Hartshorne II 3.1












2












$begingroup$


Definition: A morphism $f:Xto Y$ of schemes is locally of finite type if there exists a covering of $Y$ by open affine subsets $V_i=mathrm{Spec}(B_i)$, such that for each $i$, $f^{-1}(V_i)$ can be covered by open affine subsets $V_i=mathrm{Spec}(A_{ij})$, where each $A_{ij}$ is a finitely generated $B_i$-algebra.




Exercise 3.1: Show that a morphism $f:Xto Y$ is locally of finite type if and only if for every open affine subset $V=mathrm{Spec}(B)$ of $Y$, $f^{-1}(V)$ can be covered by open affine subsets $U_j=mathrm{Spec}(A_j)$, where each $A_j$ is a finitely generated $B$-algebra.




The 'if' direction is trivial, so I'm asking about the 'only if' direction.



Assume that $f:Xto Y$ is locally of finite type and let $V=mathrm{Spec}(B)$ an open affine subset of $Y$. I can use the covering of $Y$ given by the hypothesis to obtain a covering of $Vsubseteqbigcup_i V_i=bigcup_i mathrm{Spec}(B_i)$, so $f^{-1}(V)subseteq bigcup_i f^{-1}(V_i)subseteqbigcup_{i,j}mathrm{Spec}(A_{ij})$. Since $A_{ij}$ is a finitely generated $B_i$-algebra, it would suffice to show that $B_i$ is a finitely generated $B$-algebra.



Using this question I can find $mathrm{Spec(R)}subseteq mathrm{Spec}(B)cap mathrm{Spec}(B_i)$, so inclusions of topological spaces give rise to ring homomorphism $varphi:Bto R$ and $psi:B_ito R$. What I need to show then is that $R$ is a finitely generated $B_i$-algebra and that there exists a ring homomorphism $theta: Bto B_i$ such that $psicirctheta=varphi$.



Is my idea right? If so, how can I complete it? Is there any better way to solve this exercise? Thank you.






abstract-algebra algebraic-geometry commutative-algebra category-theory schemes







share|cite|improve this question









$endgroup$












  • $begingroup$
    Something seems not quite right. I think the hypothesis does not grant you an open cover of $V$; it only gives one for $f^{-1}(V)$.
    $endgroup$
    – Andrew
    Dec 14 '18 at 17:12






  • 2




    $begingroup$
    As for a better way to complete this exercise (and prove similar statements for many, many other properties of morphisms of schemes), look into the Affine Communication Lemma. This is the technical result that Ravi Vakil makes great use of in his notes.
    $endgroup$
    – Andrew
    Dec 14 '18 at 17:14








  • 1




    $begingroup$
    @Andrew from the definition $f$ being locally of fintie type there is a covering of $Y$ by affine subsets. Then I can take some of them to cover $V$. Thanks for there reference, I'll take a look at it.
    $endgroup$
    – Javi
    Dec 14 '18 at 17:14








  • 2




    $begingroup$
    Oh of course you are right. Next time I'll wait to finish my morning coffee to do math! ;)
    $endgroup$
    – Andrew
    Dec 14 '18 at 17:17
















2












$begingroup$


Definition: A morphism $f:Xto Y$ of schemes is locally of finite type if there exists a covering of $Y$ by open affine subsets $V_i=mathrm{Spec}(B_i)$, such that for each $i$, $f^{-1}(V_i)$ can be covered by open affine subsets $V_i=mathrm{Spec}(A_{ij})$, where each $A_{ij}$ is a finitely generated $B_i$-algebra.




Exercise 3.1: Show that a morphism $f:Xto Y$ is locally of finite type if and only if for every open affine subset $V=mathrm{Spec}(B)$ of $Y$, $f^{-1}(V)$ can be covered by open affine subsets $U_j=mathrm{Spec}(A_j)$, where each $A_j$ is a finitely generated $B$-algebra.




The 'if' direction is trivial, so I'm asking about the 'only if' direction.



Assume that $f:Xto Y$ is locally of finite type and let $V=mathrm{Spec}(B)$ an open affine subset of $Y$. I can use the covering of $Y$ given by the hypothesis to obtain a covering of $Vsubseteqbigcup_i V_i=bigcup_i mathrm{Spec}(B_i)$, so $f^{-1}(V)subseteq bigcup_i f^{-1}(V_i)subseteqbigcup_{i,j}mathrm{Spec}(A_{ij})$. Since $A_{ij}$ is a finitely generated $B_i$-algebra, it would suffice to show that $B_i$ is a finitely generated $B$-algebra.



