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in the group of Weil divisors on $C$. The book is more than 700 pages long and give all details - again you should check these details if you need this construction. In Example II.8.20 in Hartshorne they write down the following exact sequence defining the tangent bundle of projective space $X:=\mathbb{P}^n_k$:

in the group of Weil divisors on $C$. More generally if $\mathbb{P}^n$ is compex projective $n$-space it follows the global sections $H^0(\mathbb{P}^n, \mathcal{O}(d))$ is the vector space of homogeneous degree $d$ polynomials in $n+1$ variables. In Proposition II.6.4 in Hartshorne and the examples 6.5.1, 6.5.2 some examples are given.

The book is more than 700 pages long and give all details - again you should check these details if you need this construction. In Example II.8.20 in Hartshorne they write down the following exact sequence defining the tangent bundle of projective space $X:=\mathbb{P}^n_k$:

Remark: In the book of Eisenbud/Harris they speak about Weil divisors/Cartier divisors and invertible sheaves and also the projective bundle formula: For any rank $e$ locally trivial sheaf $E$ on a scheme $X$, they prove there is an isomorphism of rings

Remark: In the book of Eisenbud/Harris they speak about Weil divisors/Cartier divisors and invertible sheaves and also the projective bundle formula: For any rank $e+1$ locally trivial sheaf $E$ on a scheme $X$, they prove there is an isomorphism of rings

Note: The books referred to above are in total $496 + 633 + 813 = 1942 $ pages and you can't yourself verify all results presented in the books. The results presented are classical results on characteristic classes and most of them are correct I believe.