1. Going Projective

There are two ways in which the study of graded rings arises naturally out of the study of affine schemes. Let R be a quotient of the polynomial algebra as considered earlier. Let R_n be a filtration (increasing or decreasing) on the ring R such that R_n ^. R_m R_n+m. We obtain a graded ring = R_n.

One way to obtain an increasing filtration on a quotient of a polynomial ring is to take R_n to the subgroup (or k vector subspace) generated by polynomials of degree at most n. This process is called homogenisation and the associated ring is denoted by R^h.

A way to obtain an increasing filtration is to take R_n = Iⁿ where I is some ideal in R. The resulting graded ring _I = Iⁿ is called (for reasons that will become clear later) the “blow-up” of Spec(R) along Spec(R/I) or of R along I for short.

In each case the study of the graded ring allows us to formalise (and unify) arguments by induction. However, this is not just an algebraic trick--we will see below that the constructions have a nice geometric interpretation as well.

We note that in both cases the rings obtained are quotients of the appropriate polynomial rings. The ideal generated by all the graded elements of positive degree is called the irrelevant ideal Irr(R) of such a ring. We later encounter another ideal that has “equal right” to being called the irrelevant ideal. Perhaps this should be called the “big” irrelevant ideal.

1.1. Projective Schemes. If R is a graded finitely generated ring, then consider a morphism f : R A such that image of the irrelevant ideal generates the unit ideal in A. If A^× is a unit in A then we can modify f by defining, for each homogeneous element x of R, ( ^. f)(x) = ^deg(x)f(x). Then ( ^. f) : R A is again a homomorphism. If we write R as a quotient of the polynomial ring k[X₁,...,X_n,Y ₀,...,Y _m], with deg(X_i) = 0 and deg(Y _i) > 0, by a homogeneous ideal I, then such a homomorphism amounts to choosing elements a₁,...a_n and b₀, ...b_m in A such that they satisfy the polynomials in I and (b₀,...,b_m) is the unit ideal in A. The action of units is given by replacing b_i by ^{deg(Y _i)}b_i; since the equations are generated by homogeneous equations, this again gives a solution. An important example is the case where there are no X’s and no equations. The scheme thus obtained is called the projective space of dimension m and denoted ^m. More generally, we can see that all our schemes are subschemes (in fact closed subschemes) of ⁿ × ^m.

1.2. Segre embedding. Consider the product ^p × ^q. As a functor of points it assigns to each finite ring A the set of pairs of tuples of the form ((a₀,...,a_p),(b₀,...,b_q)) where the a_i’s and b_j’s lie in A and the a_i’s generate the unit ideal as do the b_j’s. It follows that if we define c_ij = a_ib_j, then we obtain a (p + 1)(q + 1)-tuple of elements of A which also generate the unit ideal. Moreover, this tuple satisfies the collection of equations of the form Z_ijZ_kl = Z_ilZ_kj as i,k run from 0 to p and j,l run from 0 to q. Conversely, let c_ij be a (p + 1)(q + 1)-tuple that generates the unit ideal in a finite ring A and satisfies the above equations. We can check that each such tuple is a “product” of a pair of tuples as above.

Generalising this we see that if Proj(R) and Proj(S) are two projective schemes then we can give a natural scheme structure to Proj(R) × Proj(S).

1.3. Morphisms of graded rings. If R S is a graded homomorphism of graded rings, then it need not give a natural trasformation Proj(S) Proj(R). If S A is a homomorphism such that the image of the irrelevant ideal Irr(S) generates the unit ideal in A, then it need not follow that the same is true for the image of Irr(R) under the composite homomorphism. Suppose that S is finite as an R module. The each of the generators of Irr(S) satisfies a monic polynomial with coefficients in R. Taking homogeneous components everywhere we can assume that the polynomial has the form

where the degree of r_k is d(n - k) where d is the degree of the generator under consideration. In particular, it follows that the degree of r_k is positive. Thus, some power of each generator of Irr(S) lies in the ideal Irr(R)S; hence some power of the ideal Irr(S) lies in the ideal Irr(R)S. As a consequence, if S A is a homomorphism such that the image of Irr(S) generates the unit ideal then so does the image of Irr(R) under the composite homomorphism.

