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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Spreading out a morphism of the generic fibers
Let $X$ and $Y$ be finite type schemes over $\mathrm{Spec} \mathbb{Z}$ and let $f_\xi : X_\xi \rightarrow Y_\xi$ be a morphism between the generic fibers. Then $f_\xi$ spreads out to a morphism $g_U : ...
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Applications of schemes to mathematical physics
Could anyone cite some applications or developments in mathematical physics or string theory that use schemes?
I find curious the fact that while things like derived algebraic geometry and stacks ...
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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 $...
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Fiber product of singular varieties
Let $f\colon X\to Y$ be a morphism between irreducible $\mathbb{C}$-varieties, such that $f$ is 1-1 and surjective, but $df$ fails to be injective along a subvariety $Z$. (This forces $Y$ to be ...
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$G_0(X) \cong G_0(X_{red})$ where X is a noetherian scheme
Let $\textbf {X}$ be a noetherian scheme,
$\textbf {M(X)}$ be the categroy of coherent sheaf over the scheme $\textbf {X}$.
We denote $ \textbf {K$_0$(M(X))}$ to be $ \textbf {G$_0$(X)}$.
Now I ...
user139827
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Units in the coordinate ring on a reductive group
Let $K$ be a field and $G$ a connected reductive group over $K$.
Can we describe $K[G]^{*}$?
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Relate Codimensions of Integral Schemes and their Generic Fibers
I have a question about an argument in the proof of Thm 8.2.5 (Dimension formula) from Liu's "Algebraic Geometry" (page 334):
Firstly the problem is reduced to the affine case with $Y:=Spec \text{ }\...
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Beauville-Laszlo for schemes
For a commutative ring $A$ and $f\in A$ a non-zero divisor, the Beauville-Laszlo theorem gives a gluing statement for vector bundles on $A$ in terms of a vector bundle on $A\big[\frac{1}{f}\big]$, a ...
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Other interesting notions when we change topology on $\text{Sch}/S$
Let $\text{Sch}$ be the category of schemes. Let $S$ be an object of $\text{Sch}$. Consider the category $\text{Sch}/S$.
Some interesting topologies on $\text{Sch}/S$ are Zariski, fpqc, étale, fppf.....
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Riemann–Hurwitz Formula for Normal Projective Curves
My question refers to the proof of Theorem 7.4.16 from Liu's "Algebraic Geometry" at page 290 :
QUESTION: Why does $e'_x= length_{\hat{B}}(W_{\hat{B}/\hat{A}}/\hat{B})$ hold?
During the proof we ...
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Hyperelliptic Curve (Liu's Book)
Let $Y:=\mathbb{P}^1_k$ and $X$ a hyperelliptic curve. We working with the definition 4.7 from Liu's "Algebraic Geometry and Arithmetic Curves" (page 288)
Namely there exist finite separable map $\pi:...
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Question about Immersion
Let $X,Y$ Noetherian integral schemes and assume we have an immersion
$$i:X \to Y$$
An immersion is according to https://stacks.math.columbia.edu/tag/01IO a composition of a closed immersion $c: X \...
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Do regular (but non-smooth) conics over a discretely valued field of characteristic $2$ admit a regular model over the valuation ring?
Let $K$ be a non-perfect field of characteristic $2$. Let $T \subseteq K$ be a discrete valuation ring.
Assume there exist $a,b \in K^{\times}$ such that the projective conic $C$ defined by $$aX^2 + ...
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Cohomology of sheaves on $X \cup_{Z} Y$
I am in the following situation, I have two schemes $X$, $Y$ and two closed immersions $Z \rightarrow Y$, $Z \rightarrow X$. Everything is smooth. I am interested in calculating morphisms in the ...
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How to deduce the following map between Zariski tangent spaces is surjective?
Let $f: X \rightarrow Y$ be a morphism of schemes: $X$, $Y$ are regular schemes, let $Z_1, Z_2$ be two closed regular subschemes of $Y$, let $x \in X \times_Y Z_2$ such that $y = f(x) \in Z_1 \cap Z_2$...
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Direct image functor commuting with infinite direct sum of sheaves
Normally I would think this kind of question doesn't belong on overflow, but I haven't been able to find an answer anywhere else, so perhaps it is not so trivial.
Let $f: X \rightarrow Y$ be a ...
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Road map for moduli space/moduli problem/moduli stack
I am familiar with (most of the) contents of Angelo Vistoli's notes on Descent theory (Stacks). I am also comfortable with basics of Schemes, their Cohomology (Cech), from Hartshorne's Algebraic ...
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Direct sums of invertible sheaves commuting with global sections and the functor of points approach
I am looking at the Stacks Project's treatment of the functor of points for projective space.
Let's restrict to the case that $S$ is a graded ring, generated by $S_{1}$ as an $S_{0}$ algebra. The ...
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Affine cone example
Consider the complete intersection ideal ${\displaystyle (f,g_{1},g_{2},g_{3})\subset \mathbb {C} [x_{0},\ldots ,x_{n}]}$ and let $X$ be a projective variety defined by the (product) ideal sheaf ${\...
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Factorizations of closed embeddings of smooth schemes
All schemes will be of finite type over a field $k$. Say I have a closed embedding $\iota: X \hookrightarrow Y$ of smooth $S$-schemes for some scheme $S$ (in particular it is a regular embedding). ...
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"Covering-flat" part in definition of morphism of sites
Let $(\mathcal{C},\mathcal{I})$ and $(\mathcal{D},\mathcal{J})$ be sites where $\mathcal{C}, \mathcal{D}$ are categories and $\mathcal{I}$ and $\mathcal{J}$ are Grothendieck topologies on $\mathcal{C}...
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Why care about Grothendieck topology? [closed]
Noah Schweber said here the following:
Why would you want a notion of sheaf theory for objects more general
than topological spaces? Well, the original motivation (to my
understanding) was to ...
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Locally affine varieties and du Val singularities
Let me start with an apologetic disclaimer: I am very far from an algebraic geometer, so this question might be crudely formulated.
I have a specific question about du Val singularities, but while ...
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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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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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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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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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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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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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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.
...
user142126
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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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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?
user142033
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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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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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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 ...
user141498
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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 closed ...
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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 ...
user141408
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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{...
user141356
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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.
user141316
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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 ...
user138661
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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 ...
user140820
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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 ...
user138661
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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 $X$?
user140689
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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 ...
user138661
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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}\,...
user138661