How to prove that for any faithfully flat quasi-compact (or, more generally, fpqc) morphisms are epimorphisms? My attempt: Let $f:X\to Y$ be faithfully flat quasi-compact (or, more generally, fpqc) and suppose that $g_1,g_2: Y\to Z$ are two morphisms such that $g_1\circ f=g_2\circ f$. Since $f$...

This is Exercise 11 in Section 16.1 in Dummit&Foote's Abstract Algebra. Let $V$ be an affine variety over a field $k$ and let $R = k[V]$ be its coordinate ring. Let $d_t(R)$ denote the transcendence degree of the field of fractions $k(V)$ over $k$, and let $d_p(R)$ be the Krull dimension of $R$...

The Torelli map, which maps a smooth curve to its principally polarized Jacobian, defines a morphism of algebraic stacks $$\mathscr{M}_g \longrightarrow \mathscr A_g$$ between the moduli spaces of smooth, genus $g$ curves and dimension $g$ principally polarized abelian varieties respectively....

Let $K$ be a field, and consider a morphism of locally ringed spaces (or schemes) $f: \operatorname{Spec} K \to Y$. Let $t$ be the unique topological point of $\operatorname{Spec} K$, and suppose that $f(t) \in V$, where $V$ is an affine open subset of $Y$. My question is: Can $f$ be uniquely...

I am working through exercise 14.1.B(c) in Ravi Vakil's excellent Fundamentals of Algebraic Geometry. I'll here reproduce the statement: Exercise 14.1.B(c) - Let $\pi:X \to Y$ be a morphism of ringed spaces. If $\mathscr G$ is a locally free sheaf of rank $n$ on $Y$ and $\{U_i\}$ are...

Fix a commutative ring $k$. $\def\Spec{\operatorname{Spec}}\def\Mod{\operatorname{Mod}}\def\CAlg{\operatorname{CAlg}}\def\Ab{\operatorname{Ab}}\def\Sym{\mathcal{S}}\def\V{\mathbb{V}}\def\A{\mathbb{A}}$ Let $M$ be a $k$-module and $G=\Spec H$ be an affine group scheme. After being initially...

In standard projective (P^2), the line at infinity is defined as ([X,Y,Z]) with (Z=0). However, other authors define it as ([X,Y,Z]) with (X=0). I feel some confusion. Can it be shown that this does not depend on these choices? Is the elliptic curve group structure preserved under projective...

I am tasked to prove the formal affine line $\hat{\mathbb{G}}_a$, seen as the functor from animated rings to set $$Ani(Ring)\to Ani$$ $$R\mapsto Nil(\pi_0(R))$$ taking an animated ring to the nilradical of its underlying static ring, to be an etale stack i.e. I have to show that it satisfies...

I'm trying to figure out some facts about the pullback of Cartier divisors, but honestly I'm having a hard time describing the pullback of the sheaf associated to divisor. Let $\phi\colon X \to Y$ be a dominant morphism of schemes. Suppose $X, Y$ are both integral, noetherian, separated. If...

As I mentioned in my previous post, I have been reading (Talpo and Vistoli)'s paper Infinite root stacks and quasi-coherent sheaves on logarithmic schemes. While editing that post, a second question came up, which I thought was sufficiently independent to deserve its own post. So here is the...

This problem is from the book Álgebra comutativa em quatro movimentos by Borges and Tengan. Let $k$ be a field of characteristic not equal to $2$ or $3$. Let $f(x, y) = y^2 - x^3 - 10$ and consider the ring $R = k[x, y]/(f(x, y))$. Show that $R$ is a Dedekind domain. Edit. We also have to assume...

I'm trying to solve the following problem from a commutative algebra book (Álgebra comutativa em quatro movimentos by Borges and Tengan). The question has 30 concrete examples of rings (mostly quotients), but I will restrict to two. Compute the Krull dimension of the following rings. $R =...

Let $C$ be a plane curve and suppose, for simplicity, it has a singularity at the origin. Let $\tilde{C}$ be it's strict transform after blowing up $(0,0).$ Is it possible for $\tilde{C}$ to have a singular point not at the origin?

These days I've been reading (Talpo and Vistoli)'s paper Infinite root stacks and quasi-coherent sheaves on infinite root stacks. At the moment I'm looking at their proposition 4.9....

Suppose $R$ is a reduced Noetherian ring. We know that $f \in R$ is a zero divisor if and only if $V(f)$ contains an irreducible component of $\mathrm{Spec} R$. I would like to know if one could use the irreducible components to say something about the annihilator of $f$ in $R$? (Perhaps it...

I found this problem where I think there is a mistake but am not 100% sure: Let $Z$ be a closed subset of a topological space $X$. If $Z$ is irreducible we call codim($Z,X$) as the supremum of lengths of the chains of irreducible closed subsets of $X$ which contains $Z$: $$Z\subset Z_0\subsetneq...

I am currently studying algebraic geometry and trying to understand projective spaces. let $S = \mathbb{Z}[x_0, \dots, x_n]$ so that $\mathbb{P}^n_{\mathbb{Z}} = \operatorname{Proj}(S)$. I read that if we remove the standard open affine subset $D_+(x_i) = \{ \mathfrak{p} \in...

I'm studying algebraic geometry and have a question on Kahler differentials. Let $k$ be a commutative ring with unit and $A$ a $k$-algebra. I already have the intuition if we picture $A$ as some ring of functions on $\operatorname{Spec}{A}$, the the module of relative Kahler differentials...

I'm self studying commutative algebra. Here is the question: Edit(Background and Definition): Let $k$ be a field, we define an affine variety $V\subseteq k^n$ as the set of common zeroes of some polynomials $f_1,\dots, f_m\in k[x_1,\dots, x_n]$, define its coordinate ring as $k[x_1,\dots,...

https://en.wikipedia.org/wiki/Rational_mapping The usual example is that $ \mathbb {P} _{k}^{2} $ is birational to the variety $ X $ contained in $ \mathbb {P} _{k}^{3} $ consisting of the set of projective points $ [w:x:y:z] $ such that $ xy-wz=0 $, but not isomorphic. Indeed, any two lines in...