Articles of affine schemes

Characterization of finite morsphisms over $\operatorname{Spec }O_K$

Let $K$ be a number field with ring of integers $O_K$. Now consider a finite and dominant (this condition was added later) morphism of schemes: $$\pi:X\to \operatorname{Spec }O_K$$ Do we have some results which tell us what is the scheme $X$? For instance if $L$ is a finite extension of $K$, then one possibility is […]

Question on geometrically reduced, geometrically connected.

I have a question from a book which I am trying to attempt. Let $k$ be a field not of characteristic 2 and let $a\in k$ be not a square (i.e. for all $b\in k$, $b^{2}\neq a$). I want to show that $X=\mbox{Spec}(k[U,T]/(T^{2}-aU^{2}))$ is geometrically reduced and geometrically connected. My attempt is as follows: I […]

Exercise 5.5.F. on Ravi Vakil's Notes related to associated points

This question already has an answer here: For a Noetherian ring $R$, we have $\text{Ass}_R(S^{-1}M)=\text{Ass}_R(M)\cap \{P:P\cap S=\emptyset\}$ 1 answer

Spectrum of infinite product of rings

$\def\Z{{\mathbb{Z}}\,} \def\Spec{{\rm Spec}\,}$ Suppose $R$ a ring and consider $\Spec(\prod_{i \in \mathbb{Z}} R)$. Now for the finite case, I know that holds $\Spec(R \times R) = \Spec(R) \coprod \Spec(R)$. My intutition says that this does not extend to the infinite case. Maybe $\Spec(\oplus_{i \in \Z} R) = \coprod_{i \in \Z} R$ holds, but I am […]

On Spec of a localized ring

Let $A$ be a commutative ring with unity, and $S$ a regular multiplicative subset of $A$ containing $1$. I know that given $f \in A$, Spec $A_f$ corresponds to an open subscheme of Spec $A$. I was wondering does Spec $S^{-1}A$ also correspond to an open subscheme of Spec $A$ as well (for any multiplicative […]

Questions on scheme morphisms

I have some questions on scheme morphisms. I ask pardon for posting them in one thread as they are most likely not worth to be distributed into several threads. Let $X=Spec R$ be a noetherian scheme. For a maximal ideal $m$ of $R$ one has a morphism $Spec (k(m))\to X$ induced by the ring homomorphism […]

Inverse of open affine subscheme is affine

This seems ridiculously simple, but it’s eluding me. Suppose $f:X\rightarrow Y$ is a morphism of affine schemes. Let $V$ be an open affine subscheme of $Y$. Why is $f^{-1}(V)$ affine? I noted that $V$ is quasi-compact and wrote it as a finite union of principal open sets. Because the pullbacks of principal open sets are […]

Global sections on quasi coherent sheaves on affine scheme

This is a lemma from Hartshorne’s Algebraic Geometry. Let $X=\text{Spec}(A)$ be an affine scheme $f\in A, D(f)\subseteq X$. Let $\mathcal{F}$ be a quasi coherent sheaf on $X$. If $s\in \Gamma(X,\mathcal{F})$ is such that $s|_{D(f)}=0$ then for some $n>0$, $f^ns=0$. If $t\in \Gamma(D(f),\mathcal{F})$ then for some $n>0$ $f^nt$ extends to a global section of $\mathcal{F}$ over […]

Proof of proposition 5.3.1 of Ravi Vakil's notes on algebraic geometry

I am reading the proof of proposition 5.3.1 of Ravi Vakil’s notes on algebraic geometry, and I have a problem with the last sentence : “If $g’ = g”/f^n$ ($g”\in A$) then $\textrm{Spec}((A_f)_{g’}) = \textrm{Spec} (A_{fg”})$, and we’re done.” Noting $V = \textrm{Spec} (B)$ and $V’ = \textrm{Spec} (B_g)$, and noting $D_Z (h)$ the distinguished […]