Interpretation of $Hom_{mathcal{O}_X} (mathcal{E}, mathcal{F} ) cong mathcal{E}^* otimes mathcal{F}$












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$begingroup$


Let $mathcal{E}$ be a locally free sheaf of $mathcal{O}_X$-modules of finite rank. It is a standard exercise to show that for any $mathcal{O}_X$-module $mathcal{F}$,
$$Hom_{mathcal{O}_X} (mathcal{E}, mathcal{F} ) cong mathcal{E}^* otimes_{mathcal{O}_X} mathcal{F} $$
where $mathcal{E}^*$ is the sheaf $Hom_{mathcal{O}_X}(mathcal{E}, mathcal{O}_X)$.



Let $X=mathbb{P}^1=operatorname{Proj} k[X_0,X_1]$ and $mathcal{E} cong mathcal{O}(-1)$ and let $mathcal{F} cong mathcal{O}(2)$.



Then $operatorname{Hom}_{mathcal{O}_X}(mathcal{E}, mathcal{F}) cong Gamma(X, mathcal{O}(3))$ where
$ Gamma(X, mathcal{O}(3)) $ is generated as a $Gamma(X, mathcal{O}_X)= k$-module by $X_0^3, X_0^2X_1, X_0 X_1^2, X_1^3$. This means that there exist $4$ disinct morphism $mathcal{O}(-1) to mathcal{O}(2)$ up to $Gamma(X, mathcal{O}_X^*)$.



I know each of the generators $X_0^3, X_0^2X_1, X_0 X_1^2, X_1^3$ corresponds to a morphism $phi in operatorname{Hom}(mathcal{O}(-1), mathcal{O}(2))$.



Suppose $varpi in operatorname{Hom}(mathcal{O}(-1), mathcal{O}(2))$ corresponding to the generator $X_0^3$. How is $X_0^3$ related to the image of $mathcal{O}(-1)$ under $varpi$ in $mathcal{O}(2)$ ?



At some point I was thinking that $X_0^3$ would generate (over k) all global sections of the image of $mathcal{O}(-1)$ as a subsheaf ( I guess here I would be imagining that $varpi$ is an inclusion) of $mathcal{O}(2)$






algebraic-geometry







share|cite|improve this question









$endgroup$

















    1












    $begingroup$


    Let $mathcal{E}$ be a locally free sheaf of $mathcal{O}_X$-modules of finite rank. It is a standard exercise to show that for any $mathcal{O}_X$-module $mathcal{F}$,
    $$Hom_{mathcal{O}_X} (mathcal{E}, mathcal{F} ) cong mathcal{E}^* otimes_{mathcal{O}_X} mathcal{F} $$
    where $mathcal{E}^*$ is the sheaf $Hom_{mathcal{O}_X}(mathcal{E}, mathcal{O}_X)$.



    Let $X=mathbb{P}^1=operatorname{Proj} k[X_0,X_1]$ and $mathcal{E} cong mathcal{O}(-1)$ and let $mathcal{F} cong mathcal{O}(2)$.



    Then $operatorname{Hom}_{mathcal{O}_X}(mathcal{E}, mathcal{F}) cong Gamma(X, mathcal{O}(3))$ where
    $ Gamma(X, mathcal{O}(3)) $ is generated as a $Gamma(X, mathcal{O}_X)= k$-module by $X_0^3, X_0^2X_1, X_0 X_1^2, X_1^3$. This means that there exist $4$ disinct morphism $mathcal{O}(-1) to mathcal{O}(2)$ up to $Gamma(X, mathcal{O}_X^*)$.



    I know each of the generators $X_0^3, X_0^2X_1, X_0 X_1^2, X_1^3$ corresponds to a morphism $phi in operatorname{Hom}(mathcal{O}(-1), mathcal{O}(2))$.



    Suppose $varpi in operatorname{Hom}(mathcal{O}(-1), mathcal{O}(2))$ corresponding to the generator $X_0^3$. How is $X_0^3$ related to the image of $mathcal{O}(-1)$ under $varpi$ in $mathcal{O}(2)$ ?



