Suppose I have a short exact sequence, with I an ideal and R a polynomial ring over C:
$ 0 \rightarrow I \rightarrow aI \rightarrow \frac{aI}{I} \rightarrow 0$
where $ a \in R$ and the first nontrivial map is multiplication by a. If I and aI are of grade 2, why must a be a unit in R?
My instinct is to use long exact sequences in ext to gain isomorphisms of hom( , ) but I cannot make that very useful.