Let $k$ be a field (finite if you'd like), let $(I,\le)$ be a finite directed poset with $|I|=n$, and let $(A_i,f_{ij})_{i\le j\in I}$ be an inverse system of finitely generated, graded, commutative $k$-algebras. Suppose further that we fix a presentation for each $A_i$:
$$A_i:=k[x_{i1},\ldots,x_{in_i}]/I_i$$
Is there an algorithm for determining a presentation of the inverse limit $\varprojlim_{i\in I} A_i$?
An idea: If $k$ is finite, we can determine the homogeneous components of the inverse limit because for each nonnegative integer $m$, there are only finitely many $n$-tuples $(a_1,\ldots,a_n)\in A_1\times\ldots\times A_{n}$, and we can just check which $n$-tuples are coherently related via the $f_{ij}$. However, without a bound on the degree of the generators of the inverse limit, we don't know when to stop computing homogeneous components. Even if we knew when to stop, it is still unclear to me how we might determine generators and relations from the homogeneous pieces.
I've been unable to find such an algorithm implemented in Magma. Perhaps another CAS has one?
The inverse limit $L$ of the inverse system $(A_i,f_{ij}:A_i \to A_j)$ is defined by the exact sequence
$$0 \to L \to \prod_i A_i \xrightarrow{g} \prod_{jk} A_{jk}$$
where $A_{jk} = A_j$ and where $g$ is defined by $g((a_i))_{jk} = a_j - f_{kj}(a_k)$. It is possible, in principle, to implement this directly for example in Macaulay2, but if $(I, <)$ gets large it might come out as computationally unfeasible.