2
$\begingroup$

I would like to know a reference to understand what an author means with:

"Let $A$ be a commutative algebra with unit. Regard $A$ as the algebra of functions on an affine variety $M$ (possibly singular). Actually points on $M$ are just ideals in $A$ of codimension one."

My doubt is how to regard $A$ in this way and also how to show the above equivalence with ideals.

  • 0
    Presumably $A$ is also integral and finitely generated as an algebra over a field? This should be explained in a good upper year undergraduate text, e.g. Dummit and Foote, but to get a good picture you should look at an introductory algebraic geometry text. See also: https://en.wikipedia.org/wiki/Spectrum_of_a_ring2012-10-05
  • 2
    Under the necessary assumptions about the algebra and the underlying field $k$, and once we're thinking of $A$ as an algebra of functions on $M$, the equivalence between points and codimension-1 ideals is that each point $p$ corresponds to the $k$-algebra homomorphism $A\to k$ given by $f\mapsto f(p)$, and this in turn corresponds to its kernel which is a codimension-one ideal of $A$.2012-10-05
  • 0
    @Andrew Yes, you can assume these hypotheses. I didn't see anything in this link, what should I look for to get an answer?2012-10-06

0 Answers 0