I am reading up on some algebraic geometry and came across the above definition of sheaves. Some things confuse me. When the author says in 2) of Definition 4.1, "For each inclusion of open sets $V \subset U$..", does he mean that $V$ is an open set in $X$ or in the induced topology on $U$? In...

Problem Statement Let $f: X \to Y$ be a surjective morphism of finite type between affine Noetherian schemes, where $X = \operatorname{Spec} B$ and $Y = \operatorname{Spec} A$. Suppose that for every closed point $y \in Y$ and for every $x \in X_y$ that is closed in $X_y$, the stalk map...

Let $k$ be an infinite field. Definition 1. ( Affine Hilbert function ). Let $R := k[x_1, \dots ,x_n]$ be a polynomial ring which can be viewed as a vector space over $k$. Let $R_{\le s} := k[x_1, \dots, x_n]_{\le s} $ denote the set of polynomials of total degree $\le s$ in $R$. Note that...

This question is from Proposition $6.3$ of the textbook Algebraic geometry by Daniel Perrin( Page 32). Here $k$ is a commutative field.Let $V$ be a projective algebraic set and consider a homogeneous element $f \in \Gamma_h(V)= k[X_0,...,X_n]/I_p(V)$ of degree>0. $I_p(V)$ is the ideal of...

I am self studying Algebraic Geometry from the Clader and Ross Algebraic geometry textbook: Beginnings in Algebraic Geometry. This proof is given on page 275 of the textbook and I am quite confused about it. Please help me. Theorem 9.53 (Projective Nullstellensatz) Assume that $K$ is...

For instance we have the projective space itself is a manifold, and we often talk about zero set of polynomials over the projective space in algebraic geometry. So, how do the non trivial manifold properties /algebraic topology play into the study of algebraic geometry?

Let $R = k[x_1, \dots, x_n]$ be a polynomial ring over a field $k$ equipped with a standard grading, and let $<$ be a monomial order on $R$. Let $I$ be a homogenous ideal of $R$, and let $\operatorname{in}_<(I)$ denote the initial ideal of $I$ with respect to $>$. My question is If $f\notin I$...

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...