Questions tagged [schemes]

Question 1

Let $f:X \to Y$ be a map of schemes and $y \in Y$ a point with residue field $\kappa(y)$. Let $\mathcal{F}$ be a coherent sheaf on $X$. Then there exist a comparison map $$ f_*(\mathcal{F}) \otimes \...

Question 2

Let $X$ be a scheme over base field $K$ and denote by $R[X](= \Gamma(X,\mathcal{O}_X))$ the ring of regular global functions on $X$. Question: Assume that $k$ is algebraically (resp. separably) closed ...

Question 3

I am superficially quoting the following result from Algebraic Geometry by Hartshorne. I was told the following from a brief conversion with a graduate student, and became very interested. Please be ...

Question 4

Suppose that $A \colon= {\Bbb C}[X,Y,Z,W]/f(X,Y,Z,W)$ is a three-dimensional normal ring over ${\Bbb C}$. I am seeking for $f(X,Y,Z,W)$ such that ${\mathrm{Spec}}\,A$ has singularities along a curve $...

Question 5

This must has been asked several times in this cite, so I apologize in advance if this turns out to be trivial... Let $A,B$ be commutative rings. We know that if $A\to B$ is an epimorphism in $\mathbf{...

Question 6

Definition A curve over a field $k$ is a separated scheme $C$ of finite type over $k$ which is integral of dimension 1. Let $C$ be a normal curve over a field $k$. A divisor is an element of the free ...

Question 7

An absolute beginner level question on rigid analytic spaces & their interplay with formal schemes (after Raynaud). In this minicourse by Bosch on this topic he introduced as motivation following ...

Question 8

A question about proof of Zariski-Nagata purity from Stacks project, the "easy" case $d=1$ (1st paragraph in given proof) . General statement: Let $f : X \to Y$ be a morphism of locally ...

Question 9

There is an exercise in Hartshorne, II.2.10, which asks us to describe $\mathrm{Spec}(\mathbb{R}[x])$ and its topology. This ends up being The generic point $(0)$ Closed points of the form $(x - a)$ ...

Question 10

Let $A$ be a graded commutative ring and assume that $A$ is generated by $A_1$ as an $A_0$-algebra. Define $X:= \text{Proj}(A)$ and let $\text{QCoh}(X)$ be the category of quasi-coherent sheaves on $X$...

Question 11

Say we have two (affine) group schemes, $G$ and $H$, and consider some morphism of functors $\kappa \colon h_{G}\to h_{H}$, where $h_{G}$ and $h_{H}$ are the functors of points to $\mathrm{Grp}$ ...

Question 12

Let $i:Z_0\to X_0$ be a closed immersion of locally Noetherian schemes. It is well-known (c.f. Being a regular embedding is an open condition for locally Noetherian schemes) that if i is regular at a ...

Question 13

I have few questions about explicit proof that the relative spectrum associated to invertible sheaf $\mathcal{O}_{\Bbb P^2_{\Bbb C}}(-1)$ coinsides with incident variety $I \subseteq \mathbb{P}^2 \...

Question 14

Consider a smooth DM stacks (or even a smooth scheme) $X$. Let $L$ be a line bundle and let $s$ be a section. We can consider either the root gerbe or the root stack by considering the fibre product ...

Question 15

Let $A \colon= {\Bbb C}[X_1,\cdots,X_n]$ be a polynomial ring over ${\Bbb C}$. Let us consider the action $\sigma \colon X_i \mapsto \zeta^{e_i} X_i$ for $i = 1,\cdots,n$, where $\zeta \colon= \zeta_n$...

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Questions tagged [schemes]

A Consequence from Grauert's Result on Cohomology of Fibres

Scheme over base field $k$ which is algebraically (resp. separably) closed in ring of regular functions of $X$

Conversion between Cartier Divisors, Weil Divisors, Line Bundles and Invertible Sheaves.

On hypersurface singularity.

When is $\operatorname{Spec}(B)\to\operatorname{Spec}(A)$ an epimorphism is the category of schemes?

Why is the sequence $0\rightarrow\mathscr{O}_C(C)^\ast\rightarrow k(C)^\ast\rightarrow\rm Div(C) \rightarrow Pic(C)\rightarrow 0$ exact?

Generic and Special Fibres for Rigid Analytic Model $\Bbb G_{m}^{\text{an}}$ of Multiplicative Group

Proof of Zariski-Nagata Purity Theorem for Relative Dimension $1$ in Stacks Project

What does the residue field have to do with the spectrum?

Reference for proof Serre's equivalence for quasi-coherent sheaves and quotient category

Question on morphism of group schemes

Is being regular an étale-open condition for closed immersions of locally Noetherian schemes?

Explicit Isomorphism from $\operatorname{Spec}(\operatorname{Sym}^_{\mathcal{O}_X}(\mathcal{O}_{\Bbb P^2}(-1)^))$ to an Incidence Variety

Automorphism of root gerbe/root stack

Calculating difficulty for quotient singularities.

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