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 , an -algebra (that is, a ring which is also an -module) and an -module , we define an -derivation to be a function such that for all , , and for all . (We say that an element is in if it is of the form for 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 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 -module called and a derivation such that any other derivation is given by composing with a module homomorphism . Even better, we can describe it.

We take the collection of symbols for and take formal finite sums of the form where . These are the elements, but we do identify some of them. Specifically, we identify with , with and with zero for . We define the *universal derivation* to be which takes . We call this module the module of Kähler differentials

To get a bit of a sense of this, we’ll note that if and , so we’re just looking at affine space, then is where the 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 is an -algebra and is multiplicatively closed we have . And by the way, I do apologize for the notation, using for an -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 be a topological space and sheaves of rings, with a homomorphism of sheaves of rings. This makes into an -modules. We now define to be the presheaf taking to . The restriction maps are given by taking , which is then an -derivation, and so factors through , 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 , the sheaf of relative differentials.

So now, we can specify to varieties. Let be a morphism of varieties. By definition we have a map . But we want the sheaf to be on . So recall that when we talked about morphisms of sheaves, we briefly mentioned the inverse image sheaf. Using this, we get a morphism , which makes into a -module. We define to be , and call it the relative cotangent sheaf.

As a special case, we define to be the cotangent sheaf of , and as an immediate consequence of the above definition, we can see that if is a morphism of affine varieties, then is the sheaf associated to , and this fact implies that is always a coherent sheaf.

So now, the cotangent sheaf gives us a new way to test for a point being nonsingular. If has dimension , and , then is nonsingular at if and only if is isomorphic to . This tells us that is nonsingular if and only if is a locally free sheaf of rank !

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) . We can get the sheaf of -forms by looking at , where this is the exterior algebra construction. So in that case, we see that if has dimension , then is zero, so we only get of these sheaves. Aside from , there is another special one among these: . This one always turns out to be a line bundle, and we denote it by , 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 to define one more object that is very important. We define , the dual, to be the tangent sheaf . If is a vector bundle, so is , 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.

April 16, 2008 at 12:01 pm

[...] 16, 2008 Some people might say that the natural place for this topic is before talk of differential forms and of the canonical bundle, but I disagree. Well, really it’s fine either way, but this is my blog, so I’m going [...]

July 4, 2008 at 11:51 am

[...] that for a projective smooth variety and an ample line bundle then for we have , where is the canonical bundle. Now a theorem of Grauert and Riemenschneider says that for an appropriately defined , we have the [...]

July 23, 2008 at 7:58 pm

[...] working with divisors and line bundles. Specifically, we’ll be needing linear systems and the canonical divisor. We’ll go through the proof, because I feel like it, but it isn’t really all that [...]

December 5, 2009 at 9:14 pm

Hi,

(Omega_{X})^1 is confusing me, perhaps it is so many things at the same time.

Is (Omega_{X})^1 ismorphic to the structure sheaf O_{X}. What map maps one to the other? Thank you.

December 5, 2009 at 10:25 pm

No, is not isomorphic to the structure sheaf, except in the case of elliptic curves. The structure sheaf is the trivial line bundle, whereas in the smooth variety case, is the cotangent bundle, which is rarely trivial, and of rank equal to the dimension of your base.

December 5, 2009 at 10:35 pm

I thank you for the quick response.

I think I am understanding what you have just said. In the case of elliptic

curves, how is the correspondence? How do we establish an isomorphism? I

hope I am not boring anybody with my unrigorous trivialities:)

Regards

December 5, 2009 at 11:23 pm

Well, first off, you’ll need to be on a smooth curve so that is a line bundle for there to be any hope. So then, it’s just a matter of determining which curves, if any, can have trivial cotangent bundle. For any curve, this bundle has degree , and so the genus must be 1. Now, among degree zero line bundles, there’s a unique one with a global section, so we just need to show that there’s a global section of the cotangent bundle, which is just a 1-form. But any genus 1 curve is a quotient of $\mathbb{C}$ by a lattice of rank 2. The global 1-form then descends to the quotient, and so has a global section, and is trivial. This is the proof over the complexes, but the result still works in general.

December 6, 2009 at 3:16 pm

Thi helps Charles, thank you. Simple question such as this one expose my lack of understanding. I think I need to review the the whole ‘section’, no pun intended. Thanks again

June 11, 2010 at 9:17 pm

is there a relation between the cotangent bundle in differential geometry and in algebraic geometry, and is there a relation between differential forms in differential geometry and the stuff here?

June 11, 2010 at 9:25 pm

In a word: yes. If you fix your base field to be $\mathbb{C}$, and only work on complete smooth varieties, then the global algebraic differential forms are the same thing as the global holomorphic differential forms. This is generally true, so in particular, the bundle associated to the sheaf $\Omega_X^1$ is dual to the holomorphic tangent bundle, etc. I recommend that you check out this wikipedia page for a bit more on the GAGA philosophy (and theorems, though I’m not necessarily using anything as high powered as Serre’s GAGA paper) http://en.wikipedia.org/wiki/Algebraic_geometry_and_analytic_geometry

October 13, 2010 at 12:09 am

Hi!

I’m confused about the tangent bundle and the tangent sheaf.

Let $X$ be a variety over $k$. On one hand, we have the definition of the tangent bundle $\hom(Spec k[\epsilon]/(\epsilon^2), X)$. On the other hand, we have the tangent sheaf, which is $\hom_{O_x}(\Omega_{X/k}, O_x)$.

Could you please explain the relationship between the two?

Thanks so much!

Brian

October 13, 2010 at 7:08 am

Well, you’ll want to focus on the case where is smooth so that it’s actually a bundle. Then, well, use the bundle-sheaf correspondence. Take an open subset of $X$ and show that a section of the tangent sheaf over $U$ is the same as a map from the open set to the geometric tangent bundle which is a section of the projection.