A) = MB2(R, A). Thus: Proposition 4.4. When S = 〈e, f \| e2 = e, f2 = f〉, then H2(S, A) = 0 for all A. 28 Partial resolutions in monoid cohomology 5. Example 5 is the free Burnside monoid S = M2,1,1, that is, the free band-with-identity-element on two generators e and f . The multi- plication table of S is: 1 e f ef fe efe fef e e ef ef efe efe ef f fe f fef fe fe fef ef efe ef ef efe efe ef fe fe fef fef fe fe fef efe efe ef ef efe efe ef fef fe fef fef fe fe fef S has a finite presentation S = 〈e, f \| ee = e, ff = f, efef = ef, fefe = fe〉, which we show is complete. As in Example 4, F is free on X = {e, f}, with ē = e, f̄ = f . Then R = {r, s, t, u}, where r = (ee, e), s = (efef, ef), t = (fefe, fe), and u = (ff, f) (arranged so the left hand sides are in alphabetic order). Minimal 2-cochains have four values and MC2(F, A) ∼= A⊕A⊕A⊕A. Overlaps in the left hand sides yield 12 essential words: ee efef fefe ff ee eee eefef − − efef − efefef efefe, efefefe efeff fefe fefee fefef, fefefef fefefe − ff − − ffefe fff This yields all essential diamonds: 1. eee : eee 1,r,e −−−→ ee 1,r,1 −−−→ e, eee e,r,1 −−−→ ee 1,r,1 −−−→ e. 2. eefef : eefef 1,r,fef −−−−→ efef 1,s,1 −−−→ ef, eefef e,s,1 −−−→ eef 1,r,f −−−→ ef. 3. efefef : efefef 1,s,ef −−−−→ efef 1,s,1 −−−→ ef, efefef ef,s,1 −−−−→ efef 1,s,1 −−−→ ef. 4. efefe : efefe 1,s,e −−−→ efe −→ efe, efefe e,t,1 −−−→ efe −→ efe. 5. efefefe : efefefe 1,s,efe −−−−→ efefe −→ efe, efefefe efe,t,1 −−−−→ efefe −→ efe. 6. efeff : efeff 1,s,f −−−→ eff e,u,1 −−−→ ef, efeff efe,u,1 −−−−→ efef 1,s,1 −−−→ ef. 7. fefee : fefee 1,t,e −−−→ fee f,r,1 −−−→ fe, fefee fef,r,1 −−−−→ fefe 1,t,1 −−−→ fe. 8. fefef : fefef 1,t,f −−−→ fef −→ fef, fefef f,s,1 −−−→ fef −→ fef. 9. fefefef : fefefef 1,t,fef −−−−→ fefef −→ fef, fefefef fef,s,1 −−−−→ fefef −→ fef. 10. fefefe : fefefe 1,t,fe −−−→ fefe 1,t,1 −−−→ fe, fefefe fe,t,1 −−−→ fefe 1,t,1 −−−→ fe. 11. ffefe : ffefe 1,u,efe −−−−→ fefe 1,t,1 −−−→ fe, ffefe f,t,1 −−−→ ffe 1,u,e −−−→ fe. 12. fff : fff 1,u,f −−−→ ff 1,u,1 −−−→ f, fff f,u,1 −−−→ ff 1,u,1 −−−→ f P. A. Grillet 29 In this list, efefe −→ efe stands for either of the two paths in the efefe entry, and similarly for fefef −→ fef . In particular, every essen- tial wedge can be completed to a diamond. Since R is reduced, the proof of 2.3 shows that every wedge can be completed to a diamond, and R is confluent. R is also terminating, since length decreases along every path. The list above yields 12 equalities which by 3.3 characterize minimal 2-cocycles: (1) V (r) + V (r) = e V (r) + V (r); (2) V (r) + V (s) = e V (s) + V (r); (3) V (s) + V (s) = ef V (s) + V (s); (4) V (s) = e V (t); (5) V (s) + V (s) = efe V (t) + V (s); (6) V (s) + e V (u) = efe V (u) + V (s); (7) V (t) + f V (r) = fef V (r) + V (t); (8) V (t) = f V (s); (9) V (t) + V (t) = fef V (s) + V (t); (10) V (t) + V (t) = fe V (t) + V (t); (11) V (u) + V (t) = f V (t) + V (u); (12) V (u) + V (u) = f V (u) + V (u); where efefe 1,s,e −−−→ efe, fefef 1,t,f −−−→ fef . Equation (8) reads (T) V (t) = f V (s) and shows that a minimal 1-cocycle is determined by its values on r, s, and u. The latter satisfy (A) V (r) = e V (r) and f V (r) = fef V (r), (B) V (s) = ef V (s), (C) V (u) = f V (u) and e V (u) = efe V (u) by (1) and (7), (2), (12) and (6) respectively. Conversely, (B) implies e V (s) = V (s). since π (eef) = π (ef), and it is immediate that (1) through (12) follow from (A),(B),(C), and (T). Hence MZ1(F, A) ∼= Ar ⊕As ⊕Au, where Ar = {a ∈ A \| a = ea, fa = fefa}, As = {a ∈ A \| a = efa}, Au = {a ∈ A \| a = fa, ea = efea}; the isomorphism takes V ∈ MZ1(F, A) to ( V (r), V (s), V (u) ) . If G acts trivially on A, then MZ1(F, A) ∼= Z = A⊕A⊕A. 30 Partial resolutions in monoid cohomology When U : {e, f} −→ A, δÛ [r] = Ûξ(ee)− Ûξ(e) = U(e) + eU(e)− U(e) = eU(e), δÛ [s] = Ûξ(efef)− Ûξ(ef) = U(e) + eU(f) + ef U(e) + efeU(f)− U(e)− eU(f) = ef U(e) + efeU(f), δÛ [u] = Ûξ(ff)− Ûξ(f) = U(f) + f U(f)− U(f) = f U(f); then δÛ is given by (T), since minimal 2-coboundaries are minimal 2- cocycles. Hence the isomorphism MZ2(R, A) −→ Z sends MB2(R, A) to B = {(ea, efa + efeb, fb) \| a, b ∈ A}. If S acts trivially on A, then B = {(a, a + b, b) \| a, b ∈ A}. We note that Ar = eA: indeed b = ea implies b = eb and fb = fefb, since π (ee) = π (e) and π (fefe) = π (fe). Similarly Au = fA and As = efA; hence Z = eA ⊕ efA ⊕ fA. To find Z/B we note use the isomorphism θ : A⊕A⊕A −→ A⊕A⊕A, (a, b, c) 7−→ (a, b− efea− efefc, c). Then θ(Z) = Z, since b ∈ As (b = efb) implies b − efea − efefc ∈ As. But θ (ea, efa + efeb, fb) = (ea, 0, fb), so that θ(B) = eA ⊕ 0 ⊕ fA. Hence: Proposition 4.5. When S is the free band-with-identity-element with two generators e and f , H2(S, A) ∼= efA. If S acts trivially on A, then H2(S, A) ∼= A. Note that efA ∼= feA: x 7−→ fx and y 7−→ ey provide mutually inverse isomorphisms. References [1] Brown, K., Cohomology of groups, Springer-Verlag, New York, 1982. [2] Grillet, P.A., Commutative semigroups, Kluwer Acad. Publ., Dordrecht, 2001. [3] Grillet, P.A., The commutative cohomology of finite semigroups, J. Pure Appl. Algebra 102 (1995), 25–47. [4] Grillet, P.A., The commutative cohomology of finite nilsemigroups, J. Pure Appl. Algebra 82 (1992), 233–251. [5] Guba, V.S., and Pride, J.S., Low-dimensional (co)homology of free Burnside monoids, J. Pure Appl. Algebra 108 (1996) 61–79. [6] MacLane, S., Homology, Springer-Verlag, New York, 1963. [7] Novikov, B.V., The cohomology of semigroups: a survey (Russian), Fundam. Prikl. Mat. 7 (2001) 1–18. P. A. Grillet 31 [8] Novikov, B.V., Semigroup cohomologies and applications, Algebra – representa- tion theory (Constanta, 2000), 219–234, Kluwer Acad. Publ., Dordrecht, 2001. [9] Squier, C.C., Word problems and a homological finiteness condition for monoids, J. Pure Appl. Algebra 49 (1987) 201–217. Contact information P. A. Grillet Tulane University, New Orleans, La., U.S.A. Received by the editors: 11.03.2004 and final form in 06.12.2004.

Partial resolutions in monoid cohomology

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