Using this question I can find $mathrm{Spec(R)}subseteq mathrm{Spec}(B)cap mathrm{Spec}(B_i)$, so inclusions of topological spaces give rise to ring homomorphism $varphi:Bto R$ and $psi:B_ito R$. What I need to show then is that $R$ is a finitely generated $B_i$-algebra and that there exists a ring homomorphism $theta: Bto B_i$ such that $psicirctheta=varphi$.



Is my idea right? If so, how can I complete it? Is there any better way to solve this exercise? Thank you.






abstract-algebra algebraic-geometry commutative-algebra category-theory schemes







share|cite|improve this question









$endgroup$












  • $begingroup$
    Something seems not quite right. I think the hypothesis does not grant you an open cover of $V$; it only gives one for $f^{-1}(V)$.
    $endgroup$
    – Andrew
    Dec 14 '18 at 17:12






  • 2




    $begingroup$
    As for a better way to complete this exercise (and prove similar statements for many, many other properties of morphisms of schemes), look into the Affine Communication Lemma. This is the technical result that Ravi Vakil makes great use of in his notes.
    $endgroup$
    – Andrew
    Dec 14 '18 at 17:14








  • 1




    $begingroup$
    @Andrew from the definition $f$ being locally of fintie type there is a covering of $Y$ by affine subsets. Then I can take some of them to cover $V$. Thanks for there reference, I'll take a look at it.
    $endgroup$
    – Javi
    Dec 14 '18 at 17:14








  • 2




    $begingroup$
    Oh of course you are right. Next time I'll wait to finish my morning coffee to do math! ;)
    $endgroup$
    – Andrew
    Dec 14 '18 at 17:17














2












2








2





$begingroup$


Definition: A morphism $f:Xto Y$ of schemes is locally of finite type if there exists a covering of $Y$ by open affine subsets $V_i=mathrm{Spec}(B_i)$, such that for each $i$, $f^{-1}(V_i)$ can be covered by open affine subsets $V_i=mathrm{Spec}(A_{ij})$, where each $A_{ij}$ is a finitely generated $B_i$-algebra.




Exercise 3.1: Show that a morphism $f:Xto Y$ is locally of finite type if and only if for every open affine subset $V=mathrm{Spec}(B)$ of $Y$, $f^{-1}(V)$ can be covered by open affine subsets $U_j=mathrm{Spec}(A_j)$, where each $A_j$ is a finitely generated $B$-algebra.




The 'if' direction is trivial, so I'm asking about the 'only if' direction.



Assume that $f:Xto Y$ is locally of finite type and let $V=mathrm{Spec}(B)$ an open affine subset of $Y$. I can use the covering of $Y$ given by the hypothesis to obtain a covering of $Vsubseteqbigcup_i V_i=bigcup_i mathrm{Spec}(B_i)$, so $f^{-1}(V)subseteq bigcup_i f^{-1}(V_i)subseteqbigcup_{i,j}mathrm{Spec}(A_{ij})$. Since $A_{ij}$ is a finitely generated $B_i$-algebra, it would suffice to show that $B_i$ is a finitely generated $B$-algebra.



Using this question I can find $mathrm{Spec(R)}subseteq mathrm{Spec}(B)cap mathrm{Spec}(B_i)$, so inclusions of topological spaces give rise to ring homomorphism $varphi:Bto R$ and $psi:B_ito R$. What I need to show then is that $R$ is a finitely generated $B_i$-algebra and that there exists a ring homomorphism $theta: Bto B_i$ such that $psicirctheta=varphi$.



Is my idea right? If so, how can I complete it? Is there any better way to solve this exercise? Thank you.






abstract-algebra algebraic-geometry commutative-algebra category-theory schemes







share|cite|improve this question









$endgroup$




Definition: A morphism $f:Xto Y$ of schemes is locally of finite type if there exists a covering of $Y$ by open affine subsets $V_i=mathrm{Spec}(B_i)$, such that for each $i$, $f^{-1}(V_i)$ can be covered by open affine subsets $V_i=mathrm{Spec}(A_{ij})$, where each $A_{ij}$ is a finitely generated $B_i$-algebra.




