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










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