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More precisely, consider the modules of finite length, with presentation matrix of size $m\times n$, i.e. the minimal resolution begins as: $\cdots\rightarrow R^n\rightarrow R^m$. … What is known about the "non-generic" modules? (Some stratification by the ``degeneracy type"?) …

$R=k[[x_1,\dots,x_{p>1}]]$) Given two $R$-modules, $N\subset M$, of the same (finite, non-zero) rank. Their quotient, $M/N$, is torsion. I guess not much can be said in general. …

Of course, the same question holds for modules, but then it's more difficult …

Let $R$ be a (commutative, associative, unital) ring, consider a homomorphism of some (finitely generated) free $R$-modules $E\stackrel{A}{\rightarrow}F$. Say $rank(F)=m$. … These seem to be some very natural (and simple) invariants of modules. Probably well known? References? ps. Shame on me, these are precisely the invariants described in Eisenbud's Comm.Alg. …

(at least, for regular local rings, for Gorenstein rings of low dimensions, for Cohen-Macaulay modules...) Any strengthening of such "fitting type" invariants that determines the module? upd. … Then $coker(A)$ and $coker(A^T)$ have the same fitting ideals, though the modules are non-isomorphic in general …