Exercise 3.1: Show that a morphism $f:Xto Y$ is locally of finite type if and only if for every open affine subset $V=mathrm{Spec}(B)$ of $Y$, $f^{-1}(V)$ can be covered by open affine subsets $U_j=mathrm{Spec}(A_j)$, where each $A_j$ is a finitely generated $B$-algebra.




The 'if' direction is trivial, so I'm asking about the 'only if' direction.



Assume that $f:Xto Y$ is locally of finite type and let $V=mathrm{Spec}(B)$ an open affine subset of $Y$. I can use the covering of $Y$ given by the hypothesis to obtain a covering of $Vsubseteqbigcup_i V_i=bigcup_i mathrm{Spec}(B_i)$, so $f^{-1}(V)subseteq bigcup_i f^{-1}(V_i)subseteqbigcup_{i,j}mathrm{Spec}(A_{ij})$. Since $A_{ij}$ is a finitely generated $B_i$-algebra, it would suffice to show that $B_i$ is a finitely generated $B$-algebra.



Using this question I can find $mathrm{Spec(R)}subseteq mathrm{Spec}(B)cap mathrm{Spec}(B_i)$, so inclusions of topological spaces give rise to ring homomorphism $varphi:Bto R$ and $psi:B_ito R$. What I need to show then is that $R$ is a finitely generated $B_i$-algebra and that there exists a ring homomorphism $theta: Bto B_i$ such that $psicirctheta=varphi$.



Is my idea right? If so, how can I complete it? Is there any better way to solve this exercise? Thank you.






abstract-algebra algebraic-geometry commutative-algebra category-theory schemes




abstract-algebra algebraic-geometry commutative-algebra category-theory schemes






share|cite|improve this question













share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










asked Dec 14 '18 at 17:03









JaviJavi

2,6992826




2,6992826












  • $begingroup$
    Something seems not quite right. I think the hypothesis does not grant you an open cover of $V$; it only gives one for $f^{-1}(V)$.
    $endgroup$
    – Andrew
    Dec 14 '18 at 17:12






  • 2




    $begingroup$
    As for a better way to complete this exercise (and prove similar statements for many, many other properties of morphisms of schemes), look into the Affine Communication Lemma. This is the technical result that Ravi Vakil makes great use of in his notes.
    $endgroup$
    – Andrew
    Dec 14 '18 at 17:14








  • 1




    $begingroup$
    @Andrew from the definition $f$ being locally of fintie type there is a covering of $Y$ by affine subsets. Then I can take some of them to cover $V$. Thanks for there reference, I'll take a look at it.
    $endgroup$
    – Javi
    Dec 14 '18 at 17:14








  • 2




    $begingroup$
    Oh of course you are right. Next time I'll wait to finish my morning coffee to do math! ;)
    $endgroup$
    – Andrew
    Dec 14 '18 at 17:17


















  • $begingroup$
    Something seems not quite right. I think the hypothesis does not grant you an open cover of $V$; it only gives one for $f^{-1}(V)$.
    $endgroup$
    – Andrew
    Dec 14 '18 at 17:12






  • 2




    $begingroup$
    As for a better way to complete this exercise (and prove similar statements for many, many other properties of morphisms of schemes), look into the Affine Communication Lemma. This is the technical result that Ravi Vakil makes great use of in his notes.
    $endgroup$
    – Andrew
    Dec 14 '18 at 17:14








  • 1




    $begingroup$
    @Andrew from the definition $f$ being locally of fintie type there is a covering of $Y$ by affine subsets. Then I can take some of them to cover $V$. Thanks for there reference, I'll take a look at it.
    $endgroup$
    – Javi
    Dec 14 '18 at 17:14








  • 2




    $begingroup$
    Oh of course you are right. Next time I'll wait to finish my morning coffee to do math! ;)
    $endgroup$
    – Andrew
    Dec 14 '18 at 17:17
















$begingroup$
Something seems not quite right. I think the hypothesis does not grant you an open cover of $V$; it only gives one for $f^{-1}(V)$.
$endgroup$
– Andrew
Dec 14 '18 at 17:12




$begingroup$
Something seems not quite right. I think the hypothesis does not grant you an open cover of $V$; it only gives one for $f^{-1}(V)$.
$endgroup$
– Andrew
Dec 14 '18 at 17:12




