# Questions tagged [schemes]

The first purpose of schemes theory is the geometrical study of solutions of algebraic systems of equations, not only over the real/complex numbers, but also over integer numbers (and more generally over any commutative ring with 1). It was finalized by Alexandre Grothendieck, during the 1950s and the 1960s.

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### Mistake in Hartshorne's Exercise II.1.1?

This is really an elementary question, but let me state it. Exercise 1.1 of the second Chapter of Hartshorne's Algebraic Geometry ask to prove that the sheaf associated to the presheaf sending every ...

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### Finite Picard group

Does there exist a connected scheme, smooth, proper, and positive-dimensional over $\mathbb{C}$ with finite Picard group? Note that Picard group has cardinality$>1$. Also note that this can not ...

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### Affine open with irreducible complement

Let $X$ be an integral Noetherian separated scheme. Under what conditions can we find a non-empty affine open in $X$ whose complement is irreducible?

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### Map to a given vector bundle from a split vector bundle

Let $X$ be a connected scheme, smooth and proper over $\mathbb{C}$. Let $F$ be a locally free $\mathcal{O}_X$-module of finite rank $r>1$. Suppose on a non-empty affine open $U\subset X$ whose ...

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### Size of the ring of functions on open subschemes

This question consists of two related sub-questions.
Let $X$ be a Noetherian integral affine scheme. Under what assumptions on $X$ does every open subscheme of $X$ have a Noetherian ring of global ...

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### Do codimension 1 subsets of a scheme cover it?

Let $X$ be an irreducible scheme. A point $p\in X$ is said to have codimension $n\in\mathbb{Z}_{\geq 0}\cup \{\infty\}$ if $\overline{\{p\}}$ has codimension $n$. Is it true that any point of positive ...

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### Schemes with no finite morphisms onto themselves

Let $X$ be an integral Noetherian scheme that has no automorphisms other than the identity. Is it possible to characterize (in a not completely tautological way) $X$ such that
there is no finite ...

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### Map of coherent sheaves inducing isomorphism on the stalks at the generic point

Let $f:X\rightarrow Y$ be a finite morphism between Noetherian integral schemes that is surjective on the underlying topological spaces. Does there exist an integer $n>0$ and a coherent $O_X$-...

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### depth and extension of sections

Let $S$ be an affine scheme, $X$ smooth affine over $S$ and $U$ an open subset of $X$, fiberwise of codimension at least two.
Suppose that we have a function on $U$, can we extend it to $X$?

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### Homeomorphisms coinciding on closed irreducible subsets

Let $f_1$, $f_2:X\rightarrow Y$ be two homeomorphisms of sober spaces. Assume that for any closed irreducible subset $Z\subset X$, we have $f_1(Z)=f_2(Z)$. In particular, $f_1$ and $f_2$ coincide on ...

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### Existence of bijection inducing isomorphism on stalks implies existence of isomorphism

Let $X$, $Y$ be connected smooth projective $\mathbb{C}$-schemes. Let $f:Set(X)\rightarrow Set(Y)$ be a bijection of the underlying sets. Suppose that for any $x\in X$, there exists an isomorphism $O_{...

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### Morphism of schemes with non-sober image

Let $f:X\rightarrow Y$ be a morphism of schemes. Can the image of $f$ endowed with the subspace topology not be sober?

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### Closed points on scheme locally of finite Krull dimension

Let $X$ be an irreducible scheme that has a possibly infinite cover by open sets of finite Krull dimension. Does $X$ have a closed point?

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### 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 $n$ ...

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### Irreducible of finite Krull dimension implies quasi-compact?

Let $X$ be the underlying space of a scheme.
If $X$ is irreducible of finite Krull dimension, is it necessarily
quasi-compact?
Is it necessarily Noetherian?
What if we assume not
only that Krull ...

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238 views

### One-dimensional scheme with no closed points

Can someone give a reasonably explicit example of an irreducible one-dimensional scheme with no closed points?

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### Non-flat locus for smooth schemes

Let $X$, $Y$ be connected smooth schemes of finite type over an algebraically closed field of characteristic $0$. Let $f:X\rightarrow Y$ be a non-birational morphism surjective on the underlying ...

