I'm trying to understand a comment in the introduction to chapter 18 of Eisenbud's Commutative algebra with a View Toward Algebraic Geometry:
'..is the unimxedness theorem which explains, for example, why hyperplane sections of smooth varieties do not have embedded components. This is the reason we can treat divisors as codimension 1 subvarieties. It is thus a pillar of algebraic geometry.'
I don't see the pillar =(
What's wrong with embedded components? Is this comment referring to how Cartier divisors correspond to Weil divisors (In regular schemes)? It seems like all you need for this is the codimension 1 local rings to be DVRs. What's the relation between this and primary decomposition of an ideal $(f)$ having embedded primes?
My question is not coherent but maybe someone can just elaborate the quoted text?
Thanks!