# 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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### In how many ways can one extend the zero section of the affine line with a double origin

Let $X$ be the affine line with a double origin over $\mathrm{Spec}\,\mathbb Z$. Let $X_\eta$ be its generic fibre, the affine line with a double origin over $\mathrm{Spec}\,\mathbb Q$.
Let $0$ be ...

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### Embed FPPF group scheme into smooth one

Let $A$ be a ring and $G$ be an affine commutative FPPF group scheme over $A$. Can we embed $G$ into a smooth group scheme over $A$?

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### The underlying space of an affine open dense subscheme

Let $X$ be a Noetherian scheme, $U\subset X$ be an affine open dense subscheme. Is the underlying space of $U$ necessarily homeomorphic to the underlying space of $X$?

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### Quasi-compactification of locally spectral spaces

Let $X$ be a locally spectral topological space (i.e. a space admitting an open cover by spectral spaces). Does there necessarily exist a quasi-compact locally spectral space $Y$ and an injective ...

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283 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 ...

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### Noetherian affine schemes for which localization computes the values of the structure sheaf

Is there a non-tautological characterization of the class of commutative unital Noetherian rings satisfying the following property: for any open subset $U$ of the topological space of $\mathrm{Spec}\,...

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583 views

### 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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235 views

### 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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137 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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363 views

### Schemes admitting a cover by isomorphic affine opens

Let $X$ be a Noetherian integral scheme. When does $X$ have a cover by affine open subschemes that are isomorphic as abstract schemes?
Does there exist an example when we have a cover by $n$ ...

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234 views

### 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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### Extend a Morphism of Schemes

I have a question about following argument used in Szamuely's "Galois Groups and Fundamental Groups" in the excerpt below (or look up at page 159):
Let $X,Y$ schemes which are finite and locally free ...

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342 views

### Krull dimension of the ring of global sections

Let $X$ be an irreducible scheme. Can the Krull dimension of $\mathcal{O}_X(X)$ exceed that of $X$?

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### Motivic strong bellows conjecture

There is a theorem due to Gaifullin--Ignashchenko stating that the Dehn invariant of any flexible polyhedron in the $n$-dimensional Euclidean space ($n\geq 3$) is constant during the flexion.
Is ...

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793 views

### 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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### Bijective restriction of the normalization morphism

Let $X$ be an integral separated scheme of finite type over $\mathbb{C}$. Consider the normalization morphism $f:X'\rightarrow X$. Can we always find an affine open $U\subset X'$ such that $f|_U:U\...

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### Construction of the petit Zariski topos out of the gros topos of a scheme

Let S be a scheme. Let (Sch/S) be a small category of schemes over S (including essentially all finitely presented schemes affine over S). Let E = (Sch/S)zar denote the gros Zariski topos with its ...

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### When are valuative criteria useful?

We have valuative criteria for properness and universal closedness of morphisms of schemes. I would agree that these criteria shed some light on the geometric nature of these morphisms. However, are ...

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490 views

### Can not tell colimits from limits

Proposition 71 here reads:
Let $X$ be a concentrated scheme and $F$ a quasi-coherent sheaf of modules. The
following are equivalent:
(a) The functor $\mathrm{Hom}(F, −):Qco(X)\rightarrow Ab$ ...

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414 views

### Noetherian stalks imply locally Noetherian

Is there an example of a non-Noetherian integral affine scheme with Noetherian space and Noetherian stalks? What if we replace "integral" with "reduced"?

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### Compact quasi-coherent sheaves

Let $X$ be a scheme. What are the compact objects in the category of quasi-coherent $\mathcal{O}_X$-modules? All references seem to discuss the derived category but I need the abelian category.

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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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291 views

### 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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364 views

### 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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### 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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### 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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### 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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94 views

### 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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82 views

### 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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### 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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### 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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### 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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470 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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### on flat morphisms

Let $j:U\rightarrow X$ an open immersion between k-schemes of finite type and $f:X\rightarrow S$ a surjective k-morphism of finite type.
We suppose that $f\circ j:U\rightarrow S$ is faithfully flat, ...

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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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### Ring-theoretic characterization of open affines?

Background
Recall that, given two commutative rings $A$ and $B$, the set of morphisms of rings $A\to B$ is in bijection with the set of morphisms of schemes $\mathrm{Spec}(B)\to\mathrm{Spec}(A)$. ...

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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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### Affine scheme on spec(A) of a ring A as the sheafification of a pre-sheave on spec(A)?

It is obvious that there is a parallel between the definition of structure sheaf of $\operatorname{Spec}(A)$
versus the sheafification of a pre-sheaf.
The definition of the sheaf $\mathscr F^+$ ...

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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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### 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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### 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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### on universal homeomorphisms between schemes

We are taught since when we are young that schemes are cool because they take into account "nilpotents". This means also that we can distinguish between schemes which have the same underlying ...

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### Examples of non-projective morphisms with projective fibres

Let $X\to S$ be a morphism of noetherian schemes such that, for all $s$ in $S$, the morphism $X_s\to $ Spec $k(s)$ is projective.
Then it doesn't follow that $X\to S$ is projective in general. In ...

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140 views

### 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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### 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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### 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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220 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 ...

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293 views

### 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 ...