For a torsion free sheaf $E$ on a surface $X$ we have $0\rightarrow E\rightarrow E^{**}\rightarrow Z\rightarrow 0$ where $Z$ is a zero dimensional subscheme of $X$. Let $p\in\mathbb{P}^2$ be a closed point. Let $\mathcal{I}_p$ be the corresponding ideal sheaf. We have $(\mathcal{I}_p^{\oplus 2})^{**}=\mathcal{O}^{\oplus 2}$. Hence $\mathcal{O}^{\oplus 2}/\mathcal{I}_p^{\oplus 2}$ is a zero dimensional subscheme of $\mathbb{P}^2$. […]

Let $A$ be a graded ring and $d>0$ be an integer. Define the graded ring $B$ such that $B_i=A_i$ if $d$ divides $i$ and $B_i=0$ otherwise. Is it true that a homomorphism of graded rings $B\hookrightarrow A$ induces isomorphism of schemes $\text{Proj}\,A\to\text{Proj}\,B$?

The following exercise is taken from ravi vakil’s notes on algebraic geometry. Suppose $X$ is a closed subset of $\mathbb{P}^n_k$ of dimension at least $1$, and $H$ is a nonempty hypersurface in $\mathbb{P}^n_k$. Show that $H\cap X \ne \emptyset$. The clue suggests to consider the cone over $X$. I’m stuck on this and I realized […]

The relative chinese remainder theorem says that for any ring $R$ with two ideals $I,J$ we have an iso $R/(I\cap J)\cong R/I\times_{R/(I+J)}R/J$. Let’s take $R=\Bbbk [x_1,\dots ,x_n]$ for $\Bbbk $ algebraically closed. If $I,J$ are radical, the standard dictionary tells us $R(I\cap J)$ is the coordinate ring of the variety $\mathbf V(I)\cup \mathbf V(J)$. Furthermore, […]

I am currently studying varieties over $\mathbb{C}$, i know some scheme theory. Let $f: X \rightarrow Y$ be a morphism of varieties. If we want to show flatness, is it enough to check the condition only at the closed points of $Y$? If yes, could you give an argument, if not, could you give an […]

Let $f: X \rightarrow Y$ be a finite étale covering of degree 1 of varieties over some field $k$, so $$ [K(Y):K(X)] = 1. $$ If $k=\mathbb{C}$ and the varieties are smooth, one can apply complex analytic methods to very quickly show that $f$ is an isomorphism. I would however prefer an algebraic proof of […]

The exercise 3.1 in GTM 52 by Hartshorne require to prove that $f:X \longrightarrow Y$ is locally of finite type iff for every open affine subset $V=\text{Spec}B$, $f^{-1}(V)$ can be covered by open affine subsets $U_j=\text{Spec}A_j$, where each $A_j$ is a finitely generated $B$ algebra. Now, if $f:X \longrightarrow Y$ is locally of finite type, […]

Let $X$ be a noetherian integral (separated) scheme which is regular in codimension one. Let $Y$ be a prime divisor and let $\eta$ be the generic point of $Y.$ It seems I am missing something easy but why $\mathcal{O}_{X, \eta}$ is a DVR with the quotient field the function field of $X?$ And when it […]

Let $A$ be a ring, $X=\operatorname{Spec}A$ and $f: Z \rightarrow X$ a morphism of schemes such that i) $f$ is a homeomorphism of topological spaces and ii) $f^{\#}:\mathcal{O}_X \rightarrow f_* \mathcal{O}_Z$ is a surjective morphism of sheaves. Question 1: How can we show that $Z$ is an affine scheme? I am trying to follow the […]

Let $F$ be a quasicoherent sheaf on a scheme $X$, which is supposed to be sufficiently nice. Does one then have a canonical isomorphism $Ext^1(F,F) \simeq H^1(X, \underline{End}(F))$, where with $\underline{End}(F)$ I denote the sheaf of endomorphisms of $F$. I know that this holds for $F$ locally free, but I read an article where this […]

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