Let $Q_N$ denote the cubical cell complex given by the poset of faces of the $N$-cube, and let $$ V_N=\{\pm 1\}^N $$ be its set of $0$-cells. Let $$ A_N:=V_N/\{\pm 1\} $$ be the set of antipodal pairs of $0$-cells. We have $$ |A_N|=2^{N-1} $$ For each antipodal pair $$ \alpha=\{v,-v\}\in A_N $$...

How can I show that every Projective variety in $\mathbb{P}^n$ is compact in the induced Euclidean topology? Should I consider, as customary, an arbitrary open cover in $\mathbb{P}^n$, and perhaps use the projection map $\pi:\mathbb{C}^{n+1}\setminus\{0\}\rightarrow \mathbb{P}^n$ which defines...

Let $G$ be a reductive group over a field $k$. What actually does it mean to say that $G$ is of type $A_n, B_n,\dots,G_2,{}^2A_n, {}^3D_4,...$? In case it helps, I know what the Dynkin diagrams of types $A_n, B_n,\dots,G_2$ are (but not those of types ${}^2A_n, {}^3D_4,...$). I also know how to...

This statement is given as application of earlier results on the page $22 $ of the Daniel Perrin's Algebraic Geometry textbook from which I am self studying. Here $k$ is a commutative field and $V$ is a affine algebraic set. Application $6.9$ The morphism $\phi: k \to V=V(Y^2-X^3)$ given by...

For $U \to X$ étale, define a presheaf by $\mathcal{O}_{X_{\text{ét}}}(U) = \Gamma(U,\mathcal{O}_{U})$. I want to show that this is a sheaf on $X_{\text{ét}}$. Clearly, the sheaf condition holds for all Zariski open coverings, so it is sufficient to show the sheaf condition holds for étale...

Let $k$ be an algebraically closed field and $V$ be an affine variety. From Page 19 of Daniel Perrin’s Algebraic geometry. Lemma: Let $F,G\in k [X,Y]$ be non-zero polynomials without common factors, there is a non-zero polynomial $d\in k[X]$ and polynomials $A,B\in k[X,Y]$ such that $d= AF+BG$...

I am self studying Algebraic Geometry from Daniel Perrin's Algebraic Geometry. $k$ is an algebraically closed field. $\Gamma(V)$ is image of $r: k[X_1,...,X_n]\to F(V,k)$ and $V,W$ is affine algebraic set. $\theta : \Gamma(W) \to \Gamma(V)$ be a homomorphism of $k-$algebras and $\phi: V\to k^m$...

I am self studying Algebraic Geometry from the Daniel Perrin's textbook. $k$ is an algebraically closed field. $\Gamma(V)$ is image of $r: k[X_1,...,X_n]\to F(V,k)$ and $V,W$ is affine algebraic set. $\theta : \Gamma(W) \to \Gamma(V)$ be a homomorphism of $k-$algebras and $\phi: V\to k^m$ is the...

$k$ is an algebraically closed field. $\Gamma(V)$ is image of $r: k[X_1,...,X_n]\to F(V,k)$ and $V$ is affine algebraic set. I am self studying Algebraic Geometry from Daniel Perrin's Algebraic Geometry and have a question in proof of Proposition $4.8 $ of Chapter $1$ on Page $17$. Proposition...

Let $G$ be a group object in an $\infty$-topos $\mathcal T$, and let $P \to X$ be a $G$-torsor as in the accepted answer by Daniël Apol to this question; equivalently, as in the answer, such a $G$-torsor is given by a map $\varphi \colon X \to BG$, and $P$ is the pullback of $\varphi$ along the...

Let us consider a prime number $p$ and three distinct positive integers $n_1<n_2<n_3$ less than $p$. Let us assume that the triple $(n_1, n_2, n_3)$ satisfies the following condition $$k+[kn_1]+[kn_2]+[kn_3]=2p, \ \ for \ all \ 1\leq k\leq p-1$$ Where $[kn_i]$ denotes the rest of the division of...

Let $\varphi: X\to Y$ be a dominant morphism of affine varieties over algebraically closed field $k$, $\varphi*: k[Y]\to k[X]$ be the induced $k$-algebra monomorphism. Let $\mathfrak{m}_y$ be the ideal of a point $y\in Y$. I believe the ideal $I(\varphi^{-1}(y))$ of the fiber $\varphi^{-1}(y)$...

I'm working through Vakil and encountered the following exercise. 5.1.B. EXERCISE. Exercise 3.7.F showed that there is a bijection between irreducible closed subsets and points for affine schemes (the map sending a point p to the closed subset $\overline{\{p\}}$ is a bijection). Show that this...

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...