be finite. The homology of $G$ coincides with the homology of $M$. Garside theory provides two useful resolutions of $\Z$ by free $\Z M$-modules.

The first one was defined in [@CMW]. Another one, with more complicated differential but a smaller number of cells, has been defined in [@DL].

The Dehornoy-Lafont complex
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Let $M$ be a Garside monoid with a finite set of atoms $\mathcal{X}$. We choose an arbitrary linear order $<$ on $\mathcal{X}$. For $m \in M$, denote $\md(m)$ denote the smaller element in $\mathcal{X}$ which divides $m$ on the right ($m = a\md(m)$ for some $a \in M$). Recall that $\lcm(x,y)$ for $x,y \in M$ denotes the least common multiple on the left, that is $v = gx = h y$ implies $v = j \lcm(x,y)$ for some $j \in M$. If $A = (x,B)$ is a list of elements in $M$ we define inductively $\lcm(A) = \lcm(x, \lcm(B))$.

A $n$-cell is a $n$-tuple $[x_1,\dots,x_n]$ of elements in $\mathcal{X}$ such that $x_1 < \dots < x_n$ and $x_i
= \md (\lcm(x_i,x_{i+1},\dots,x_n))$. Let $\mathcal{X}_n$ denote the set of all such $n$-cells. By convention $\mathcal{X}_0
= \{ [\emptyset] \}$. The set $C_n$ of $n$-chains is the free $\Z M$-module with 