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

This means that we are able to reconstruct $I G$ from its retracted affine functions on the resolution. Moreover, in this case, the retracted module complex can also be recoverd from the quotient composition, i.e.,

I I I DnG(G ) = nG(ADGI /DnG(G ))|/DnG(G )

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so that we are now left with the question of charactrizing those module complexes of affine functions in $D * A n$ which could give rise to compositions having this free object as their resolution.

Under assumption 3, we may proceed to the analysis of the following type of module complexes $N < nG( D ) A n$ : they are finitely generated projective (i.e., their zero-simplex modules are so), and contain the constant functions; call these complexes representative.

4.4 Orbit Spaces and Classifying Schemes

By the universal property of the standard compositions, the automorphism group $SA,n*$ of the standard composition $ADn*$ identifies to a subgroup of the symmetric group $Sm+1$ of permutations of $ADn*$ if the standard covering is defined on the integer interval $[0,m]$ as discussed above. By retraction, this group acts from the right on the set $RepA,n$ of representative module complexes on $ADn*$

ret : RepA,n* × SA,n* --> RepA,n*: (N, g) '--> N |g.

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The orbit space of this action has this role (Mazzola, 2001b):

Theorem 6 The orbit space $RepA,n*/SA,n*$ is in bijection with the set of isomorphism classes of $A$ -addressed global compositions with projective functions and finitely generated projective atlases which have a covering complex isomorphic to $* n$ . This bijection is induced by the retraction of the function module complex to the resolution $ADn*$ , in one direction, and by the quotient composition on a given representative module complex on $ADn*$ , in the other.

In particular, this classification result is valid for the global compositions having as their address a module $A$ over a semi-simple commutative ring $R$ .

A more in-depth discussion of the action of the automorphism group of the standard composition on module complexes yields this geometric classification spaces (Mazzola, 2001b):

Theorem 7 For an addresse $A$ which is locally free of rank $m$ over the commutative ring $R$ , there is a subscheme $Jn*$ of a projective $Spec(R)$ -scheme of finite type such that its $S$ -valued points $Jn*(Spec(S))$ for a $R$ -algebra $S$ are in bijection with the classifying orbits of module complexes $N$ in $D * S ox RA n$ which are locally free of defined co-ranks on the zero-simplexes of $n*$ .

In particular, if the ground ring $R$ is semi-simple, this theorem gives the classification of any global composition which are addressed in a finitely generated $R$ -module $A$ .