    At some point I was thinking that $X_0^3$ would generate (over k) all global sections of the image of $mathcal{O}(-1)$ as a subsheaf ( I guess here I would be imagining that $varpi$ is an inclusion) of $mathcal{O}(2)$






    algebraic-geometry







    share|cite|improve this question









    $endgroup$















      1












      1








      1





      $begingroup$


      Let $mathcal{E}$ be a locally free sheaf of $mathcal{O}_X$-modules of finite rank. It is a standard exercise to show that for any $mathcal{O}_X$-module $mathcal{F}$,
      $$Hom_{mathcal{O}_X} (mathcal{E}, mathcal{F} ) cong mathcal{E}^* otimes_{mathcal{O}_X} mathcal{F} $$
      where $mathcal{E}^*$ is the sheaf $Hom_{mathcal{O}_X}(mathcal{E}, mathcal{O}_X)$.



      Let $X=mathbb{P}^1=operatorname{Proj} k[X_0,X_1]$ and $mathcal{E} cong mathcal{O}(-1)$ and let $mathcal{F} cong mathcal{O}(2)$.



      Then $operatorname{Hom}_{mathcal{O}_X}(mathcal{E}, mathcal{F}) cong Gamma(X, mathcal{O}(3))$ where
      $ Gamma(X, mathcal{O}(3)) $ is generated as a $Gamma(X, mathcal{O}_X)= k$-module by $X_0^3, X_0^2X_1, X_0 X_1^2, X_1^3$. This means that there exist $4$ disinct morphism $mathcal{O}(-1) to mathcal{O}(2)$ up to $Gamma(X, mathcal{O}_X^*)$.



      I know each of the generators $X_0^3, X_0^2X_1, X_0 X_1^2, X_1^3$ corresponds to a morphism $phi in operatorname{Hom}(mathcal{O}(-1), mathcal{O}(2))$.



      Suppose $varpi in operatorname{Hom}(mathcal{O}(-1), mathcal{O}(2))$ corresponding to the generator $X_0^3$. How is $X_0^3$ related to the image of $mathcal{O}(-1)$ under $varpi$ in $mathcal{O}(2)$ ?



      At some point I was thinking that $X_0^3$ would generate (over k) all global sections of the image of $mathcal{O}(-1)$ as a subsheaf ( I guess here I would be imagining that $varpi$ is an inclusion) of $mathcal{O}(2)$






      algebraic-geometry







      share|cite|improve this question









      $endgroup$




      Let $mathcal{E}$ be a locally free sheaf of $mathcal{O}_X$-modules of finite rank. It is a standard exercise to show that for any $mathcal{O}_X$-module $mathcal{F}$,
      $$Hom_{mathcal{O}_X} (mathcal{E}, mathcal{F} ) cong mathcal{E}^* otimes_{mathcal{O}_X} mathcal{F} $$
      where $mathcal{E}^*$ is the sheaf $Hom_{mathcal{O}_X}(mathcal{E}, mathcal{O}_X)$.



      Let $X=mathbb{P}^1=operatorname{Proj} k[X_0,X_1]$ and $mathcal{E} cong mathcal{O}(-1)$ and let $mathcal{F} cong mathcal{O}(2)$.



      Then $operatorname{Hom}_{mathcal{O}_X}(mathcal{E}, mathcal{F}) cong Gamma(X, mathcal{O}(3))$ where
      $ Gamma(X, mathcal{O}(3)) $ is generated as a $Gamma(X, mathcal{O}_X)= k$-module by $X_0^3, X_0^2X_1, X_0 X_1^2, X_1^3$. This means that there exist $4$ disinct morphism $mathcal{O}(-1) to mathcal{O}(2)$ up to $Gamma(X, mathcal{O}_X^*)$.



      I know each of the generators $X_0^3, X_0^2X_1, X_0 X_1^2, X_1^3$ corresponds to a morphism $phi in operatorname{Hom}(mathcal{O}(-1), mathcal{O}(2))$.



      Suppose $varpi in operatorname{Hom}(mathcal{O}(-1), mathcal{O}(2))$ corresponding to the generator $X_0^3$. How is $X_0^3$ related to the image of $mathcal{O}(-1)$ under $varpi$ in $mathcal{O}(2)$ ?



      At some point I was thinking that $X_0^3$ would generate (over k) all global sections of the image of $mathcal{O}(-1)$ as a subsheaf ( I guess here I would be imagining that $varpi$ is an inclusion) of $mathcal{O}(2)$






      algebraic-geometry




      algebraic-geometry






      share|cite|improve this question













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      asked Jan 23 at 20:04









      JadwigaJadwiga

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          $begingroup$

          $mathcal O(-1)$ doesn't actually have any global sections, but it has lots of local sections. For example, $s = (X_0 + 4X_1) / X_1^2$ is a local section of $mathcal O(-1)$, which is defined on the open set $mathbb P^1 setminus V(X_1)$. Under the action of $varpi = X_0^3 in {rm Hom}(mathcal O(-1), mathcal O(2))$, $s$ maps to $varpi(s) = X_0^3(X_0 + 4X_1) / X_1^2$, which is a local section of $mathcal O(2)$ defined on the same open set.



