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.

Filter by
Sorted by
Tagged with
2
votes
0answers
378 views

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 ...
4
votes
0answers
124 views

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 $A$?
3
votes
2answers
329 views

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 $X$?
2
votes
1answer
134 views

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 ...
0
votes
0answers
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 ...
2
votes
0answers
106 views

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}\,...
5
votes
2answers
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 $O_X$-...
2
votes
1answer
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 ...
0
votes
0answers
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 ...
5
votes
0answers
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 $n$ ...
5
votes
0answers
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 ...
1
vote
0answers
137 views

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 ...
3
votes
1answer
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 $X$?
3
votes
0answers
172 views

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 ...
6
votes
2answers
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 $f$ ...
0
votes
0answers
57 views

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\...
16
votes
2answers
1k views

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 ...
5
votes
2answers
639 views

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 ...
3
votes
1answer
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 $\mathrm{Hom}(F, −):Qco(X)\rightarrow Ab$ ...
2
votes
1answer
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"?
5
votes
0answers
310 views

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.
0
votes
0answers
380 views

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 ...
4
votes
1answer
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?
2
votes
0answers
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 ...
2
votes
0answers
130 views

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 ...
3
votes
2answers
218 views

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 ...
1
vote
0answers
82 views

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 $X$?
0
votes
0answers
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_{...
0
votes
1answer
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?
1
vote
0answers
154 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 $n$ ...
3
votes
2answers
269 views

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 ...
6
votes
1answer
261 views

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 ...
-2
votes
2answers
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?
-1
votes
1answer
299 views

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, ...
2
votes
1answer
270 views

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 ...
43
votes
2answers
4k views

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 $\mathrm{Spec}(B)\to\mathrm{Spec}(A)$. ...
1
vote
0answers
257 views

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 ...
22
votes
1answer
3k views

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 $\mathscr F^+$ ...
1
vote
0answers
51 views

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 ...
1
vote
0answers
116 views

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?
0
votes
2answers
407 views

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 ...
1
vote
1answer
219 views

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 $\...
2
votes
1answer
525 views

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 ...
3
votes
1answer
474 views

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 ...
2
votes
1answer
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 ...
13
votes
1answer
471 views

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?
7
votes
0answers
229 views

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 ...
0
votes
0answers
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 ...
4
votes
2answers
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 $...
0
votes
0answers
133 views

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 ...

1 2 3
4
5
12