Describe $H^0(\mathcal O_X(m-2))\to H^1(I_X(m-2))$ induced by exact short sequence of sheaves Let $i:X\subset\mathbb P^1$ be a set of $m$ points and let $I_X\subset\cal O_{\mathbb P^n}$ be its sheaf of ideals. Then the sequence
$$(1)0\to I_X(m-2)\to\mathcal O_{\mathbb P^1}(m-2)\to\mathcal i_*O_X(m-2)\to0$$ is exact (here $\mathcal O_X$ denotes $\mathcal O_{\mathbb P^1}/I_X)$.
I'd like to describe explicitly a homomorphism $H^0(\mathcal O_X(m-2))\to H^1(I_X(m-2))$ induced by the exact short sequence. I've got no ideas how to deal with this. Think it'd be helpful to identify $I_X(m-2)$ and $\mathcal O_{\mathbb P^1}(-2).$ Could you please help me?
 A: The short exact sequence $(1)$ is known as the "twisted" Koszul resolution of $I_X$. It is a resolution of $I_X$ by locally free sheaves that is used to compute the local cohomology of $I_X$.
To describe the map $H^0(\mathcal O_X(m-2))\to H^1(I_X(m-2))$ induced by the short exact sequence $(1)$, we can use the long exact sequence in cohomology associated with the short exact sequence $(1)$.
The long exact sequence in cohomology associated with $(1)$ is given by:
$$\cdots\to H^0(\mathcal O_X(m-2))\to H^1(I_X(m-2))\to H^1(\mathcal O_{\mathbb P^1}(m-2))\to H^1(\mathcal O_X(m-2))\to\cdots$$
The map $H^0(\mathcal O_X(m-2))\to H^1(I_X(m-2))$ is the connecting homomorphism in this long exact sequence. It is induced by the short exact sequence $(1)$ and is given by the boundary map $H^0(\mathcal O_X(m-2))\to H^1(\mathcal O_{\mathbb P^1}(m-2))$.
To compute this boundary map, we need to understand the map $\mathcal O_{\mathbb P^1}(m-2)\to\mathcal O_X(m-2)$. This map is given by the natural surjective map $\mathcal O_{\mathbb P^1}\to\mathcal O_X$ composed with the map $\mathcal O_X\to\mathcal O_X(m-2)$ given by multiplication by an element of $\mathcal O_X(m-2)$.
Since $I_X$ is the sheaf of ideals of $X$, the map $\mathcal O_{\mathbb P^1}\to\mathcal O_X$ is surjective and its kernel is $I_X$. Therefore, the map $\mathcal O_{\mathbb P^1}(m-2)\to\mathcal O_X(m-2)$ is surjective and its kernel is $I_X(m-2)$.
This gives us the short exact sequence:
$$0\to I_X(m-2)\to\mathcal O_{\mathbb P^1}(m-2)\to\mathcal O_X(m-2)\to0$$
which is the sequence $(1)$.
Using the long exact sequence in cohomology associated with this short exact sequence, we can now compute the map $H^0(\mathcal O_X(m-2))\to H^1(I_X(m-2))$ induced by the short exact sequence $(1)$.
Since $\mathcal O_{\mathbb P^1}(m-2)$ is a locally free sheaf, $H^1(\mathcal O_{\mathbb P^1}(m-2))=0$. Therefore, the map $H^0(\mathcal O_X(m-2))\to H^1(I_X(m-2))$
To describe explicitly the homomorphism $H^0(\mathcal O_X(m-2))\to H^1(I_X(m-2))$ induced by the exact short sequence $(1)$, we can start by considering the definition of the homomorphism induced by an exact sequence.
Given an exact sequence of sheaves of abelian groups
$$0\to A\xrightarrow{\alpha} B\xrightarrow{\beta} C\to 0$$
the homomorphism induced by the exact sequence is defined as the map
$$H^0(C)\to H^1(A)$$
such that for any $c\in H^0(C)$, $c$ is mapped to the element $a\in H^1(A)$ that satisfies the following conditions:
$a$ is in the kernel of the map $H^1(A)\to H^1(B)$ induced by $\alpha$.
There exists a $b\in H^0(B)$ such that $\beta(b)=c$ and $b$ is mapped to $a$ under the map $H^0(B)\to H^1(A)$ induced by $\alpha$.
To apply this definition to the exact sequence $(1)$, we can start by identifying the sheaves $A$, $B$, and $C$ in the sequence.
In this case, we can take $A=I_X(m-2)$, $B=\mathcal O_{\mathbb P^1}(m-2)$, and $C=\mathcal O_X(m-2)$.
Then, the homomorphism induced by the exact sequence $(1)$ can be defined as the map
$$H^0(\mathcal O_X(m-2))\to H^1(I_X(m-2))$$
such that for any $c\in H^0(\mathcal O_X(m-2))$, $c$ is mapped to the element $a\in H^1(I_X(m-2))$ that satisfies the following conditions:
$a$ is in the kernel of the map $H^1(I_X(m-2))\to H^1(\mathcal O_{\mathbb P^1}(m-2))$ induced by the inclusion map $I_X(m-2)\hookrightarrow \mathcal O_{\mathbb P^1}(m-2)$.
There exists a $b\in H^0(\mathcal O_{\mathbb P^1}(m-2))$ such that the quotient map $\mathcal O_{\mathbb P^1}(m-2)\to \mathcal O_X(m-2)$ sends $b$ to $c$ and $b$ is mapped to $a$ under the map $H^0(\mathcal O_{\mathbb P^1}(m-2))\to H^1(I_X(m-2))$ induced by the inclusion map $I_X(m-2)\hookrightarrow \mathcal O_{\mathbb P^1}(m-2)$.
I hope this helps clarify the definition of the homomorphism induced by the exact sequence $(1)$. Let me know if you have any other questions.
