5
$\begingroup$

Let $A$ be a commutative algebra over a field $k$ which is finite dimensional as a vector space over $k$. Let $M$ be a faithful $A$-module. Does it follow that $dim_k(M)\geq dim_k(A)$?

$\endgroup$
8
$\begingroup$

$A$ is a product of local Artinian rings, so the question is local. The best result I know is in this paper of Gulliksen, who proved that if the socle dimension is at most $3$, then the length of any faithful module is at least the length of the ring. So the answer is yes if $A$ is the product of local Artin rings of socle dimension at most $3$. He also gave counter example when the socle dimension is bigger than $3$.

| cite | improve this answer | |
$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

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