Mazzola, Guerino / Noll, Thomas / Lluis-Puebla, Emilio: Perspectives in Mathematical and Computational Music Theory

in Mazzola (1985). Those results also included recursive algorithms to calculate classes of global compositions in modules of finite length. They have been generalized to compositions of variable address (Mazzola, 2001b,a). In the following subsections, we want to give a series of concepts and results which describe the recent classification theorems on variable addresses.

Assumption 1 In the following discussion of classification theory, we shall always assume that the global compositions are $A$ -addressed for a fixed address $A$ , objective, and that we work over the category $ModR$ of modules over a commutative ring $R$ . We also assume that the supports $G$ of our global compositions $GI$ are finite sets, in other words, we are situated in the category $RObGlobfiAnite$ .

Let us start with the standard compositions of the theory. They represent compositions with »notes in general position«, i.e. their configuration is as »free« as possible from >occasional< coincidences. In fact, the standard composition is a geometric realization deduced from the nerve $I n(G )$ of the composition, i.e., of its covering $I$ , and thus depends only on combinatorial information. There is a natural projection from the standard object onto the generating composition. Here is the formal construction:

Given a module $A$ in $RMod$ and a natural number $0 < n$ , we denote $|~| n A$ the affine $n + 1$ -fold coproduct $|_| n+1 A$ of $A$ . By construction, there is an isomorphism $~ A~ || n --> Rn o+ An+1$ . We denote the canonical basis of $Rn$ by $(e1,...en)$ , and for any element $a (- A$ and $0 < i < n$ , we set $ai = (0,...,a,...0)$ for the $n+ 1$ -tuple in $An+1$ having $a$ at its $i +1$ -th position and zero else; the zero element is denoted by $e0$ . We have the inclusion morphisms

si : A --> A |~| n

(3)

for $0 < i < n$ , with

{ si(a) = (0,a0) (linear) if i = 0, (ei,ai) (affine) if i > 0.

(4)

This defines a local, $A$ -addressed composition $ADn < A@RA~n ||$ which is called the $A$ -addressed local standard composition of dimension $n$ . By construction, it has the following property: If $M$ is any $R$ -module, and if $s.= (s0,...sn)$ is any sequence of $A$ -addressed points⁹

⁹: In the Yoneda language, these are the morphisms in $A@RM$ .

in $M$ , with associated local composition $S = {s0,...sn} < A@RM$ , then there is exactly one morphism of local compositions

(s.) : ADn --> S : si '--> si for i = 0,...n.

(5)

This morphism is in fact defined by the universal property of the coproduct and is mediated by the following affine function $|~| n f : A --> M$ : Write $ti si = e .si,0$ . Then we have

f(e) = t , 0 0 f(ei) = ti- t0 (linear) for i > 0, f(ai) = si,0(a) (linear) for i > 0,