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Ever seen a ringed group?

A locally ringed space is a common generalization of schemes and various manifolds. I am wondering about a locally ringed group which should be a common generalization of group schemes and various Lie ...
Bugs Bunny's user avatar
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4 votes
0 answers
317 views

Is the "naive" version of Chevalley's theorem still true?

Reposting from math.se in case more people are interested here. Chevalley's theorem says that if $f \colon X \to Y$ is a morphism of finite presentation of schemes and $C \subset X$ is constructible, ...
Spencer Dembner's user avatar
4 votes
0 answers
483 views

A slightly canonical way to associate a scheme to a Noetherian spectral space

Let $C$ be the category whose objects are Noetherian spectral topological spaces and whose morphisms are homeomorphisms. Let $\mathrm{AffSch}$ be the category of Noetherian affine schemes (morphisms ...
user avatar
3 votes
0 answers
197 views

Topological properties of Noetherian affine schemes that do not hold for general Noetherian spectral spaces

I used to think that the only reason why an affine scheme with a Noetherian space can fail to be Noetherian is nilpotents. It turns out that this is not true. This leads me to the following question: ...
user avatar
2 votes
0 answers
215 views

A closed point in the closure of any point in the closure of any point of an irreducible scheme

Let $X$ be the underlying space of an irreducible scheme. In particular, $X$ is non-empty. Note that the closure of any point is a closed irreducible subset. We say that a point has codimension $n$ ...
user avatar
2 votes
0 answers
217 views

Constructible sets, I (Morphisms)

I have a number of questions on constructible sets. The first one is on morphisms: suppose $X$ and $Y$ are constructible sets, respectively in projective spaces $\mathbf{P}_1$ and $\mathbf{P}_2$ over ...
THC's user avatar
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1 vote
0 answers
92 views

Topological space modeled by special topological structures

Let $X$ be a topological space. Suppose it is "modeled by" topological spaces of the form $\text{Spec}(A)$ for some commutative ring $A$, then, (along with some other conditions/structure), we call $...
Praphulla Koushik's user avatar
1 vote
0 answers
298 views

Fully faithful functor from schemes to spaces

Is there a fully faithful functor from the category of schemes to the category of topological spaces and continuous maps (or some other sufficiently topological objects, e.g. smooth manifolds and ...
user avatar
1 vote
0 answers
127 views

Explicit description of the scheme obtained by relative gluing data over a base scheme

I have recently been trying to get a better understanding of the projective space bundle of a quasi-coherent sheaf of graded algebras over a scheme $X$. The key idea is the following construction ...
Luke's user avatar
  • 453
0 votes
0 answers
144 views

Making a quasi-compact open into an affine open

Let $X$ be a spectral topological space, $U\subset X$ be a quasi-compact open subspace. Is there necessarily some scheme structure on $X$ (we do not require it to be affine) such that $U$ endowed with ...
user avatar
0 votes
0 answers
146 views

A scheme whose underlying space is the product of the underlying spaces of schemes

We know that the product of two spectral topological spaces is spectral. If $X$ is the underlying space of the scheme $\mathrm{Spec}\,\mathbb{Z}[x]$, what is a simple example of an affine scheme ...
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0 votes
0 answers
325 views

Grothendieck topology on a scheme equivalent to the circle

Suppose a class of morphisms of schemes is reasonable enough so that we can associate a small site to any scheme (just like small étale site). Is there a simple class of morphisms such that there is a ...
user avatar
0 votes
0 answers
238 views

Pro-constructible subset of scheme intersects very dense subsets?

Let $X$ be a scheme, let $D$ be a very dense subset of $X$ and let $Y$ be a pro-constructible subset of $X$. Is it true that $Y \cap D \neq \emptyset$? If $Y$ is just constructible, this is true. ...
user68570's user avatar