An analogue of the Aluffi algebra for modules

P. Aluffi introduced in [1] a new graded algebra in order to conveniently express characteristic cycles in the theory of singular varieties. This algebra is attached to a surjective ring homomorphism $A\text{surjects}B$ by taking a suitable inverse limit of graded algebras, one for each representation of $A$ as a residue ring of a given “ambient” ring $R\mathrm{.}$ Since giving a ring surjection $A\text{surjects}B$ is tantamount to giving an ideal $I\subset A$, it would seem natural to ask for an analogous notion for $A$-modules. This is the central purpose of this work. Since a given module may not admit any embedding into a free module, a preparatory toil includes dealing with this technical point at the outset. On the bright side, the intrusion of modules raises a few algebraic questions interesting on their own. It is to expect that this extension to modules may be transcribed in terms of coherent sheaves, thus possibly providing an answer to a question by Aluffi in this regard. Two main bodies of examples are treated in detail to illustrate how the theory works and to show the relation to finer properties of other algebras.