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...

$\require{AMScd}$I am in the process of understanding how to work with (quotient) stacks. In my case, it is usually helpful to get my hands dirty so I like to come up with examples where maybe using stacks can make an argument more transparent. I recalled an exercise that I did when preparing...

Consider functions from $R^n$ to a subset of $R^n$. So $f(x_1,x_2,...,x_n) = (y_1,y_2,...,y_n)$. where $x_i,y_i$ are all real. More specific consider $$f(x_1,x_2,...,x_n) = (Q_1(x_1,x_2,...,x_n),Q_2(x_1,x_2,...,x_n),...,Q_n(x_1,x_2,...,x_n))$$ where the $Q_i$ are all Quadratic forms. Even more...

Suppose that $$ x=u^{-d},\qquad y=f(u), $$ where $$ d\in \mathbb Z_{>0},\qquad f(u)\in \mathbb C((u)), \qquad y=o(x). $$ Equivalently, $\operatorname{ord}_u f(u)>-d$. Let $ (P,Q)\in \operatorname{Aut}_{\mathbb C}\mathbb C[x,y]$ be a polynomial coordinate system, and write $$ X=P(x,y),\qquad...

I am currently working through Fulton's Algebraic Curves and I have attempted the following problem: $$\text{Let P be a simple (non-singular) point on }F\text{ . Show that the tangent line to }F \text{ at } P\text{ has the equation }F_X (P )X + F_Y (P )Y + F_Z (P )Z = 0$$ My solution thus far...

Let $k$ be a commutative ring and let $A$ be a commutative $k$-algebra. The cotangent complex $\mathbf{L}_{A \mid k}$ can be computed using a simplicial resolution of $A$ as follows: choose a simplicial commutative $k$-algebra $P$ and an augmentation $\epsilon : P_0 \to A$ (i.e. a $k$-algebra...

Vakil, Definition 20.1.1: Let $X$ be a variety (reduced separated finite type scheme, actually I'm not sure if we need all this. I think we can get away with dropping "reduced" and "separated" and just assume $X$ is a finite type scheme) over a field $k$ (not necessarily algebraically closed)....

I am trying to work through the following problem: Let $k$ be a field, and let $X = \operatorname{Spec} k[x, y, z, w]/(xy−zw)\subseteq \mathbb{A}^4_ k.$ (a) Show that $D= V (x, z)$ is a prime (Weil) divisor on $X$ and that $\operatorname{Cl} X\simeq \mathbb{Z}$ is generated by the divisor class...