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

We now review global affine functions as patchworks of affine functions on charts of atlases. Suppose that we are given a global composition $GI$ . For a simplex $s$ of $n(GI)$ , we have the canonical local composition $/~\ s$ , and a morphism $f : /~\ s --> A@ R R$ is an affine function on $/~\ s$ . The set $nG(s)$ of these functions is provided with the structure of a $R$ -module by the pointwise addition and scalar multiplication.

For an inclusion of simplexes $s < t$ of $n(GI)$ , the ambient spaces of the charts of these simplexes are the same, i.e., the inclusion of local compositions $/~\ t < /~\ s$ are in bijection with an inclusion of local compositions $K < K < A@ N t s R$ for a specific module $N$ . Since an affine function on $/~\ t$ is the restriction of an affine morphism $A@h : A@ N --> A@ R R R$ , $f$ evidently extends to the restriction $A@h | Ks$ , so the transition morphisms by restriction of affine functions are surjective. The corresponding complex of affine functions is denoted by $nG(GI)$ . The subcomplex $C = C I G$ of constant functions is defined by $C(s) = {f (- nG(GI)(s),f = constant on /~\ s}$ . The set of global sections of the function complex is denoted by $G(GI)$ and clearly identifies to the previous definition of $G(GI)$ .

Let $fi : GI --> HJ$ be a morphism of global compositions. Take a simplex $s$ in $n(GI)$ , and its image $s'$ under the associated simplicial map. Then, each restricted morphism

f| : /~\ s --> /~\ s' /~\ s

gives rise to a map

f i *nG(HJ )(s) = nG(HJ )(s')-- > nG(GI)(s)

by right composition with this restricted morphism. Moreover, the map is $R$ -linear. Therefore, if $M$ is any subcomplex of $nG(HJ )$ , its induced complex $f/iId *M A$ is mapped $R$ -linearly onto what is called the retracted module complex

M |fi/Id < nG(GI). A

(11)

In particular, if $M = CHJ$ , we have $CHJ|fi < CGI$ .

With these techniques in mind, the resolution functor $resA$ and its associated natural transformation $dA$ give rise to a module complex of affine functions $DnG(GI) = nG(GI)| resGI$ in $nG(DGI )$ , for each global composition $GI$ . Call this complex the resolution complex of composition $GI$ . Moreover, this assignment commutes with the morphism of the resolution functor, i.e., for a morphism $fi : GI --> HJ$ , we have a canonical inclusion

nG(DHJ )| resfi < nG(DGI)

(12)

of the the retracted resolution complex of $HJ$ in the resolution complex of $GI$ . The next step deals with the reconstruction of $GI$ from $nG(DGI )$ and the related question of classification of global compositions by use of the resolution complex which is suggested by the functorial relation (12).

The generic situation from the preceding constructions is that we are given a module complex $M < nG(GI)$ , containing the constants $C$ , and that we would like to construct a kind of »quotient« composition the affine functions of which are those of $M$ . We first look at the local situation.