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### Obstructions to abelian sheaf being quasi-coherent

Let $X$ be a Noetherian spectral topological space. A necessary condition for an abelian sheaf on $X$ to be quasi-coherent with respect to some affine scheme structure on $X$ is that its higher ...

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### Cohomology of the pullback of a coherent sheaf (considered as an abelian sheaf)

Let $X$ be an affine Noetherian scheme, $Y$ a separated Noetherian scheme, $f:X\rightarrow Y$ a morphism of schemes inducing a homeomorphism on the underlying topological spaces.
Let $F$ be a ...

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### Non-constructible conditions on the fibers that lift from the generic point to a non-empty open

Let $f:X\rightarrow Y$ be a flat morphism of schemes, with an irreducible locally Noetherian target. Call a condition on the fibers of $f$ "good" if the condition holds at the generic point of $Y$ iff ...

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### The underlying space of a scheme remembers its affineness?

Let $f:X\rightarrow Y$ be a morphism of schemes. We know that if $Y$ is affine and $f$ induces homeomorphism on the underlying spaces then $X$ is affine. Is it true that if $X$ is affine and $f$ ...

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### Epimorphisms from an affine scheme?

Let $X$ be an affine scheme. Let $f:X\rightarrow Y$ be an integral morphism that is an epimorphism in the category of schemes. Is $Y$ affine?

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### Doing scheme theory with Hausdorff spaces

Suppose I have an allergy to non-Hausdorff spaces but I really want to do, say, arithmetic geometry. I wonder if there is some perverse way I could develop scheme theory that would accomodate for my ...

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### Closed immersion inducing isomorphisms on stalks

Does there exist a closed immersion of schemes that induces isomorphisms on stalks, has a non-empty source, an irreducible (possibly non-reduced) target, and is not an isomorphism? Such $f$ is ...

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### Subschemes of the affine line over PID

Let $R$ be a PID with infinitely many prime ideals. Suppose we have two integral locally closed subschemes of $\mathrm{Spec}\,R[x]$ such that
both have non-empty intersection with the affine open $\...

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### Schemes monomorphing into affine scheme of dimension 1

Let $Y$ be an affine scheme of Krull dimension 1. Let $X\rightarrow Y$ be a monomorphism in the category of schemes. If $X$ is connected, is $X$ necessarily affine? What if we assume that $Y$ is a ...

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### Voisin examples in $p$-adic geometry

Let $K$ be an algebraic closure of p-adic rationals. Does there exist a proper smooth rigid-analytic variety over $K$ whose etale homotopy type is not isomorphic to etale homotopy type of a proper ...

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### Embedding smooth proper schemes into smooth proper schemes

Do there exist connected proper smooth $\mathbb{C}$-schemes $X_i$ ($\forall i\in \mathbb{Z}_{>0}$) with $\mathrm{dim}_{\mathbb{C}}X_i=i$ such that $X_i$ admits an immersion into $X_{i+1}$ and any ...

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### Weil homotopy theory

In algebraic geometry, we have something called Weil cohomology theories, which formalize the notion of a "good" cohomology theory of smooth projective varieties. I believe that for $l$-adic ...

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### Is there an analogue of projective spaces for proper schemes?

Does there exist a countable set of connected proper smooth $\mathbb{C}$-schemes such that any connected proper smooth $\mathbb{C}$-scheme admits a $\mathbb{C}$-immersion into one of them?

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### Sufficient condition for quasi-compactness of scheme morphism

I wonder what kind of conditions on a morphism of schemes imply, in a non-trivial fashion, quasi-compactness of the morphism. Some examples
Finiteness of surjective etale morphisms
Is a universally ...

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### A proper flat family with geometrically reduced fibers

Let $R$ be a Noetherian commutative unital ring. Let $f:X\rightarrow Y=\mathrm{Spec}\,R$ be a proper flat morphism of schemes with geometrically reduced fibers. We want to prove that the induced map $...

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### Linear Morphism of Schemes

Let $U$ be an arbitrary scheme and we consider the schematic fiber product $\mathbb{A}^1 _U = \mathbb{A}^1 _{\mathbb{Z}} \times_{\mathbb{Z}} U$.
My question referer to Bosch's "linear morphisms" (of ...

