# 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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### Is a universally closed monomorphism a closed immersion?

The question is essentially in the title: $f\colon X \rightarrow Y$ is a monomorphism of schemes that is universally closed; does this imply that $f$ is a closed immersion? Any such $f$ is quasi-...

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### Glueing modules over $\{x\}\times \operatorname{Spec} R$

Let $k$ be a field and $(C,\mathcal{O}_C)$ be a smooth geometrically irreducible projective curve over $k$ of function field $k(C)$ and let $x$ be a closed point on it. From Laszlo-Beauville's lemma, ...

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170 views

### Proving the representability of a functor that is covered by open subfunctors

I already posted this on math.stackexchange some while ago, but haven't received any answers yet. (https://math.stackexchange.com/questions/3221006/proving-the-representability-of-a-functor-that-is-...

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### Coproduct in the category of affine schemes, functorial point of view

$\let\opn=\operatorname$An affine scheme is defined as a covariant representable functor $X:\opn{CRing} \to \opn{Set}$. The Yoneda embedding implies that the category of affine schemes, $\opn{...

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### Finding a cover of the Hilbert functor, that corresponds to a cover of the Grassmannian

Let $\mathrm{Grass}_{k+1,n+1}:\mathrm{Sch}^{\circ}\rightarrow\mathrm{Set}$ be the Grassmann functor, which maps a scheme $S$ to the set:
$$\left\{\mathscr{U}\subseteq\mathscr{O}_S^{n+1}:\,\,\mathscr{...

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### Equivalence between categories of affine schemes over $X$ and representable functors $\operatorname{Points}(x) \to \operatorname{Sets}$

I am reading Strickland - Formal schemes and formal groups. For a functor $X\colon \operatorname{Rings}\to \operatorname{Sets}$, he defines (2.14) the category of $\operatorname{Points}(X)$ in the ...

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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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160 views

### Restriction of a Cartier Divisor

Let $X$ be a surface (so $2$-dimensional proper $k$-scheme)
$D \subset X$ an effective Cartier divisor of $X$ which corresponding to an invertible sheaf $\mathcal{L}=O_X(D)$ and
$C \subset X$ a ...

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150 views

### Smallest non-vanishing index of Local-Cohomology and Ext functor, as in the definition of depth, for not necessarily affine schemes

Let $(Z,\mathcal O_Z)$ be a closed subscheme of a Noetherian scheme $(X,\mathcal O_X)$. Then there is an ideal sheaf $\mathcal J$ on $X$ such that $i_*(\mathcal O_Z) \cong \mathcal O_X/\mathcal J$ , ...

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111 views

### Sheaf of Kähler Differentials is Invertible in Dense Open Subset

Let $f:S→B$ be an elliptic fibration from an integral surface $S$ to integral curve $B$
.
Here I use following definitions:
A surface (resp. curve) is a $2$
-dim (resp. $1$-dim) proper k scheme ...

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166 views

### Tangent Space of Picard Scheme

Let $X$ be a scheme and it's Picard scheme $\underline{Pic}(X)$ exist. Denote by $\underline{Pic}^0(X)$ the connected component of the origin of
the Picard scheme. My question is what the geometric ...

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112 views

### Locus of trivialization of an extension of a vector bundle

Let $X$ be a normal noetherian affine scheme. Let $j:U\rightarrow X$ be a codimension two open of $X$ and $\mathcal{E}$ a vector bundle on $U$.
We assume that $j_*\mathcal{E}$ is a vector bundle. In ...

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### Relation between $\mathrm{Proj}(A[tI])$ and $\mathrm{Proj}(A[tJ])$ for ideals $I$ and $J$ of $A$ with $I^2 \subset J \subset I$

Let $k$ be a field and $A$ a noetherian local $k$-algebra.
Let $I$ and $J$ be two ideals of $A$ with $I^2 \subset J \subset I$.
Let $A[tI] = \bigoplus\limits_{i\geq 0} t^i I^i$ and $A[tJ] = \...

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### Algebraic geometry “over the function field” of the base

This is vaguely similar to, but quite different from, this question.
In the above linked question the focus is on fields of large cardinality per se. Here we fix a base field $k$ (say algebraically ...

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### True on stalks, false on affine opens [closed]

In scheme theory, there are some properties that can be specified purely on the stalks of the structure sheaf but they "lift" to the properties of the values of structure sheaf on affine opens, e.g.
...

