Last time, we talked about \mathrm{Spec}, which takes rings and gives us affine schemes in a manner analogous to affine varieties. Now, though it is safe to say that projective varieties are schemes (well, with the extra points taken into account) because they are covered by affine varieties (and so affine schemes), they’re actually MUCH nicer than arbitrary schemes. So, we’re going to discuss \mathrm{Proj} which takes graded rings to special types of schemes.

The analogies between projective varieties and \mathrm{Proj}(R) for a graded ring R are the same as those between affine varieties and \mathrm{Spec}(R). To start with, we take R=\oplus_{d\geq 0} R_d a graded ring. There is a distinguished ideal in R called the irrelevant ideal, it is given by R_+=\oplus_{d\geq 1} R_d. So we take the set \mathrm{Proj}(R) to be the set of homogeneous prime ideals not containing the irrelevant ideal. Now, why do we throw out this irrelevant ideal? Let’s look at the case of k[x_0,\ldots,x_n]. We want Proj of this to be projective space. The irrelevant ideal is then (x_0,\ldots,x_n), which requires that all the homogeneous coordinates vanish. However, this isn’t a point in projective space, so we want to ignore it in general. Additionally, the primes that are maximal among those not containing the irrelevant ideal will correspond to the points of a projective variety. In \mathbb{P}^n, we take the point (a_0:\ldots:a_n), then the ideal of the point is given by the linear equations of the form a_ix_j-a_jx_i, and is maximal among this set.

So we have our point set. We’ll get our closed sets in the same way as before: let I be a homogeneous ideal, then V(I)=\{P\in \mathrm{Proj}(R):P\supset I\} is a closed set. This gives us a topology on \mathrm{Proj}(R). All that remains is to define the structure sheaf. We define the distinguished open sets of \mathrm{Proj}(R) as follows: given f a homogeneous element of R of degree one, we define the distinguished open given by f to be \mathrm{Proj}(R)-V(f). The distinguished opens cover the space, and so we can define a sheaf by giving it on these open sets. To the distinguished open U_f determined by f, we assign the ring (R_f)_0, that is, the degree 0 elements of R_f. Similarly to the affine case, this procedure will give for projective varieties the original structure sheaf.

So now we’ve defined Proj. We step back and work with both affine and projective schemes for a moment. Let R be a ring. Then we make the convention that \mathrm{Spec}(R[x_1,\ldots,x_n]) is called \mathbb{A}_R^n and that \mathrm{Proj}(R[x_0,\ldots,x_n]) is called \mathbb{P}_R^n. These two schemes are related just as we would expect from varieties: \mathbb{P}^n_R can be thought of as \mathbb{A}^n_R\cup \mathbb{P}_R^{n-1}.

We’ve spoken about subvarieties, so now what about subschemes? We’ll need a subset which has, in the subspace topology, a sheaf, and we’ll even want it to relate to the original one in some nice manner. Let X and Y be schemes. We define a morphism Y\to X of schemes to be a morphism of locally ringed spaces. Now, an open subscheme of X will just be an open subset U\subset X whose structure sheaf is the restriction of that of X. So then an open immersion will just be a morphism f:Y\to X such that it induces an isomorphism of Y with an open subscheme of X.

More interesting is the case of closed subschemes. Here, we must first define a closed immersion, which is a map f:Y\to X such that it induces a homeomorphism of Y onto a closed subset of X and such that the induced map on sheaves is surjective. Then a closed subscheme is an equivalence class of closed immersions, where f:Y\to X and f':Y'\to X are equivalent if we have i:Y'\to Y an isomorphism such that f'=f\circ i. Now this sounds complicated, but really this just says that we’re identifying the subset up to automorphism. After all, what’s the difference between parameterizing the x-axis from left to right and from right to left? We’ll just treat them as the same. As for closed immersion, it’s saying that a closed immersion is given by a sheaf of ideals, so this really is a good definition of closed subscheme to have generalizing the notion of a subvariety.

This, in fact, tells us that the closed subschemes of \mathrm{Spec}(A) are of the form \mathrm{Spec}(A/I) for an ideal I, and that the same holds for \mathrm{Proj}(A) with the caveat that I must be homogeneous.

Now, we’ll talk about the properties that let us distinguish our abstract varieties from arbitrary schemes. I’ll start out by just saying what a variety is, and then we’ll go for each word individually. A variety is an integral separated scheme of finite type over an algebraically closed field k. The first word is easy: an integral scheme is any scheme whose structure sheaf consists only of integral domains. This has the nice property that a scheme is integral if and only if it is irreducible as a topological space and all the rings have no nilpotents.

Now we’re going to skip to the “over k” part. Let S be a scheme. Then we say that X\to S is a scheme over S, that is, a scheme with a map to S. The ability to do things like this is extremely important, and we call S the base that we are working over. Many powerful theorems are proved using a method called base change. The idea there is that we have X\to S and S'\to S and we’d like to work over S' instead. So we take the fiber product X\times_S S' and call it X', and we have a map X'\to S'. Then the problem is reduced to checking the property for X'\to S' and checking that it is preserved by base change. Now, to make sense of the notation, whenever S happens to be of the form \mathrm{Spec}(A), we often say that X is a scheme over A as a convention. So this just means that a variety comes with a map to \mathrm{Spec}(k), that is, a point with structure sheaf k. The map to the point is easy, and the last part says that the structure sheaf consists of k-algebras, and that’s what we’d prefer to have.

Next up is separated. This will be quick, because we’ve discussed it before as the difference between prevarieties and varieties (this is why people used to call arbitrary schemes preschemes: they kept the word scheme for the separated ones). Now we can actually say what it means for a morphism to be separated. A morphism f:X\to Y is separated if the diagonal morphism \Delta:X\to X\times_Y X, which is the unique morphism whose composition with both projections is the identity map on X, is a closed immersion. So what we really mean above is that X is separated over k, that is, the morphism X\to \mathrm{Spec}(k) is separated, which means that X\to X\times_k X is a closed immersion. The truth of the matter is, we’ve been secretly taking fiber products over k the whole time, because we have a unique map there for varieties, so this is precisely the condition we started with.

Now the tricky one: of finite type. This is again a property of the morphism X\to \mathrm{Spec}(k). We define a morphism of schemes f:X\to Y to be locally of finite type if there is a covering of Y by open affine subsets \mathrm{Spec}(B_i) whose preimages are covered by open affines \mathrm{Spec}(A_{ij}) where A_{ij} is a finitely presented B_i-algebra. The morphism if of finite type if we can do this with finitely many A_{ij}‘s for each B_i. So this says that the preimage of this structure map, which is the whole variety, can be covered by finitely many affine vareities.

Taken all together, these conditions say the following: of finite type over k tells us that our space is covered by finitely many affine varieties, then the integral condition requires that there be no nilpotent functions and that it be irreducible, and finally separatedness moves our prevariety into the realm of being an actual variety, so we have our old class singled out in this way from the more general collection of schemes. We even can get back down to projective varieties by requiring that we look at the Proj of a graded ring on top of things, so inside of the big category of schemes we can find the varieties that we know and love.

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