Suppose that we are given an integral $k$-algebra $A$ of finite type explicitly, by which I mean that we are given the generators of the defining ideal $J$ where $A = k[x_1,...,x_n]/J$. What kinds of tools are out there to compute the integral closure of $A$? I would like the answer as explicitly as possible i.e. generators of the defining ideal.
While suggestions of computer programs are welcome, I want to be able to do these on my own, so I am looking for results which let me prove the answer. I'm asking from the perspective of someone who knows very little computational commutative algebra.
As a related question, if I have two such rings $A$, $B$ given explicitly as above, together with an explicit homomorphism between them, how can I go about determining the kernel and cokernel explicitly? Also, how about if we localize everything at a maximal ideal?
Broadly speaking, I would like to know about what kinds of computational methods are available for rings which arise from studying complex algebraic varieties. Have people out there settled these kinds of computations completely, or is this a hard question in general?