Standard short exact sequences in Algebraic Geometry

Question

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.

Answer 1

Let $X$ be a scheme of finite type over a field, $Y\subset X$ a closed subscheme and $U=X\setminus Y$ its open complement.
Then we have (Fulton,Prop. 1.8, page 21) for the Chow goups of $k$-cycles the exact sequence $$CH_k(Y)\to CH_k(X)\to CH_k(U)\to 0 $$ This powerful tool immediately implies, for example, that for any closed irreducible hypersurface $Y=Y_d\subset X=\mathbb P^n_k$ of degree $d$ of projective space over a field $k$ we have for $U=\mathbb P^n_k\setminus Y$ : $$\operatorname {Pic}(U)=CH_{n-1} (U)=\mathbb Z/d\mathbb Z$$ allowing us to brag that for every cyclic group we know a smooth algebraic variety whose Picard group is that cyclic group! ( For the infinite cyclic group we show off $\mathbb Z=\operatorname {Pic}(\mathbb P^n_k) $).

Answer 2

The Kummer sequence is another useful one, especially in studying Brauer groups and twisted sheaves: $$ 0 \to \mu_n \longrightarrow \mathbb{G}_m \overset{n}{\longrightarrow} \mathbb{G}_m\to 0 $$ where $\mu_n$ is the set of $n^\text{th}$ roots of unity.

Answer 3

One of the most important sequence is the "tautological sequence". This is very useful when computing Chern classes.

Assume $X$ is the space parametrizing some vector subspace of a fixed vector space $V$. Then, above $x \in X$ there is a corresponding $V_x \subset V$. Then, there will be a corresponding exact sequence $$ 0 \to V_x \to V \to V/V_x \to 0$$

called the "tautological sequence".

The typical example is the Grassmanian $X = G(k,V)$. The sequence is $$ 0 \to S \to V \to Q \to 0 $$

where $V$ is the trivial vector bundle $V \times X$, $S$ is the vector bundle which fiber over $x \in X$ is the corresponding subspace $x \subset V$ and $Q$ is the vector bundle with fiber $Q_x = V/x$.

Similar sequences exist when $X$ is an Hilbert scheme or a flag variety. For more details see the book by Fulton and Harris, "3264 and all that".

Another sequence is pretty useful. If $D \subset X$ is a smooth divisor in a projective variety then there is a long exact sequence $$ \dots \to H^i(X) \to H^i(U) \to H^{i-1}(D) \to H^{i+1}(X) \to \dots $$

which is the algebraic version of the Gysin exact sequence. For a reference this is probably in Fulton's book on intersection theory.