We’re going to need to start out the day with a bit of algebra, because we’re going to talk about differential forms. Once we have forms, we’ll make a sheaf out of them, and then we’ll use this sheaf to construct other things.

We start out by needing the notion of a derivation. Given a ring $R$, an $R$-algebra (that is, a ring which is also an $R$-module) $S$ and an $S$-module $M$, we define an $R$-derivation to be a function $D:S\to M$ such that $D(fg)=fD(g)+D(f)g$ for all $f,g\in S$, $D(f+g)=D(f)+D(g)$, and $D(f)=0$ for all $f\in R$. (We say that an element $f\in S$ is in $R$ if it is of the form $f\cdot 1_S$ for $f\in R$ using the module structure.)

So really, what this is is just a function that acts like taking a derivative. The product rule is there, we think of $R$ as the set of constants, so they have derivative zero, it’s additive, etc. There is, in fact, a universal such derivation. Specifically, there’s an $S$-module called $\Omega_{S/R}$ and a derivation $S\to \Omega_{S/R}$ such that any other derivation is given by composing with a module homomorphism $\Omega_{S/R}\to M$. Even better, we can describe it.

We take the collection of symbols $d(f)$ for $f\in S$ and take formal finite sums of the form $\sum_{k=1}^n g_i d(f_i)$ where $g_i,f_i\in S$. These are the elements, but we do identify some of them. Specifically, we identify $d(g+f)$ with $d(g)+d(f)$, $d(fg)$ with $fd(g)+gd(f)$ and $d(f)$ with zero for $f\in R$. We define the universal derivation to be $d_{S/R}:S\to \Omega_{S/R}$ which takes $f\mapsto d(f)$. We call this module the module of Kähler differentials

To get a bit of a sense of this, we’ll note that if $R=k$ and $S=k[x_1,\ldots,x_n]$, so we’re just looking at affine space, then $\Omega_{k[x_1,\ldots,x_n]/k}$ is $\sum_{i=1}^n f_i dx_i$ where the $f_i$ are polynomials. For those who have done a bit of differential geometry, this should be looking familiar: it is an algebraic analogue of 1-forms.

Now, Kähler differentials have an extremely nice property: they commute with localization. That is, if $S$ is an $R$-algebra and $U\subset S$ is multiplicatively closed we have $\Omega_{U^{-1}S/R}=U^{-1}\Omega_{S/R}$. And by the way, I do apologize for the notation, using $S$ for an $R$-algebra is standard, as far as I know, and in the localization post I used it for a multiplicatively closed set.

So now we must come up with a sheaf theoretic version of this, because varieties are only locally equivalent to rings. Let $X$ be a topological space and $\mathscr{R},\mathscr{S}$ sheaves of rings, with $\mathscr{R}\to\mathscr{S}$ a homomorphism of sheaves of rings. This makes $\mathscr{S}$ into an $\mathscr{R}$-modules. We now define $\mathrm{pre}-\Omega_{\mathscr{S}/\mathscr{R}}$ to be the presheaf taking $U$ to $\Omega_{\mathscr{S}(U)/\mathscr{R}(U)}$. The restriction maps are given by taking $\mathscr{S}(U)\to\mathscr{S}(V)\to\mathrm{pre}-\Omega(V)$, which is then an $\mathscr{R}(U)$-derivation, and so factors through $\mathrm{pre}-\Omega(U)$, giving is the desired restriction maps. Now, once we’ve done all of this, we sheafifiy, and now we have a sheaf which we will call $\Omega_{\mathscr{S}/\mathscr{R}}$, the sheaf of relative differentials.

So now, we can specify to varieties. Let $f:X\to Y$ be a morphism of varieties. By definition we have a map $\mathscr{O}_Y\to f_*\mathscr{O}_X$. But we want the sheaf to be on $X$. So recall that when we talked about morphisms of sheaves, we briefly mentioned the inverse image sheaf. Using this, we get a morphism $f^{-1}\mathscr{O}_Y\to \mathscr{O}_X$, which makes $\mathscr{O}_X$ into a $f^{-1}\mathscr{O}_Y$-module. We define $\Omega_{X/Y}$ to be $\Omega_{\mathscr{O}_X/f^{-1}\mathscr{O}_Y}$, and call it the relative cotangent sheaf.

As a special case, we define $\Omega_X=\Omega_{X/\{pt\}}$ to be the cotangent sheaf of $X$, and as an immediate consequence of the above definition, we can see that if $X\to Y$ is a morphism of affine varieties, then $\Omega_{X/Y}$ is the sheaf associated to $\Omega_{k[X]/k[Y]}$, and this fact implies that $\Omega_{X/Y}$ is always a coherent sheaf.

So now, the cotangent sheaf gives us a new way to test for a point being nonsingular. If $X$ has dimension $r$, and $P\in X$, then $X$ is nonsingular at $P$ if and only if $\Omega_{X,P}$ is isomorphic to $\mathscr{O}_{X,P}^{\oplus r}$. This tells us that $X$ is nonsingular if and only if $\Omega_X$ is a locally free sheaf of rank $r$!

So this gives us a cotangent bundle! Now, we’re going to change notation slightly, and call this bundle (and the sheaf associated to it) $\Omega_X^1$. We can get the sheaf of $k$-forms by looking at $\Omega_X^k=\bigwedge^k\Omega_X^1$, where this is the exterior algebra construction. So in that case, we see that if $X$ has dimension $n$, then $\Omega^{n+1}_X$ is zero, so we only get $n$ of these sheaves. Aside from $\Omega_X^1$, there is another special one among these: $\Omega_X^n$. This one always turns out to be a line bundle, and we denote it by $\omega_X$, because it is so important. In fact, we call it the canonical bundle, and it will be showing up in the future regularly.

Before we stop for the day, we’re going to use $\Omega^1_X$ to define one more object that is very important. We define $\mathscr{H}om(\Omega_X^1,\mathscr{O}_X)={\Omega_X^1}^\vee$, the dual, to be the tangent sheaf $\mathscr{T}_X$. If $\Omega_X^1$ is a vector bundle, so is $\mathscr{T}_X$, and it is called the tangent bundle. We’ve now brought over a lot of valuable objects from differential geometry, and they’ll be quite important in our further study of varieties.