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### 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 ...

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### When does the image of a morphism of schemes support scheme structure?

Let $Y$ be a qcqs scheme, $f:X\rightarrow Y$ be a quasi-compact morphism locally of finite presentation. Are there any conditions on $f$ which
do not force $f(X)$ be open or closed but force it to be ...

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### An injection from curve to projective plane is subscheme inclusion

Let $X$ be a connected scheme, $X\rightarrow \mathrm{Spec}\,\mathbb{C}$ be a morphism of finite type and relative dimension 1. Assume that there is a $\mathbb{C}$-morphism $f:X\rightarrow \mathbb{P}^2$...

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### Theorem on formal functions and cohomological flatness

Let $f:X\rightarrow S$ be a proper morphism of schemes with Noetherian target. The theorem on formal functions says that for any point $s\in S$ there is an isomorphism between inverse limits of $(f_*...

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### Morphisms such that the inverse image of every affine open is contained in an affine open

Is there a name/description in standard terms of the class of morphisms of schemes defined by the following property: the inverse image of any affine open is contained in an affine open?
It should ...

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### Given a semi-abelian scheme, is the set of points such that the fibres are abelian varities open?

Let $\pi:\mathcal{A}\rightarrow C$ be a semi-abelian scheme, i.e. $\mathcal{A}$ is a smooth separated commutative group scheme over $C$ via $\pi$ with geometrically connected fibres, such that each ...

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### Global functions on a flat proper family

Let $R$ be an integral domain. Let $f:X\rightarrow \mathrm{Spec}\,R$ be a flat proper morphism of schemes. Is it possible that $O_X(X)$ is not a flat $R$-module?

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### Irreducible Smooth Proper one-dimensional Schemes isomorphic to $\mathbb{P}^1$

I have a curious question about an argument/hint given in following thread:
https://math.stackexchange.com/questions/3062986/irreducible-smooth-proper-one-dimensional-mathbbz-schemes
The OP asked if ...

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### Projective subvarieties of blow-ups of affine varieties

Let $X$ be an integral affine scheme of finite type over a field $k$. Let $Y\subset X$ be an integral closed subscheme of codimension $n>0$. We blow up $X$ along $Y$ and get a $k$-scheme $X'$. Is ...

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### Why is, for a group scheme of finite type, “smooth” (resp. irreducible) equivalent to “geometrically reduced” (resp. geometrically irreducible)?

I have some questions about two statements from Bosch's "Algebraic Geometry and Commutative Algebra" about algebraic varieties (page 479). Since I still don't have the permission to add images I quote ...

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### Valuative criterion over non-locally Noetherian base

Let $X$ be an irreducible scheme and $f:X\rightarrow S$ be a morphism of finite type. Let $\eta$ be the generic point of $X$. Assume that for any (not necessarily discrete) valuation ring $A \subset K=...

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### Finite Group acting on a Quasiprojective Scheme [closed]

I was trying to do this problem. I was able to proove it in the case where X embeds in an open affine of $\mathbb{P}^n_k$, but I have no idea about how to proceed in the general case. In particular I ...

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### Schemes for conditional distributions (monads)

(Note: This is a heuristic question. I'm trying to work out if this idea makes sense. I don't have much background in this area, so apologies if I'm wide of the mark.)
Suppose you have a monad ...

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### Are higher etale homotopy groups topological groups in a natural way?

Since etale fundamental group of a scheme $X$ is the group of natural automorphisms of the fibre functor of the category of finite etale covers of $X$, it comes with structure of a topological group. ...

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### Quasi-compactness of irreducible separated scheme locally of finite type

Is an irreducible separated scheme locally of finite type necessarily quasi-compact?

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### Finiteness of surjective etale morphisms

Is every surjective etale morphism from a connected separated scheme to $A^n_{\mathbb{C}}$ of finite type? Is it finite? We use Stacks project's definitions.
EDIT: From Jason Starr's answer, we ...

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### Invariants of etale topological type that are not homotopy invariants

Artin--Mazur theory attaches etale homotopy type to reasonable schemes. Associated to this homotopy type are certain invariants of the scheme, such as etale fundamental group and higher homotopy ...