10
$\begingroup$

Let $A$ be a finite dimensional commutative algebra. We can assume that it is local.

Question: Which such $A$ have the property that every finite dimensional $A$-module has complexity at most 1? (This should be equivalent to the simple module having complexity equal to one or equivalently bounded Betti numbers)

Note that a module has complexity at most one if and only if the terms in a minimal projective resolution have bounded dimensions.

Examples include $A=K[x]/(x^n)$. Do you know other examples?

$\endgroup$

1 Answer 1

11
$\begingroup$

There are no other examples. This property is equivalent to $A$ being a hypersurface (see Avramov's note "Infinite Free Resolutions"). By Cohen Structure Theorem, an Artinian local hypersurface (which is automatically complete) that contains a field must be isomorphic to $k[[x]]/(f)$, which is equal to $k[[x]]/(x^n) = k[x]/(x^n)$ with $n$ being the smallest power of $x$ in $f$.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.