URL: http://www.elsevier.nl/locate/entcs/volume81.html 14 pages

Rewriting Systems and Hochschild-Mitchell

Homology

Philippe Malbos 1

Laboratoire Géométrie-Topologie-Algèbre, UMR 5030 Université Montpellier II F-34095 Montpellier, France

Abstract

This paper forms part of a project focusing on the development of homological and simplicial methods in rewriting. The purpose of this contribution is to generalize Kobayashi's theorem for monoids to "monoids with several objects". Following Squier's theorem, Kobayashi constructs a resolution for monoids presented by convergent rewriting systems. We construct a free acyclic resolution for fcC, as a C-bimodule over a commutative ring k, where C is a small category provided with a convergent presentation. This resolution, associated to the Knuth-Bendix completion algorithm, reflects the combinatorial properties of C. In particular, categories admitting finite convergent presentations by graphs and relations have a finite type Hochschild-Mitchell homology.

1 Introduction

Rewriting is a method for studying equational theories from a computational point of view. Oriented equations, called rewriting rules, provide equational theories with decision procedures, [3]. Many decision problems arising in equational theories, for instance the fundamental word problem or the combinatorial enumeration problems, require specific presentations by rewriting systems. Convergent presentations, both confluent and terminating, are methods to solve these problems in an algorithmic way. D. Knuth and P. Bendix, [13], proved that there exists for every rewriting system an equivalent convergent system which may be infinite. In general, an equational theory does not admit a presentation by a finite convergent rewriting system, even if the equational theory is finite. Our purpose is to construct homological finiteness necessary conditions for the existence of such presentations for various structures involved in rewriting theory.

1 Email: malbos0math.univ-montp2.fr

@2003 Published by Elsevier Science B. V.

1.1 Rewriting on monoids in monoidal categories

Rewriting occurs in many contexts, rewriting acts on words in algebraic structures (monoids, multisets, groups, algebras), on paths in oriented structures (trees, graphs, small categories) or on terms in equational theories (algebraic theories, operads). All these rewriting processes lie in a common framework given by the notion of internal monoid in a monoidal category on which concepts of rewriting, such as confluence or finite convergent presentations, can be generalized. The word problem has a natural extension to internal monoids in monoidal categories defined by generators and relations. One overgoal is to generalize homological finiteness conditions FP^, related to rewriting on words in monoids in the category of sets, to rewriting on words in internal monoids in an arbitrary monoidal category. This categorical framework unifies numerous algebraic structures involved in computer science and includes various kinds of monads such as algebraic theories, [12], and operads, [6].

The aim of this paper is to study the case of small categories presented by graphs and relations which can be viewed as monoids in the monoidal category of graphs. The methods of rewriting on paths in a category are related to the representation theory of finite dimensional algebras,

1.2 Homological finiteness conditions

How can one detect the obstruction to the presentation of a monoid by a finite convergent system? In rewriting theory, only a few criteria giving necessary conditions for the existence of such presentations are known. From an homo-logical point of view a solution consists in constructing a projective resolution reflecting the combinatorial properties of the rewriting systems. According to the works of C.C, Squier, [20], Y, Kobavashi, [14], J.E.J Groves, [10], and D.J. Anick, [1], [2], it is known that convergent presentations for monoids can be characterized by using a free acyclic resolution of the involved monoid.

By extending Fox's differential calculus, Squier, [20], showed that if a monoid admits a finite convergent presentation then it is of type FP3 over Z. In particular, its third homology group is of finite type. In addition, he constructed a monoid having a decidable word problem but not being of type FP3. Consequently, a finitely presented monoid with a decidable word problem does not admit in general a finite convergent presentation. Numerous generalizations of this result have been made. Kobavashi, [14], proved, under the same hypothesis, that such monoids are of type FP^ over any commutative ring. He constructed an effective free acyclic resolution of modules over the algebra of the monoid whose chains are given by paths in the graph of reductions. These chains are a particular case of chains defined by Anick, [1], constituting a general method, based on antichains, to construct free resolutions generated by overlap ambiguities.

