I'm reading Harteshorne chapter 2.8. In theorem 8.13, he shows there is a short exact sequence: $$ 0\rightarrow \Omega_{X/A}\rightarrow \mathcal O_{X}(-1)^{n+1}\rightarrow\mathcal O_X\rightarrow 0 $$ where $X = \mathbb P_A^n$ is the projective space over $A$.
I guess I can understand the proof line by line. But I have no idea what's going on in this proof. Do these computations have any geometric interpretations?
Also, this short exact sequence seems to be very important. Is there any initiative way to understand this exact sequence? Does it have geometric meaning? Thank you in advance.
Let $S:=Spec(A)$ be an affine scheme and let $E:=A\{e_0,..,e_n\}$ be the free left $A$-module on the $n+1$ elements $e_i$, and $E^*:=A\{x_0,..,x_n\}$, with $x_i$ the dual basis. Let $\mathcal{E}$ be the sheafication of the $A$-module $E$. There is by Hartshornes book (Chapter II.7) a relative projective space bundle $\mathbb{P}(\mathcal{E}^*)$, and a canonical surjective morphism $\pi: \mathbb{P}(\mathcal{E}^*) \rightarrow S$ with projective spaces as fibers. For any point $s \in S$ it follows the fiber $\pi^{-1}(s) \cong \mathbb{P}^n_{\kappa(s)}$, where $\kappa(s)$ is the residue field of the point $s$ and where $\mathbb{P}^n_{\kappa(s)}$ is projective $n$-space over the field $\kappa(s)$. The morphism $\pi$ is locally trivial in the Zariski topology. The tautological sequence reflects the fact that we may define the projective space bundle using representable functors: The projective space bundle represents the following functor. Define the functor $F:Sch(S) \rightarrow Set$ from the category of schemes over $S$ to the category of sets as follows: Let $F(X, g)$ be the set of pairs $(\mathcal{L}, \phi)/\cong$. Where $\phi: g^*\mathcal{E} \rightarrow \mathcal{L}$ is a surjective map onto an invertible sheaf $\mathcal{L}$ on $X$, modulo the following equivalence relation: Two pairs $(\mathcal{L}, \phi)$ and $(\mathcal{L}', \phi')$ are equivalent iff there is an isomorphism of sheaves $\psi: \mathcal{L} \rightarrow \mathcal{L}'$ making the obvious diagram commute. We may prove that there is an isomorphism of functore $Hom_S(-, \mathbb{P}(\mathcal{E}^*) ) \cong F(-)$ (this is proved in Hartshorne Prop II.7.12). Hence the functor $F$ is represented by the projective space bundle $\mathbb{P}(\mathcal{E}^*)$. If $S:=Spec(k)$ is the spectrum of a field $k$, it follows $\mathbb{P}(\mathcal{E}^*)(k):=Hom_S(S, \mathbb{P}(\mathcal{E}^*) = F(S, id)$, and from this it follows the $k$-rational points $\mathbb{P}(\mathcal{E}^*)(k)$ of the projective space bundle $\mathbb{P}(\mathcal{E}^*)$ equals the set of lines in the $k$-vector space $E$. Hence the tautological sequence reflects the fact that the projective space bundle parametrizes "lines" in $E$.
More generaly if $\mathcal{E}$ is a coherent $\mathcal{O}_S$-module, the EGA book series defines the projective space bundle $\mathbb{P}(\mathcal{E}^*)$ and a surjective morphism $\pi: \mathbb{P}(\mathcal{E}^*) \rightarrow S$ with projective spaces as fibers. If $i: U\rightarrow S$ is an open subscheme where $\mathcal{E}$ is locally trivial it follows the fiber product $U\times_S \mathbb{P}(\mathcal{E}) \cong \mathbb{P}(i^*\mathcal{E}^*) \cong \mathbb{P}(\mathcal{E}_U^*)$ defines a fibration $\pi_U: \mathbb{P}(\mathcal{E}_U^*) \rightarrow U$ which is a locally trivial fibration in the Zariski topology with projective spaces as fibers. A good exercise is to fill in all details of HH, Prop II.7.12 and then try to generalize to the grassmannian bundle. One may construct the grassmannian bundle $\pi: \mathbb{G}(k,\mathcal{E}) \rightarrow S$ of a rank $d$ locally trivial $\mathcal{O}_S$-module $\mathcal{E}$. The map $\pi$ is a surjective map with the property that for any point $s \in S$ it follows the fiber $\pi^{-1}(s)$ is isomorphic to the classical grassmannian $\mathbb{G}(k,\mathcal{E}(s))$, where $\mathcal{E}(s)$ is the fiber at $s$. It follows the fiber $\pi^{-1}(s)$ parametrize $k$-dimensional $\kappa(s)$-sub spaces of the fiber $\mathcal{E}(s)$. The map $\pi$ is locally trivial in the Zariski topology. The following book gives all details on this construction. It also defines and studies flag bundles.
A. Grothendieck, Éléments de géométrie algébrique. I. (English) Zbl 0203.23301 Die Grundlehren der mathematischen Wissenschaften. 166. Berlin-Heidelberg-New York: Springer-Verlag. IX, 466 p. (1971).