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### Sections of Normal Sheaf and Tangent Space of Hilbert Scheme [duplicate]

Let $X= \mathbb{P}^n$ the projective space (interpreted as scheme) and $Y \subset X$ a closed subspace.
The existence of a called Hilbert scheme $H_X(Y)$ is well known which parametrizes closed ...

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168 views

### Number of distinct scheme structures on a set [closed]

Given a cardinal number $|X|$, how many isomorphism classes of schemes with the cardinality of the set of points equal to $|X|$ are there?

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### Representability of Grassmannian functor by a scheme

I am having some trouble following a proof that the Grassmannian functor is representable by a scheme. I am following the proof in EGA 9.7.4. It is only a small step that I am stuck on. For reference, ...

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### 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 ...

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### Affine open with an irreducible complement

Let $X$ be an integral scheme such that the morphism to $\mathrm{Spec}(\mathbb{Z})$ is proper. Assume the morphism to $\mathrm{Spec}(\mathbb{Z})$ has well-defined relative dimension and that the ...

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### Strongly abnormal schemes

Call a scheme $Y$ proper positive-dimensional over $\mathrm{Spec}\,\mathbb{C}$ abnormal if there exists an irreducible scheme $X$ affine of finite type over $\mathrm{Spec}\,\mathbb{C}$ and a $\mathbb{...

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354 views

### Why are algebraic schemes called algebraic?

In scheme theory, an algebraic scheme is the data of a scheme + a morphism of finite type to the spectrum of a field. Where does the term "algebraic scheme" come from? It does not seem intuitive to me ...

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### Henselian Schemes

I have a question about properties of Henselian ring/schemes exploited in M. Artin's article "On Isolated Rational Singularities of Surfaces":
Here the relevant excerpt:
In the excerpt we start with ...

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149 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:...

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419 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 ...

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200 views

### Is there a flat proper morphism that is not finitely presented?

Is there an example of a flat proper morphism of schemes $X\rightarrow S$ whose fibers are geometrically connected, reduced and have dimension 1, but which is not itself finitely presented? What ...

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### The ring of global sections of a regular scheme

Let $X$ be a Noetherian regular scheme. Is $\mathcal{O}_X(X)$ a regular ring? For affine schemes this is true, see 02IU on the Stacks project.

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### Quasi-compactifying schemes

Let $X$ be a scheme. Does there exist an open immersion $X\rightarrow Y$ with $Y$ quasi-compact?

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### Krull dimension of schemes locally of finite type over PID

Let $R$ be a commutative unital ring that is a PID. Assume that $R$ is not a DVR. Let $X$ be an integral scheme locally of finite type over $\mathrm{Spec}\,R$. Can the Krull dimension of $\mathcal{O}...

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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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### Functorial subscheme structure on non-locally closed subsets

If we have a scheme and a locally closed subset of the underlying topological space, then there is a canonical way to put a scheme structure on it so that the inclusion map can be upgraded to a ...

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### Are schemes obtained from reduced schemes by gluing infinitesimal stuff along a reduced closed subscheme?

Let $X$ be a scheme of finite type (say over the complex numbers). The set of points for which the local ring is reduced is then an open subset $U\subseteq X$.
Is it true that there is a closed ...

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### Does the structure morphism matter in GAGA?

Let $X$ be an integral Noetherian separated scheme. Can there exist two morphisms of finite type $X\rightarrow \mathrm{Spec}\,\mathbb{C}$ such that the corresponding complex-analytic spaces are not ...

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### Do Grothendieck universes matter for an algebraic geometer?

I recently learned that some parts of SGA require axioms beyond ZFC. I am just a simple algebraic geometer so I am trying to understand how can this fact impact my life (you may have engaged in a ...

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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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### Learning to appreciate Gabber's result

There is a theorem in scheme theory due to Gabber which makes our life easier when setting up the theory of cohomology of quasi-coherent sheaves.
I am trying to understand why should I appreciate ...

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### 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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### Proper variety containing no affine open with irreducible complement

Does there exist an integral positive-dimensional scheme proper over $\mathbb{C}$ that contains no non-empty affine open subscheme with irreducible complement?

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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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### A curve is proper iff the space of global sections is finite-dimensional

Let $k$ be a field, $X\rightarrow \mathrm{Spec}\,k$ be a separated morphism of finite type of relative dimension$\leq 1$ (as defined here). Is it true that $f$ is proper iff $f_* \mathcal{O}_X$ is ...

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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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### 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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### 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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### 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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329 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 $X$?