In this paper, using Kobavashi's construction, [14], we construct an acyclic free resolution for small categories admitting a convergent presentation. The

resolution is constructed using the additive Kan extension of Anick's an-tichains generated by the set of normal forms. This construction can be adapted to the construction of the analogous resolution for internal monoids in monoidal categories provided by a convergent presentation. Finally, we prove that if a small category admits a finite convergent presentation then its Hoehsehild-Mitehell homology is of finite type in all degrees,

1.3 Conventions and notations

Throughout the paper, k denotes a commutative ring, A /¿-category C is a small category provided with a structure of a A;-module on its hom sets and where the composition is A;-bilinear, According to [18], Z-categories will be called additive categories, Ce will denote the enveloping /¿-category C° ®k C. A C-module (resp, C-bimodule) over A; is a /c-funetor from C (resp, Ce) to Mod(/c), the category of left-Zc-modules, and we will denote by Modfe(C) the category of C-modules over k. The category of A;-funetors from C to Mod(A;) is isomorphic to the category of additive functors from C to the category of abelian groups Ab, If C is a small category, we will denote by |C| its set of objects, by Ca its underlying discrete category and by Ce = C° x C the enveloping category. We let kC denote the /¿-category with the same objects as C and whose hom set kC(p,q) is the free /c-module generated by C(p,q), p,qE |C|, For a morphism w in C, we will denote by o(w) the source of w and by t(w) its target,

2 Rewriting Systems for Small Categories

Small additive categories presented by graphs and relations and path algebras give both a way to compute projective resolutions of modules over rings, [4], [2], In algebra, the main interest of small additive categories and their quotients is that they are natural generalizations of rings, [18], There are applications of path algebras for the study of free algebras, group algebras, but the main interest lies in the representation of finite dimensional algebras through Morita theory. Another reason for studying rewriting on path algebras is the investigation of applications of related algebraic methods in computer science areas, such as process algebras, data structures like Giavitto's Group-Based Field, [9], or reduction graphs in the diagrammatic presentation of the axiomatic rewriting systems, [15], [17],

2.1 Notation and Rewriting Systems

For monoidal categories and internal monoids in monoidal categories we refer to [6], Let Q be a directed graph, we will denote Q0 and Qi its sets of vertices and arrows respectively. Let us consider the slice category Set/Q0 x Qo denoted by Gph(Q0); whose objects are morphisms Q : Qi Q0 x Q0 in Set and a morphism from Q to Q' : Q[ Qo x Qo is a map / : Qi Q[ such that

Q = Q' ° /■ For Q : Qi —Q0 x Qo and Q' : Q[ —Q0 x Qo in |Gph(Q0)|, the product Q Xq0Q' is defined as the composition: 7rop* : Qx Xq0 Q[ ^ QqxQo where tt is the first projection and p* is given by the following pullback diagram:

Qi XQ0 Q'I---"Qo XQqxQQ

Ql x Q[-x Qo x Qo x Q0.

The product Xq0 endows the category Gph(Q0) with a structure of a non-symmetric monoidal category where the unit is given by the diagonal 8 '■ Qo Qo x Q0. Let Mon(Gph(Q0)) be the category of internal monoids in (Gph(Qo), Xq0,S). The forgetful functor

U : Mon(Gph(Q0)) —► Gph(Q0)

has a left adjoint F. For every directed graph Q e |Gph(Q0)|, (F(Q),o, 1) is the free category generated by Q, F{Q) is the morphism F(Qi) Q0 x Q0 where F(Qi) = J [„ Q„ and Qn is the n-ith iterate : Qn = Q„_i xQo Qi, n > 2, The product o : F(Qi)xq0F(Qi) F{Qi) is given by pullback and the unit 1 : Q0 F{Qi) is obvious.

Let (T = UF, fj,_, rj_) be the monad on Mon(Gph(Q0)) given by the above adjunction. For Q 6 |Gph(Q0)| the free T-algebra (T(Q)^q) is called the path algebra of Q, where ¡jlq is the concatenation of paths. We use A to denote the empty path in T(Q). The slice category Gph(Q0) has finite limits given by pullback over Q0 x Q0, and the category of free T-algebras on Q has the same finite limits,

A rewriting system < Q \ R > on Gph(Q0) consists of an object Q e |Gph(Q0)| together with a subset R C T(Q) Xq0Xq0 T(Q). The reduction relation on T(Q) is defined by w r u/ if there exist (u, v) e R and i,|/G T(Q) such that w = fj,g(x, Hq(u, y)) and w' = Hq(x, Hq(v, y)). The relation and ^*R are respectively the transitive reflexive and symmetric transitive reflexive closure of —A path w e T(Q) is said to be in normal form iff there is no path w' such w w'.