2




2




$begingroup$
As for a better way to complete this exercise (and prove similar statements for many, many other properties of morphisms of schemes), look into the Affine Communication Lemma. This is the technical result that Ravi Vakil makes great use of in his notes.
$endgroup$
– Andrew
Dec 14 '18 at 17:14






$begingroup$
As for a better way to complete this exercise (and prove similar statements for many, many other properties of morphisms of schemes), look into the Affine Communication Lemma. This is the technical result that Ravi Vakil makes great use of in his notes.
$endgroup$
– Andrew
Dec 14 '18 at 17:14






1




1




$begingroup$
@Andrew from the definition $f$ being locally of fintie type there is a covering of $Y$ by affine subsets. Then I can take some of them to cover $V$. Thanks for there reference, I'll take a look at it.
$endgroup$
– Javi
Dec 14 '18 at 17:14






$begingroup$
@Andrew from the definition $f$ being locally of fintie type there is a covering of $Y$ by affine subsets. Then I can take some of them to cover $V$. Thanks for there reference, I'll take a look at it.
$endgroup$
– Javi
Dec 14 '18 at 17:14






2




2




$begingroup$
Oh of course you are right. Next time I'll wait to finish my morning coffee to do math! ;)
$endgroup$
– Andrew
Dec 14 '18 at 17:17




$begingroup$
Oh of course you are right. Next time I'll wait to finish my morning coffee to do math! ;)
$endgroup$
– Andrew
Dec 14 '18 at 17:17










0






active

oldest

votes











Your Answer





StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");

StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);

StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});

function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});


}
});














draft saved

draft discarded


















StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3039639%2fhartshorne-ii-3-1%23new-answer', 'question_page');
}
);

Post as a guest















Required, but never shown

























0






active

oldest

votes








0






active

oldest

votes









active

oldest

votes






active

oldest

votes
















draft saved

draft discarded




















































Thanks for contributing an answer to Mathematics Stack Exchange!


  • Please be sure to answer the question. Provide details and share your research!

But avoid …



  • Asking for help, clarification, or responding to other answers.

  • Making statements based on opinion; back them up with references or personal experience.


Use MathJax to format equations. MathJax reference.


To learn more, see our tips on writing great answers.




draft saved


draft discarded














StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3039639%2fhartshorne-ii-3-1%23new-answer', 'question_page');
}
);

Post as a guest















Required, but never shown





















































Required, but never shown














Required, but never shown












Required, but never shown







Required, but never shown

































Required, but never shown














Required, but never shown












Required, but never shown







Required, but never shown







-

Popular posts from this blog

Probability when a professor distributes a quiz and homework assignment to a class of n students.

Image
0 0 $begingroup$ Need help with this problem. Suppose our lazy professor collects a quiz and a homework assignment from a class of n students one day, then distributes both the quizzes and the homework assignments back to the class in a random fashion for grading. Each student receives one quiz and one homework assignment to grade. (a) What is the probability that every student receives someone else's quiz to grade, and someone else's homework to grade? (b) What is the probability that no student receives both their own quiz and their own homework assignment to grade? In this case, some students may receive their own quiz, and others may receive their own homework assignment. (c) Compute the limiting probability as n approaches infinity in each case. ...
Read more

Aardman Animations

Image
Aardman Animations, Ltd. Aardman Animations Tipo privada Atividade Animação em stop-motion , Animação em CGI Fundação 1972 Fundador(es) Peter Lord David Sproxton Sede Bristol, Inglaterra  Reino Unido Proprietário(s) Dreamworks Pessoas-chave Peter Lord David Sproxton Nick Park Divisões Aardman Features Aardman Digital Aardman Commercials Aardman Broadcast Aardman International Aardman Rights Aardman Effects Aardman 3-D Systems Aardman Nathan Love AardBoiled Website oficial aardman.com ...
Read more

Are they similar matrix

Image
0 $begingroup$ Do $begin{bmatrix} 0&i&0\0&0&1\0&0&0 end{bmatrix} $ and $begin{bmatrix} 0&0&0\-i&0&0\0&1&0 end{bmatrix} $ are similar.Is this True/false Clearly both are nilpotent and one is conjucate transpose of other but how to know if they are similar.i'm stuck. Please help me linear-algebra vector-spaces eigenvalues-eigenvectors generalizedeigenvector share | cite | improve this question edited Dec 10 '18 at 6:38 Vasanth Kris asked Dec 10 '18 at 6:19 ...
Read more