Some questions about Fitting ideals Let $R$ be a ring and $M$ a finitely presented $R$-module. Given a free presentation
$$ R^{\oplus m} \to R^{\oplus n} \to M \to 0 $$
we define $Fitt_k(M)$, the $k$-th Fitting ideal of $M$, to be the ideal generated by all the $(n-k)\times (n-k)$ minors of the matrix representation of the map
$$ R^{\oplus m} \to R^{\oplus n}. $$
This definition makes sense because one can easily show that $Fitt_k(M)$ is independent on the presentation of $M$.
I guess this can be be defined using the exterior algebra instead of minors, could you give me some clues on how I should do that?
Moreover, do you know any good and modern book/paper in which the theory of Fitting ideals is treated? 
In particular I'm very interested in statements about the Krull dimension of $Fitt_k(M)$. 
Any suggestion is welcome!
 A: As Martin suggested, a nice reference is 

Eisenbud - Commutative Algebra with a view towards algebraic geometry

The definition of Fitting ideals that can be found there is indeed in term of exterior algebra, and precisely is the following (it consists of 2 parts)

Def: Let $\varphi:F\to G$ be a map of free modules over a ring $R$. We define the ideal $I_k(\varphi) \subset R$ to be the image of
  the canonical map $$ \Lambda^kF\otimes\Lambda^kG^*\to R $$ induced by
  $\varphi$ (see the note below for details).

Now we are ready to define Fitting ideals:

Def: Let $M$ be a finitely presented module over $R$, and consider a free presentation $$ \varphi: F\to G \to M \to 0 $$ of $M$ such that
  $G$ is a finitely generated $R$-module of rank $r$. For every integer
  $k\geq 0$ we define the $k$-th Fitting ideal of $M$ to be $$ Fitt_k(M) := I_{r-k}(\varphi). $$

The above definition makes sense, since $Fitt_k(M)$ doesn't depend on the chosen presentation, i.e. if we have another presentation
$$ \varphi': F'\to G' \to M \to 0 $$
with $G'$ of rank $r'$ then Hans Fitting showed in 1936 that $I_{r-k}(\varphi) = I_{r'-k}(\varphi') $.

Note: The canonical map involved in the definition of $I_k(\varphi)$ is obtained in the following way: first by the universal property of exterior algebra, given $\varphi:F\to G$, we have a unique map
$$ \Lambda^k\varphi : \Lambda^k F \to \Lambda^k G $$
and we can thus define the above canonical map as
$$ \Lambda^kF\otimes\Lambda^kG^*\to R, \qquad a\otimes b \mapsto b(\Lambda^k\varphi(a)) $$