In order to fix notation, we recall some facts about rewriting systems, A rewriting system < Q | R > is said to be convergent if the following two conditions hold:

i) R is terminating, that is, the relation is well founded,

ii) R is confluent, that is, each congruence class of contains exactly one element which represents the normal form of the elements of the class.

Definition 2.1 A convergent presentation for a small category C is a convergent rewriting system < Q | R > such that C is the quotient of the free category F{Q) by the relation

Thus, if < Q | R > is convergent, each path w in F{Q) can be rewritten

into a unique normal form, denoted by w, representing the image in C of w by the projection from F{Q) to C,

A critical pair is a pair (a, /3) G—x —*R describing one of the following situations:

i) inclusion ambiguities: uwv w' and w —w", w ^ A;

ii) overlap ambiguities: uw w' and wv —w", w ^ X.

A resolution of a critical pair (a, /3) is a pair (a', /3') G—x —*R such that r(a) = a (a') and r(/3) = o(P') and r( a') = r(/3'),

A presentation < Q \ R > is said to be finite if Q is a finite graph and the set R is finite, A terminating presentation is confluent if and only if there exists a resolution for all its critical pairs. In particular a terminating presentation having a finite set of critical pairs is confluent. The obstruction to the existence of a finite convergent presentation to presentation lies in the infinity of its critical pairs and multi-critical overlaps, Anick's chains allow us to compose with them,

2.2 Anick 's Chains

Let < Q | R > be a rewriting system, we denote by < the order on the set of paths defined by u < w iff u = l^) or it = V1V2 • • • vs and w = V1V2 • • • vn for some 1 < s < n and Vi E F(Q). Such a u is called a (pre-)subpath of w. The set I = {w E F\Q) | w is in normal form} is an order ideal of paths pointed by source, i.e., w E I and u < w implies u E I. The set of antichains for I is defined by:

Aj(Q, R) = {v E F(Q) - I | u < v and u # v imply u E I}.

Let us note that this notion of antichains, in the sense of [1], could be defined with another order ideal in order to adapt it to other processes of completion. From the definition we deduce the two following lemmas.

Lemma 2.2 A path w E F\Q) is reducible if and only if there exists an antichain u E A^Q^R) such that u < w.

Lemma 2.3 Let v% be in I and x be a normal form in L{Q). If vlx is reducible, then there exists an unique decomposition vlx = vlvl+ly where v'vt+l E Ar(Q, R) and y E L(Q).

We denote by:

Cn(Q, R) = {(V, • --,vn) E ¡v'EQu uV+1 e AT(Q, R)},

the set of n-ehains for the rewriting system < Q | R >, We set by convention C0 = Qo and C\ = Qi f~l I. If R is left-reduced, that is the left-hand side of each rule in R is not a subpath of a left-hand side of another rule, then C2 corresponds to the set of left-hand side of all the rules in R. Moreover, if R

is left-reduced, then there are no inclusion ambiguities, and the set of critical pairs corresponds to the set of 3-ehains, The set of n-ehains corresponds to the n-overlap ambiguities. We can associate to every rewriting system an equivalent left-reduced, therefore in the rest of this paper we always assume that R is left-reduced.

Let C be a small category and < Q \ R > a presentation of C, for each n > 0 and p,p' E |C|, let Cn(p,p') be the free abelian group generated by Cn(Q,R) fl C(p,p'). Thus Cn defines an additive functor from the discrete enveloping ZCJi to Ab,

The module of Anick's n-chains of < Q \ R > is the left Z-linear Kan extension of Cn along the inclusion functor i : ZCJi M- ZCe, i.e, the bifunctor:

ZC[C„] = Cn ZCe,

such that the following diagram commutes:

ZC§-i--zce

y/'KCyCn]

Let us note that Kan extensions can be thought as a way of extending the domain of a functor from "one object" to "several objects", The C-bimodule ZC[C„] can also be defined by:

ZC[C,](_,_)= 0 G\(p,p>)®zZC%p,p>),(-,-))^ 0 ZC%p,p>),(-,-))■

(p,p')e|C|x|C| c*(p,p')

(p,p')€|C|x|C|

Thus, for any q,q' E |C|, there is a decomposition:

Ze[C*](W) = 0 ZC(p',g')®za(p,p')®zZC(P!g),

(p,p')e|C|x|C|

The action of (v, v') E Ce on ZC [C#] is given by:

(w' 0 (v1, ■ ■ ■ ,vn)0 w).(v, v') = v'w' 0 (vi, - • • , vn) 0 wv. We will denote by [cn] a generator 1 ® cn 0 1 in ZC[C„],

3 Finitely Generated Resolutions for Small Categories

3.1 Categories of type FPn

Let C be a small ^-category. The set {C(p, _) | p E \C\} forms a set of small projective generators in Modfe(C), thus every projective in Modfe(C) is a retract or a coproduct of such representables, A resolution P* —y M in

Modfe(C) is said to be of finite type if each Pi is finitely generated, that means that it is a quotient of finite eoproduets of representables, A C-module M is said to be of type FP„, n > 0, if there exists a partial projective resolution of finite type —^ - - • —^ —M in Modfe(C), By adapting the proof in [8], VIII prop, 4,3, we get:

Lemma 3.1 Let M be a C-module, and n > 1. The following conditions are equivalent:

i) M is of type FP„,

ii) M is finitely generated, and for every partial projective resolution of finite

type Pk Pk-1 —• • • Po —M —0, k < n, the kernel ker 8k is finitely generated as a C-module.

Thus, a C-module M is said to be of type FPoo if it is of type FP„ for all integers n > 0,

Definition 3.2 A small category C is said to be of type (right-left-)FPn over k, if kC is of type FP„ as a C-bimodule over k.

If C is of type FP„ over Z then, by tensoring by k, C is of type FP„ over any commutative ring k. We will say that C is of type FP„ if it is of type FP„ over Z.

Remark 3.3 The difference of the definition 3,2 with the FP„ notion for groups, recall that a group G is of type FP„ if the trivial G-module Z is FP„ ([8], VIII, 5), lies in the fact that Extc0><c(ZC,JL4) = Extc(Z,A) for every small category C and every right C-module A : C —Y Mod (A;) where p is the pull back of the projection : C° x C C, see ([6], 8,5), Consequently, the FP„ finiteness condition for a category C is related to its category of coefficients, that is Modfe(C),

Proposition 3.4 The following assertions characterize the FP condition up to 1.

i) A category C is FP0 if and only if it is finitely generated.

ii) A category C is FPX if it is finitely presented.

Proof.

A derivation d : C —Y M, M being a C-bimodule, is a family of applications: dp.q : C(p,q) —Y M(p,q), p, q E |C|, satisfying:

d(xy) = d(x)y + xd{y), p4g4rGC.

Let ¡jl : ZC®Zq0ZC —Y ZC be the augmentation: fj,(w®w') = ww'. The augmentation ideal of C is the C-bimodule Ic defined by the exact sequence:

0 —Y Ic —> ZC ®ZQ0 ZC —Y 0, (1)

The ideal Ic as a C-bimodule is generated by the set{l<g)w; —w;<g)l, !«eC}, Let < Q | R > be a presentation of C, Then there exists an unique derivation d : C —y Ic such that: d(f) = 1 ® f — f ® 1, for every / 6 Qi. By the universal property of the derivation, we get that Ic is a free C-bimodule with basis (d(f))f£qi. Thus C is finitely generated if and only if C is FPi_

In order to prove ii), let < Q \ R > be a presentation of C, There is an exact sequence :

ZC[R] A- ZC[Qi] A- ZC ®ZQo ZC ^ ZC where di and d2 are the C-bilinear morphisms defined by :

—idixj^xj^i • •' xn,

<k([l,r]) = ^(1)-T^r),

where d\ is obtained by composing d\ by the projection tt:

F(Q)^ZF(Q)[Qi]

Thus, if < Q | R > is finite, then ker(d) is a finitely generated C-bimodule,□

For n > 2, the FP„ condition constitutes a strengthening of the finite presentation reflecting combinatorial properties of C,

3.2 Convergent presentation implies FP^ for categories

We are going to construct a finite type projective resolution for a small category admitting a convergent presentation. The construction is in the same spirit as the Kobavashi resolution for monoids, [14], In this section we suppose that < Q | R > is a convergent left-reduced rewriting system, and that C is a small category presented by < Q \ R >.

