I realized there was one more result I probably should have included last time. Oh well. Here goes:
Let be integral, a prime ideal in R and prime ideals in lying over . If , then .
Proof: Recall that is integral over by last time for any multiplicative set, and also that prime ideals are preserved in rings of fractions. Thus the hypotheses still hold if we localize at . Thus is integral over , and are prime ideals. Thus we can WLOG replace and by their localizations and hence assume they are local. So now is a maximal ideal in . Thus by last time is maximal. Since , we have .
Now we are ready for the two big theorems. Here is the “Lying Over” Theorem. Let be an integral extension. If is a prime ideal in R, then there is a prime ideal in lying over , i.e. .
Proof: First note that and form the two sides of a commutative diagram. By last time is integral over . Choose a maximal ideal in . Thus is maximal in . But is local with unique max ideal , so . But the preimage of a prime ideal is prime, so is a prime ideal in .
Now we just diagram chase: . And also: .
Thus lies over .
Our other big theorem is the one about “Going Up”: If is an integral extension and are prime in R, and lies over , then there is a prime ideal lying over with .
Proof: By last time is an integral extension where is embedded in as . Now we just replace and R by these rings so that both and are . Now we just apply the Lying Over Theorem to get our result.
So as we see here integral extensions behave extremely nicely. These theorems guarantee that se always have prime ideals lying over ones in the lower field. This has some important applications to the Krull dimension that we’ll start looking at next time.