Exact sequences seem to have a key role throughout Algebraic Geometry in general. For example to deduce vanishings in cohomology and so on. And there seems to be a series of standard short exact sequences which are widely and frequently used again and again. So the motivation for this post should be clear: I thought it would be very nice to have a list of as many standard short exact sequences as possible, providing if possible some short motivation or intuition for each of them.
I have looked for such a list already but I haven't found anything. If anyone knows of a reference with such a list, it would also be very appreciated.
Here are the standard short exact sequences that I have come across so far:
- The Euler sequence and its dual $$ 0\to \mathcal{O}_{\mathbb{P}^{n}} \to \mathcal{O}_{\mathbb{P}^{n}}(1)^{\oplus n+1 } \to \mathcal{T}_{\mathbb{P}^{n}} \to 0$$
- Sequence associated to an effective Cartier divisor $$ 0\to \mathcal{O}_{X}(-D) \to \mathcal{O}_{X} \to \mathcal{O}_{D} \to 0$$ and the analog sequence for differential $k$-forms $$ 0\to \Omega^{k}_{X}(-D)\to \Omega^{k}_{X}\to \Omega^{k}_{D} \to 0 $$
- Relative cotangent sequence and its dual (I only write the easier to remember geometric version that can be pictured esily for $\pi \colon X\to Y$ smooth varieties and $Z$ a point, as suggested by Vakil in his notes) $$ 0\to \mathcal{T}_{X/Y} \to \mathcal{T}_{X/Z}\to \pi^{*}\mathcal{T}_{Y/Z} \to 0$$
- Relative conormal sequence and its dual (again, the easier to picture/remember geometric version when $i\colon Y\to X$ is a closed immersion of smooth varieties) $$ 0\to \mathcal{T}_{Y} \to \mathcal{T}_{X}\mid_{Y} \to \mathcal{N}_{Y/X} \to 0$$
- The exponential sequence (in the analytic topolgoy) $$ 0\to \mathbb{Z} \to \mathcal{O}_{X} \to \mathcal{O}_{X}^{\times} \to 0 $$
P.S. I am not sure this if this is a valid question or not. One could say "don't be lazy and go trhough the literature yourself to find the answer". But experience so far shows that this takes a huge amount of time and that invariantly new "standard" short exact sequences keep popping out every now and then. I know that this last impression is just because I only started studying Algebraic Geometry recently, and hence I still have a lot to learn (including a lot of exact sequences). But I feel like my study of this area would be much more efficient if I had always with me such a list from the beginning, and maybe other people share this opinion too.