Definition 3.5 Let q,q' E |C|, x' ® (V,--- , vn) ® x E ZC[Cn](q,q') and ^»(li1,-- - ,um)®yEZC[Cm](q,q'). We define:

x' <g> (v\ ■ ■ ■ , vn) <g> x ]f ® (u\ ■ ■ ■ , um) ® y,

if there exists a reduction path x'vn • • • vlx —tfum • • • uly. Furthermore, we define: x' ® (V, • • • , vn)®x >- tf ® (ul, • • • , um) ®y, i(x'®(v1,--- , vn) ® x ^ y' ® (u1, • • • , um) ® y and x'v1 • • • vnx tfu1 • • • umy in L(Q).

Now we need to determine the induced ordering on the sums, let

x = E E ZC[Cn](q,q>) and Y = E %®d^®% E ZC[Cm]{q,q'),

i=0 j=Q

we define X y Y if there exists i E {0,...,/} such that x'i 0 cf 0 Xi y y'j 0 df 0 yj, for every j E {0,,,,, n}.

Lemma 3.6 For any q,q' E \C\, y is a noetherian order on (J ZC[Cn](q, q')

compatible with the left action ZC.

According to [14], the acyclic resolution (ZC[C*], is constructed inductively, The C-bilinear morphisms Sn : ZC[C„] —Y ZC[C„_i] and a left-C-linear contracting homotopv in : ZC[C„_i] —Y ZC[C„], satisfying in) and iin), are defined inductively on n relatively to a reduction map rn = inSn satisfying ivn). So we consider the induction hypothesis %n defined as the conjunction of the four following assertions:

in) Sn-iSn = 0;

iin) in-l$n-l + Snin = lzC[C„_i];

iiin) Sn(x' 0 cn 0x) =<! x' 0 cn 0 x and in(x' 0 cn 0 x) =<! x' 0 cn 0 x;

. . , , . _ J -< x' 0 (vl, ■ ■ ■ , vn) 0 x if vnx is reducible,

1V„) rn(x' 0 (v , • • • , vn) 0 X) < ^ ^

1 = x' 0 (v , • • • , vn) 0 x if vnx is irreducible.

Definition 3.7 We define S0 = ¡jl as the augmentation defined in 3,4, and ¿o : ZC —Y ZC[Q0] as the left-C-linear morphism defined by io(x) = x<S)lT(x)-Let : ZC[Ci] —Y ZC[Q0] be the C-bilinear morphism defined by:

¿>l(M) = 1<t(V) 0V -V0 1t(„).

R is convergent so any x in C has an unique decomposition in normal form paths in Ci : tx' — tx' * * * x yj. ix' j Ci. We define the left-C-linear morphism i\ : ZC[Q0] —> ZC[Ci] as a formal derivation on paths by:

¿l(l 0x)= Xi ■ ■ ■ Xi_i 0 [xj] 0 xi+1

, - l "J - , - ' ^n-

Lemma 3.8 The previous Sn and in, n = 0,1, satisfy Hi. Proof. i0) and iii0) are trivially checked. For any x E ZC,

5i?i(l 0x)= xi ■ ■ -x^i(l <g> Xi - Xi0 l)xi+i ■ ■ - xn, i=1

= 1 <g> Xi • • • xn — Xi ■ ■ ■ xn 0 1, = 1 ® x — ¿0^0(1 ® x).

