Theorem 2.2. Auslander-Buchsbaum, Sheaf Version.
If \(X\) is a smooth variety, then every coherent \(\mathcal O_X\)-module has a finite resolution by vector bundles.
Section 2 Let’s do some geometry!
Subsection 2.1 Working with schemes.
Macaulay2 is built with the algebraic geometry in mind. Many of the commutative algebra functionalities have a natural extension to schemes. For example, let’s build our beloved projective space \(\mathbb P^3\text{.}\)
In \(\mathbb P^3\text{,}\) let’s make a rational quartic curve given in local coordinates as \((t, t^3, t^4)\text{.}\)
Was eine Kurve ist, glaubt jeder Mensch zu wissen, bis er so viel Mathematik gelernt hat, daß ihn die unzähligen möglichen Abnormalitäten verwirrt gemacht haben.Everyone knows what a curve is, until he has studied enough mathematics to become confused through the countless number of possible exceptions.―Felix C. Klein
We can make the sheaf of regular functions \(\mathcal{O}_C\) with ease.
... and the ideal sheaf \(\mathcal{I}_{C/\mathbb P^3}\text{.}\)
We can do most things we want with coherent sheaves. For example, we can construct the line bundles \(\mathcal O_C(d)\text{.}\)
and do direct sums \(\oplus\) or tensor products \(\otimes\text{.}\)
We can do all sorts of homological algebra with sheaves. Just do a few lines of code instead of calculating everything by hand! (which can be either good or bad, depending on what your goals are ..)
Even the tangent/cotangent sheaves are built-in.
... and of course, we are tempted to calculate some sheaf cohomology!
... and we have an instance of Serre duality.
Subsection 2.2 The Koszul Complex
Many standard +notions in commutative algebra are built-in, including regular sequences. Let’s ask M2 to check for us whether the following sequences are regular in the ring \(\mathbb Z/2[x,y,z]\text{.}\)
For regular local rings, the Koszul complex is exact, so it provides a finite free resolution. Let’s try to carry this out for our projective curve \(C\text{.}\) Recall that \(S = \mathbb Q[x,y,z,w]\) and \(I = (y z-x w, z^3-y w^2, x z^2-y^2 w, y^3-x^2 z)\text{.}\)
What does this mean geometrically? If we sheafify everything, then the Koszul complex becomes
\begin{equation*}
0 \to O_{\mathbb P^3}(-5) \to \cdots\to \mathcal O_{\mathbb P^3}(-2)^{\oplus 4}\to \mathcal O_{\mathbb P^3}(-1)\to \mathcal O_{\mathbb P^3}\xrightarrow{\texttt{ gens } I} \mathcal O_C \to 0.
\end{equation*}
Here, \(\mathcal O_C\) means the proper pushforward \(i_*\mathcal O_C\) along the inclusion \(i: C\to \mathbb P^3\text{.}\) Therefore, the Koszul resolution upgrades to a locally free resolution of a coherent sheaf.
