Let $R$ be an integral domain, "nice" (regular for instance). Consider a homomorphism $$ f: R^m \to R^m $$ of two rank $m$ free $R$ modules. Assume that $\ker f =0$ and that the cokernel is $M$. Now let $I$ be the second Fitting ideal of $f$, i.e. the ideal generated by $(m-1) \times (m-1)$ minors of $f$ (write $f$ in terms of an $m \times m$ matrix). The homomorphism $f$ induces $ (R/I)^m \to (R/I)^m$ whose kernel is denoted by $K$ and cokernel is $M/IM$.

The question is: can you construct a canonical homomorphism $$ Hom_{R/I}(I/I^2, R/I) \to Hom_{R/I}(K, M/IM) $$

This question is extracted from the algebro-geometric situation concerning the normal bundle of the determinantal varieties.


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