          It is true that the sheaf morphism $varpi : mathcal O(-1) to mathcal O(2)$ is injective on all stalks. The sheaf morphism $varpi$ is also surjective on all stalks except for the stalk at $p := [0 : 1]$, and the cokernel of $varpi$ is a skyscraper sheaf of rank $3$ supported at $p$.






          share|cite|improve this answer









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            $begingroup$

            $mathcal O(-1)$ doesn't actually have any global sections, but it has lots of local sections. For example, $s = (X_0 + 4X_1) / X_1^2$ is a local section of $mathcal O(-1)$, which is defined on the open set $mathbb P^1 setminus V(X_1)$. Under the action of $varpi = X_0^3 in {rm Hom}(mathcal O(-1), mathcal O(2))$, $s$ maps to $varpi(s) = X_0^3(X_0 + 4X_1) / X_1^2$, which is a local section of $mathcal O(2)$ defined on the same open set.



            It is true that the sheaf morphism $varpi : mathcal O(-1) to mathcal O(2)$ is injective on all stalks. The sheaf morphism $varpi$ is also surjective on all stalks except for the stalk at $p := [0 : 1]$, and the cokernel of $varpi$ is a skyscraper sheaf of rank $3$ supported at $p$.






            share|cite|improve this answer









            $endgroup$


















              1












              $begingroup$

              $mathcal O(-1)$ doesn't actually have any global sections, but it has lots of local sections. For example, $s = (X_0 + 4X_1) / X_1^2$ is a local section of $mathcal O(-1)$, which is defined on the open set $mathbb P^1 setminus V(X_1)$. Under the action of $varpi = X_0^3 in {rm Hom}(mathcal O(-1), mathcal O(2))$, $s$ maps to $varpi(s) = X_0^3(X_0 + 4X_1) / X_1^2$, which is a local section of $mathcal O(2)$ defined on the same open set.



              It is true that the sheaf morphism $varpi : mathcal O(-1) to mathcal O(2)$ is injective on all stalks. The sheaf morphism $varpi$ is also surjective on all stalks except for the stalk at $p := [0 : 1]$, and the cokernel of $varpi$ is a skyscraper sheaf of rank $3$ supported at $p$.






              share|cite|improve this answer









              $endgroup$
















                1












                1








                1





                $begingroup$

                $mathcal O(-1)$ doesn't actually have any global sections, but it has lots of local sections. For example, $s = (X_0 + 4X_1) / X_1^2$ is a local section of $mathcal O(-1)$, which is defined on the open set $mathbb P^1 setminus V(X_1)$. Under the action of $varpi = X_0^3 in {rm Hom}(mathcal O(-1), mathcal O(2))$, $s$ maps to $varpi(s) = X_0^3(X_0 + 4X_1) / X_1^2$, which is a local section of $mathcal O(2)$ defined on the same open set.



                It is true that the sheaf morphism $varpi : mathcal O(-1) to mathcal O(2)$ is injective on all stalks. The sheaf morphism $varpi$ is also surjective on all stalks except for the stalk at $p := [0 : 1]$, and the cokernel of $varpi$ is a skyscraper sheaf of rank $3$ supported at $p$.






                share|cite|improve this answer









                $endgroup$



                $mathcal O(-1)$ doesn't actually have any global sections, but it has lots of local sections. For example, $s = (X_0 + 4X_1) / X_1^2$ is a local section of $mathcal O(-1)$, which is defined on the open set $mathbb P^1 setminus V(X_1)$. Under the action of $varpi = X_0^3 in {rm Hom}(mathcal O(-1), mathcal O(2))$, $s$ maps to $varpi(s) = X_0^3(X_0 + 4X_1) / X_1^2$, which is a local section of $mathcal O(2)$ defined on the same open set.



                It is true that the sheaf morphism $varpi : mathcal O(-1) to mathcal O(2)$ is injective on all stalks. The sheaf morphism $varpi$ is also surjective on all stalks except for the stalk at $p := [0 : 1]$, and the cokernel of $varpi$ is a skyscraper sheaf of rank $3$ supported at $p$.







                share|cite|improve this answer












                share|cite|improve this answer



                share|cite|improve this answer










                answered Jan 23 at 22:46









                Kenny WongKenny Wong

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                19.1k21440






























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