# How to check whether a module is an n-th syzygy

Given a finite dimensional algebra $A$, define $\Omega^{n}(mod-A)$ (modules here are always finite dimensional) to be the full subcategory of projective modules or modules $M$ such that $M \cong \Omega^{n}(N)$ for some other module $N$. How can one check in an easy way to see whether a given module $M$ (we can assume it is indecomposable) is in $\Omega^{n}(mod-A)$? For special subcategories there is sometimes an easy answer. Example: Let $A$ have dominant dimension $d \geq 1$, then $\Omega^{n}(mod-A)$ is equal to the full subcategory of modules having dominant dimension at least $n$. I'm interested in calculating the intersection $\Omega^{\infty}$ of all $\Omega^{n}$ for $n \geq 1$ for special algebras (which can be assume to be representation-finite if that helps) with the GAP package QPA.

## 1 Answer

A method for this is now implemented in QPA as IsNthSyzygy. For a given indecomposable module $$M$$, one checks whether $$M$$ is a direct summand of $$\Omega^n(\Omega^{-n}(M)) \oplus P(M)$$ when $$P(M)$$ is the projective cover of $$M$$.

• Does that command test whether $M$ is an nth syzygy? Or whether it is a summand of an nth syzygy? Dec 1 '20 at 10:32
• @JeremyRickard I think it actually just checks whether it is a direct summand. But for many algebras this is the same at least for many n. Im not sure how to check really whether a module is an NthSyzygy but I didnt think about this for some time.
– Mare
Dec 1 '20 at 10:49