a generalization of the annihilator of cokernel ideal (some new invariants of modules?) [closed]

Question

Let $R$ be a (commutative, associative, unital) ring, consider a homomorphism of some (finitely generated) free $R$-modules $E\stackrel{A}{\rightarrow}F$. Say $rank(F)=m$. The basic invariants of $A$ are the Fitting ideals, $I_j(A)$ being generated by all the $j\times j$ minors of $A$.

A more delicate invariant is the annihilator-of-cokernel ideal, $ann.coker(A):=ann\frac{F}{Im(A)}$. This ideal is the counterpart of the minimal among the Fitting ideals.

What are the counterparts of the other Fitting ideals?

There is the standard definition: $ann.coker_k(A)=ann.coker(\overset{k}{\wedge}A)$. e.g. in Eisenbud's book on Comm.Alg. This notion is good in various senses, but has the following drawback. The ordinary $ann.coker(A)$ is the "maximally reduced" version of the ideal $I_m(A)$. While for $ann.coker_k(A)$ as above one looses this notion of reducedness. (As can be seen already for the diagonal matrices.)

For my work I need the following, "more refined" version, which follows this reducedness idea. Consider all the possible free direct summands $F_k\subseteq G$, of rank $k$. Define $a.c._{m-k}(A):=\sum\limits_{F_k\subseteq F} ann\frac{F}{Im(A)+F_k}$. Eventhough the sum is infinite, only a finite number of embeddings are relevant. (At least for rings which are Noetherian or close to them.)

For what follows assume that $R$ is local, Noetherian.

One gets immediately that $a.c._k(A)$ depends on the module $coker(A)$ only, (and not on the resolution) thus is a meaningful invariant. Moreover, these invariants extend the ordinary ann.coker(A) and resemble the Fitting ideals, as follows.

1. $ann.coker(A)=a.c._m(A)\subseteq a.c._{m-1}(A)\subseteq\cdots\subseteq a.c._1(A)=I_1(A)$.
2. $a.c._k(A)\supseteq I_k(A)$ and it seems: $I_k(A)\supseteq (a.c._k(A))^k$.
3. It seems that for square matrices $a.c._k(A)=I_k(A):I_{k-1}(A)$.
4. If $A=diag(\lambda_1,\dots,\lambda_m)$, with $(\lambda_1)\supseteq\cdots\supseteq(\lambda_m)$, then $a.c._k(A)=(\lambda_k)$.
5. $a.c._{m-k}(A)=0$ iff $rank(coker(A))>k$.
6. (Some Kueneth type formula) $a.c._{m-k}(A_1\oplus A_2)\supseteq \sum\limits_{k_1+k_2=k}a.c._{m_1-k_1}(A_1)\cap a.c._{m_2-k_2}(A_2)$. (Probably the equality holds.)

(And many other properties.)

One can define a 'dual object'. For any free direct summand of rank $k$, $F_k\subset F$, define $\widetilde{a.c._k}(A)=\sum\limits_{F_k\subset F}ann\frac{F_k}{F_k\cap Im(A)}$. In many cases these ideals coincide with $a.c._k(A)$. (Probably the two definitions are equivalent?)

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. Exercise 20.9. Closing this question.