Maximal Ideals/Domination/Local Rings

If anyone is interested in the context of this question, I am reading through the proof of Lemma 6.5 in Hartshorne's Algebraic Geometry.

We have fields such that . Also, assume is algebraically closed.

Let , and let be the integral closure of the ring in . It can be shown that is a Dedekind domain which is finitely generated as a -algebra (and I'm fine with that).

Now, suppose is a discrete valuation ring between and such that . Since is integrally closed (being a DVR), we must have .

Here's where I get lost: they say that if is the (unique) maximal ideal of , then is a maximal ideal of , and that is *dominated* by . What I don't see is (1) how we know that is a maximal ideal of (instead of just prime), and (2) why they are saying that is local. They seem to be claiming this, as the relation of domination is defined only for two local rings.

Re: Maximal Ideals/Domination/Local Rings

Anyone? Sorry if it seems like there are too many details; I just didn't know how many details would be necessary to solve this. Hartshorne just skips right over this as if it's blatantly obvious (who knows, maybe it is... but I don't see it).

Re: Maximal Ideals/Domination/Local Rings

Quote:

Originally Posted by

**topspin1617** Anyone? Sorry if it seems like there are too many details; I just didn't know how many details would be necessary to solve this. Hartshorne just skips right over this as if it's blatantly obvious (who knows, maybe it is... but I don't see it).

In my humble opinion, Hartshorne's book is a terrible, awful, anguishing, sickening one for learning algebraic geometry. It though can be a very good one for advanced students in the subject and/or as a reference book.

Somewhere in the net (google it) solutions to some of this book's ridiculously hard (sometimes) exercises. give it a shot.

Tonio

Re: Maximal Ideals/Domination/Local Rings

Quote:

Originally Posted by

**topspin1617** If anyone is interested in the context of this question, I am reading through the proof of Lemma 6.5 in Hartshorne's Algebraic Geometry.

We have fields

such that

. Also, assume

is algebraically closed.

Let

, and let

be the integral closure of the ring

in

. It can be shown that

is a Dedekind domain which is finitely generated as a

-algebra (and I'm fine with that).

Now, suppose

is a discrete valuation ring between

and

such that

. Since

is integrally closed (being a DVR), we must have

.

Here's where I get lost: they say that if

is the (unique) maximal ideal of

, then

is a maximal ideal of

, and that

is

*dominated* by

. What I don't see is (1) how we know that

is a maximal ideal of

(instead of just prime), and (2) why they are saying that

is local. They seem to be claiming this, as the relation of domination is defined only for two local rings.

well, suppose that is not a maximal ideal of and let be a maximal ideal of which contains . now, we get a chain of prime ideals of , contradicting this fact that the Krull dimension of a Dedekind domain is at most .

by the way, understanding Hartshorne without having a good commutative algebra background is impossible. the author assumes that background.

Re: Maximal Ideals/Domination/Local Rings

Quote:

Originally Posted by

**NonCommAlg** well, suppose that

is not a maximal ideal of

and let

be a maximal ideal of

which contains

. then we will have a chain of prime ideals

of

, contradicting this fact that the Krull dimension of a Dedekind domain is at most

.

by the way, understanding Hartshorne without having a good commutative algebra background is impossible. the author assumes that background.

Oh wow..... thank you. Now I feel like a complete moron lol, this is so obvious.

I do have a very good commutative algebra background. I just wasn't even thinking about Krull dimension. I was thinking more in the geometric direction, trying to figure out if there was something about the geometry here that forced that ideal to be maximal.

That still leaves my other question though... do we know that must be local? Again, it's just the way that the author mentions " dominates " which is only defined for local rings. Though I SUPPOSE that could be a typo/oversight/mistake.

Re: Maximal Ideals/Domination/Local Rings

Re: Maximal Ideals/Domination/Local Rings

Re: Maximal Ideals/Domination/Local Rings

Quote:

Originally Posted by

**topspin1617** You're right in that there doesn't seem to be any reason that

should be local.

If that's how he meant domination, then he really shouldn't use it in this context. I'm just reading his own definition:

"If

are local rings contained in a field

, we say that

*dominates* if

and

."

yes, i saw the definition but that's not a big deal. the author is trying to prove that and this tells you that is not necessarily local.

Re: Maximal Ideals/Domination/Local Rings

i just realized that, in my proof to your first question, we need to explain why . i think the author assumes from the beginning that , which implies that and so .

you might have another way to show this though.

Re: Maximal Ideals/Domination/Local Rings

Quote:

Originally Posted by

**NonCommAlg** yes, i saw the definition but that's not a big deal. the author is trying to prove that

and this tells you that

is not necessarily local.

I know it's not that big of a deal. I was just wondering if he purposely meant something there, in which case that would have mattered.

Quote:

Originally Posted by

**NonCommAlg** i just realized that, in my proof to your first question, we need to explain why

. i think the author assumes from the beginning that

, which implies that

and so

.

you might have another way to show this though.

Well, he doesn't assume that from the beginning. He begins the following paragraph with "if, in addition, ". I'll have to think about why the intersection is nonzero in general.