Thus ii0) is satisfied. In order to prove iv0) let us consider iiSi(x' 0 [w] ® x) = x'{ii{l 0 vx) — vii{l 0 x)). If vx is irreducible, then ¿i( 1 0 vx) = 1 <S> [w] ® x and consequently ¿i^i (x' 0 [w] 0x) = x' 0 [w] ® x. If vx is reducible,

then vx Vi-'-Vn, Vi e Ci, and h(l 0 vx) = Vi''' yi-iNi/i+i-f«-

Then ¿i(l ® vx) =<! 1 ® [w] ® x, and since vii(l ® x) =<! 1 ® [w] ® x, we have iiSi(x' ® [w] ®x)=^x'® [w] ®x. □

Definition 3.9 Let us define rn : = inSn as the reduction map on lzc[c„]- The boundary map is the C-bilinear morphism defined by :

<Jn+i(l®(ü\ ■■■ ,vn, vn+1)®l) = l®(u\ • • • , vn)®vn+l^rn{l®{v\ • • • , vn)®vn+1),

and the contracting homotopv is the left C-linear morphism:

¿„+i(l®(uV-- ,vn)®x)

0 if vnx is irreducible,

1® (V,-- - , vn, vn+l) ® у +

i„+i(r„(l ® (vl, • • • , vn ® vn+1)).y) if vnx is reducible,

where vnx = vnvn+1y is the unique decomposition with (vn, vn+1) E Aj(Q, R) given by the lemma 2,3,

Lemma 3.10 Under the hypothesis %n, 6n+i and in+1 satisfy %n+\.

Proof. in) and iin) imply that Snrn = Sn and thus SnSn+1 = 0 that is in+i)-In order to prove iin+i) let us consider 1 ® (V, • • • , vn)®x E ZC[C„], If vnx is irreducible, then in+\{l ® (V, • • • ,vn)®x) = 0, and r„( 1 ® (V, • • • , vn)®x) = 1 ® (V, ■■■ ,vn)®xby ivn). Thus in8n + Sn+iin+i = lzc[c,j-

If vnx is reducible, then according to lemma 2,3 there exists a decomposition vnx = vnvn+1y with vnvn+1 e Aj(Q,R), and consequently:

5n+lin+l(l ®(v\---,vn)®x) = 5n+i(l ® (v\ • • • , vn, vn+l) ® у

+in+l(rn(l®(v1,--- ,vn®vn+l)).y)),

with r„( 1 ® (V, • • • , vn ® vn+1)).y =<! 1 ® (V, • • • , vn) ® x. Thus applying iin) we have :

tjn{rn{ 1 ®(v1,--- ,vn® vn+l)).y) + 6n+lin+l(rn( 1 ® (V, ■■■ ,vn® vn+l)).y) = rn{\ ® {v\ ■ --,vn ® vn+1)).y.

Sn+lln+l(l ® (v\ ■ ■ ■ ,Vn) ® x) = 5n+i(l ® (v\ ■ ■ ■ , vn, vn+l) ® y)-inSn( 1 ® (У, ■■■ ,vn) ® vn+ly) + rn( 1 ® (V, ■■■ ,vn ® vn+l)).y.

1 ® (V, • • • , vn) ® vn+ly - rn{ 1 ® (v\ ■■■ ,vn) ® vn+l).y,

we have :

Sn+iin+i(l ® (v1, ■ ■ ■ , vn) ® x) =

1 ® (V, ■■■ ,vn) ® vn+ly - inSn( 1 ® (V, ■■■ ,vn) ® vn+ly), which concludes.

By definition of and in+i, iiin+i) is trivially cheeked. In order to prove ivn+i) let x' 0 (v1, • • • ,vn, vn+1) 0 x be in ZC[Cn+i]. We have

Sn+i@® ,vn,vn+l)0x) =

x' 0 (v1, ••• ,vn)0 tf^x - rn(x' 0 (v1, ••• ,vn)0 vn+l)x.

As vnvn+l is reducible, then according to ivn), rn(x' 0 (vl, • • • , vn) 0 vn+1)x -< i'®^1, ■ ■ ■ ,vn, vn+1)0x, and if vn+1x is reducible, we have Sn+i(af'®(w1, • • • ,vn, vn+1)0 x) -< x' 0 (v1, • • • , vn, vn+1) 0 x. Thus,

rn+i(x' 0 (v\ ■■■ ,vn, vn+1) 0x)^x'0 (v\ ■ ■ ■ , vn, vn+1) 0 x.

