Articles of projective schemes

Existence of a morphism of schemes from $X$ to $\mathbb P_A^n$?

In exercise §II.III.VIII. of Algebraic geometry and arithmetic curves by Qing.Liu. one is asked the following: Let $X$ be a scheme over a ring $A.$ Let $f_0, \cdots,f_n\in\mathcal O_X(X)$ be such that the $f_{i,x}$ generate the unit ideal of $\mathcal O_{X,x}$ for every $x\in X.$ Show that $X=\cup_i X_{f_i}$ and that we have a morphism […]

Why are projective spaces over a ring of different dimensions non-isomorphic?

Let $A$ be a nonzero commutative ring with unit. Define $\mathbb P_A^n$ to be the scheme $\operatorname {Proj} A[T_0,\dots,T_n]$, where the grading on the polynomial ring is by degree. Why is it true that $\mathbb P_A^n\not\cong \mathbb P_A^m$ when $n\neq m$? (I admit I am just guessing here. It seems like it should be true. […]

When are complete intersections also local complete intersections?

First, let us recall some definitions. Let $P=\mathbb P^d_k$ be a projective space over a field $k$. Let $X$ be a closed subvariety of $P$ of dimension $r$. We say that $X$ is a complete intersection if it is defined (as a variety) by $d-r$ homogeneous polynomials $F_1,\dots,F_{d-r}$. This notion depends on the embedding $X\rightarrow […]

Proj construction and ample dualizing sheaf

my question concerns a smooth projective variety $X$ with dualizing sheaf $\omega_X$: if I have that this dualizing sheaf is ample, then I have read you can conclude that $X\simeq Proj(\oplus_{k} H^{o}(X,\omega_X^{k}))$ as projective varieties. Can someone explain why this is the case and perhaps give me some reference apart from Hartshorne where these projective […]

Isomorphism of Proj schemes of graded rings, Hartshorne 2.14

This question is based on exercise $2.14$ of chapter $2$ of Hartshorne. Suppose $\varphi:S\rightarrow T$ is a graded homomorphism of graded (commutative, unital) rings such that $\varphi_d := \varphi|_d$ is an isomorphism for all $d$ sufficiently large. Then I want to show that the natural morphism $ f: $Proj $T \rightarrow $Proj $S$ is an […]

Tangent space in a point and First Ext group

Let $X$ be an abelian variety over an algebraically closed field $k$. I have read that one has for an arbitrary closed point $x$ on $X$ a canonical identification $$T_x(X)\simeq \operatorname{Ext}^1(k(x),k(x))$$ where on the left we have the tangent space in the point $x$ and on the right I take the first $\operatorname{Ext}$-group of the […]