mm-ag2

[mm-ag2]

In the proof of proposition 4.1 of lecture 5, how does Prop 1.3 of lecture 4 finish the proof here?

You just have to use that it is surjective, hence the maps on the rings is of the form A\to A/I, for which you may apply the result cited.

[mm-ag2]

In the proof of corollarly 1.5 of lecture 4 from (i) to (ii) we say that f determines a closed subscheme of X, hence a closed subscheme of every affine open, why is this the case? Or in other words, why is the closed subscheme of X intersected with an affine open again a closed subscheme of the affine open? In advance, thank you for your help.

This is equivalent to say that closed immersions are stable under base change (in this case, the base change you consider is the inclusion of the open affine subscheme)

[mm-ag2]

In the proof of proposition 3.1 of lecture 5 it is stated that to prove that a morphism of schemes from V to W is a closed immersion, it is enough to show that for an open covering {U_{i}} the morphism f^{-1}(U_{i}) to U_{i} is a closed immersion. Why is this true?

Look for a hint in this section: there are different equivalent conditions for being a closed immersion, one of which is yours.

https://stacks.math.columbia.edu/tag/01QN

[mm-ag2]

When your scheme is covered by affine opens Spec(R) with R a noetherian ring, then why is this scheme locally factorial? Is it true that noetherian rings are always UFDs? If so, then the first question would follow from the fact that all local rings of this scheme are noetherian rings.

Could you please point out in which point this fact is given? In general no, I don’t think all noetherian rings are UFD.

[mm-ag2]

Why is the fiber product of two separated schemes again separated? I tried to show this using universal properties, but couldn't get the desired result.

https://stacks.math.columbia.edu/tag/01KU try to work out the details following this proof as a guide

[mm-ag2]

Is the pushforward of the structure sheaf under a morphism f again a quasi-coherent sheaf? This question comes from exercise 5 of lecture 7.

In general it is not. Look here for a counterexample and for conditions to be satisfied for that to be true https://stacks.math.columbia.edu/tag/070A

[mm-ag2]

Are we allowed to use that in a Noetherian ring every element is contained in at most finitely many maximal ideals?

When you use a fact of commutative algebra which you are not sure how to prove but it is “general knowledge”, it is good to put a reference for it.

[mm-ag2]

Do we also have to prove that O_X(U) injects into K(X) by sending f to (U,f) in the hand-in, or are we allowed to use this just as we are allowed to use that O_X,x injects into O_X,\eta?

I think you can give it for granted

[mm-ag2]

Is the singleton {x} always irreducible in a topological space X? If not, would it be if we assume X to be irreducible?

Being irreducible is a topological property of the space, not a notion relative to being subset of another space. Hence singletons are irreducible topological spaces

[mm-ag2]

Does every irreducible closed subset of an irreducible scheme X also have a unique generic point?

It would be an irreducible scheme, hence it would have only one generic point.

[mm-ag2]

In a scheme X of finite type over k is the generic point also always a closed point of that scheme? Since one could always take U=Spec(k) as an open neigbourhood of this generic point. And in this open it is a closed point and thus it is also closed in X since X is of finite type over k. The thing that I can think of that goes wrong in this reasoning is that with saying that this open U exists I basically say that {generic point} is open in X.

The generic point of a scheme is not a closed point(in general). For example, you may try to see what goes wrong in definition II.5.2 of Mumford. Otherwise, notice that by definition of generic point, its closure it’s not just itself.

[mm-ag2]

When viewing O_X(U) as a subset of the function field K(X), is it right to denote the elements of O_X(U) as (U,f) for f in O_X(U). I know that they often just use f as a notation since the map of O_X(U) into K(X) is injective, but I find it harder to work with if I forget about the open set.

What is the question in the end? If you want you can denote it like that, if that is more useful for you.

[mm-ag2]

I tried to show that the dimension as defined in AG1 for any open (affine) U in an irreducible scheme X has dimension smaller than or equal to X, but I failed doing this. Is this a fact that we may use in the hand-in? If so, could you still help me solving it, since I am really curious how this is proven.

Look here for the proof of your claim

https://stacks.math.columbia.edu/tag/0054

[mm-ag2]

It is not explicitly defined in the lecture notes, but with a flasque sheaf resolution of a sheaf F do we just mean a complex consisting of only flasque sheaves?

Yes. Of course, to be a resolution you need the complex to be exact as well.

[mm-ag2]

In exercise 1 iv) we have to show that there are natural isomorphisms, but do we also have to give the ismorphisms explicitly?

No, as long you can argue they are the “natural” iso’s you are looking for.

[mm-ag2]

In exercise 2 i) of the hand in, do we really need to take the disjoint union?

Yes.

[mm-ag2]

In mumfords on page 90 it is stated that a point in a scheme X of finite type over k is closed if it is closed in some open neighbourhood of itself. I want to use this in exercise 2 i) of the hand-in, but I cant find any statement about closed points of schemes of finite type over k in the lecture notes. Are we still allowed to use this fact?

Yes, you may use it.