On the other hand, if vn+lx is irreducible, then :

rn+l(x' 0 (v\ ■ ■ ■ , vn, vn+1) ® x) = ln+l(x' 0 (v\ ■■■ ,vn) 0 vn+1x)

- in+i (rn(x' 0 (v1, „, vn) 0 vn+l).x), = x' 0 (v1,--- ,vn,vn+1)0x,

which concludes ivn+i). □

Therefore we have shown the following theorem.

Theorem 3.11 Let C be a small category and let < Q \ R> be a convergent presentation of C. Then, with the above notations, (ZC[C#], 5*, is a free acyclic resolution of ZC by Q-bimodules.

If the rewriting system < Q \ R > is finite and convergent, then it has a finite number of critical pairs and for every n > 0 the set of n-ehains Cn is finite and ZC[C„] is a finitely generated C-bimodule, which implies :

Corollary 3.12 If a small category C has a presentation by a finite convergent rewriting system < Q \ R> then C is of type FP,over any commutative ring k.

4 Hoehsehild-Mitehell Homology for Convergent Categories

The category Modfe(Ce) is abelian with enough projeetives. Let Torfe (resp, Ext^e) be the derived functor of the bifunctor -0kce - (resp, HomfeCe (-, -))■ They can be computed as usually using projective resolutions. For any pair of C-bimodule (F,G), we have Toif (F.G) = //„(/•' 0kCe P*) where P* is a projective resolution of G in Ab°e and Ext^(F, G) = iP(Homfeee (P*, G)) where -P* is a projective resolution of F in Modfc(Ce), The Hoehsehild-Mitehell homology of a small category C with coefficients in a C-bimodule M over k ([18], sec, 12) is defined by:

Hn(C,M) = Torf (kC,M).

We refer to [5] for its relations with other homologieal theories.

If the presentation < Q \ R > is finite, then the set of n-ehains is finite for each n > 0, and the complex obtained by tensoring by kC the resolution of Theorem 3,11 over kCe is a finitely generated complex. Consequently we have:

Corollary 4.1 If a small category C admits a finite convergent presentation < Q | R >, then its Hochschild-Mitchell homology groups Hn{C, k) are finitely generated.

Several consequences can be deduced from the property FP^, In particular, let us consider T(Q|ii), the graph of irreducibles of < Q \ R >: its set of vertices is the set I of irreducibles and its edges are pairs (vi, v<¿) such that ViV2 G Aj(Q,R). Let us denote by m(Q\R) the maximal length of a path in r(Q|i?) beginning in J fl Qi. Then, if C admits a finite convergent presentation, its Hochschild-Mitchell dimension dimfe C = Sup{n | Ext^Ce(A;C, _) ^ 0} is finite and bounded by m(Q\R).

5 Conclusion and Further Work

Hochschild-Mitchell (co)homologieal theory for categories is closely related to (co)homologieal theory for associative algebras. In the same spirit, M, Jibladze and T, Pirashvili, [19], define the eohomologv for algebraic theories closely to Mac-Lane eohomologv of rings. It has an interpretation in terms of the Hochschild-Mitchell eohomologv, thus we hope that our construction gives a way to extend the homologieal finiteness conditions from monoids to equational theories presented by term rewriting systems.

Since we have noticed in 3,3, the homologieal finiteness conditions as FP„ are related to the category of coefficients involved, Baues define in [6] natural systems in Ab as general coefficients for computing eohomologv of categories. We have to explicit the finiteness conditions relatively to this category.

Following the framework defined in [6], for the eohomologv of monoids in an arbitrary monoidal category, we have to construct the same resolution as in section 3,2 for such monoids provided with a convergent presentation. Thus, this construction should be adapted to monads over various kind of categories in order to obtain this condition for algebraic theories and term rewriting systems, which can be respectively modeled by finitarv monads over the categories Set and Pre, the category of preordered sets, [12], [16],

The Kobavashi construction is inductive, we have to work out a construction for the contracting homotopv involved in (3,11), We will then be able to construct a simplicial object associated to a convergent presentation reflecting geometrically the combinatorial property of confluence. This question is still open.

We can also adapt the techniques of Anick's antichains to other procedures of completion than Knuth-Bendix's one. On the other hand, the constructive

resolution used in the present paper gives a way to compute the (co)homologv

of a small category. For a finitely presented category, we have to implement

the algorithm which performs such computations.

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