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Hi Let $R = K[X_1,\ldots, X_n]$ where $K$ is a computable field. Suppose we are given two modules with presentations

$$ R^n \rightarrow R^m \rightarrow M \rightarrow 0 $$ and $$ R^l \rightarrow R^p \rightarrow N \rightarrow 0 $$

Then is it possible to verify whether $M$ is isomorphic to $N$ (using a computer algebra software)?

Longtime ago (in 2003) this was not possible. I do not know whether it is possible now.

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Are they arbitrary modules? I believe that if they are finitely generated and graded, then there is some hope. – Youngsu Jun 21 '13 at 14:42
they are finitely generated and graded – Tony Puthenpurakal Jun 28 '13 at 10:18
up vote 2 down vote accepted

You can almost do. There is a theorem (due essentially to Bongartz, but in this form perhaps can be found in Yongwei Yao's thesis) that if $\rm{length}(M\otimes L)) = \rm{length}(N\otimes L))$ for all finite length module $L$, then $M\cong N$ (this is a local result, but under reasonable assumptions, for example if both $M,N$ are presented by matrices in $m=(x_1,\dots,x_n)$, it still should be OK).

So, one can generate a bunch of random modules $L$ supported at the origin and compare the lengths. If they are equal for say, 1000 of them, then the universe has to really conspire against you for the modules not to be isomorphic! $\text{}$

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