We have thus shown that a graded homomorphism R S of graded rings gives a morphism Proj(S) Proj(R) if the homomorphism is finite.

Conversely, suppose that R S is a graded homomorphism of graded rings such that R₀ S₀ is finite and for which there is a positive integer N such that Irr(S)^N is contained in Irr(R)S. There is a a finite collection w_d1, ..., w_dkd of homogeneous elements of degree d in S which generate all homogeneous elements of degree d in S over S₀. Expanding this collection as necessary and using the finite-ness of S₀ over R₀, we can assume that the same generators work over R₀ as well. Let e₁, ..., e_r denote the union of such collections for all degrees less than or equal to N. Any homogeneous element a of S of degree greater than N can be written as a linear combination of elements of Irr(R) with coefficients from S; such coefficents must have degree smaller than the degree of a since elements of Irr(R) have positive degree. By induction, we can thus write any element of S as a linear combination over R of the e₁,..., e_r. Hence, S is finite as an R module.

Thus, in order that a graded homomorphism R S be finite it is necessary and sufficient that R₀ S₀ is finite and that the image of the irrelevant ideal of R generates an ideal in S that contains a power of the irrelevant ideal of S.

1.4. Homogeneous primary decomposition. Let R be a finitely generated graded ring and Q a primary ideal in R that is not necessarily homogeneous. Let Q' be the sub-ideal of Q generated by its homogeneous elements. We assert that Q' is primary as well. To prove this we can go modulo Q'; in this case we are assuming that Q is a primary ideal such that it has no non-zero homogeneous elements. We must then show that if fg = 0 and g is not 0, then f is nilpotent. Now if f and g are homogeneous and g is not zero then g does not lie in Q either; by the primariness of Q, if follows that fⁿ must lie in it for some n and hence fⁿ = 0. In the general case, we write f = f₀ + + f_s-1 + f_s + ... and g = g_r + ... where f_i and g_i are homogeneous of degree i. Further suppose that s and r are chosen so that f_i are nilpotent for i < s and g_r is the lowest degree term of g which is non-zero. Let N be chosen so that (f₀ + + f_s-1)^N = 0 and consider the binomial expansion of (f - (f₀ + + f_s-1))^Ng. The only term that does not contain fg as a factor is (f₀ + + f_-s)^Ng. Thus (f_s + ...)^Ng = 0; in particular, the lowest degree term f_s^Ng_r = 0. Now applying the homogeneous case proved above we see that f_s is nilpotent as well. By induction we thus see that f is nilpotent as required.

Now any homogeneous ideal I in R has a primary decomposition I = Q_i. Since I is homogeneous we see that I = Q_i^h as well. So that we have a homogeneous primary decomposition of I. We shall only deal with such primary decompositions below. We note in passing that the radical of a homogeneous ideal is also homogeneous and so the associated and minimal primes for a homogeneous ideal are also homogeneous.

1.5. The “small” irrelevant ideal. Let R be a finitely generated graded ring. Let N_k = Ann(Irr(R)^k)). This is an increasing collection of homogeneous ideals for k sufficiently large; by the Noetherian-ness condition, this sequence must terminate. Let N_R = N_k for k sufficiently large. Under any morphism R A where the image of the irrelevant ideal generates the unit ideal, the annihilator of any power of this ideal must go to zero; N_R must go to 0 under such a homomorphism. Thus it is truly “irrelevant”! To summarise the “big” irrelevant ideal of R is irrelevant since under all homomorphisms under consideration its image generates the full ring, while the “small” irrelevant ideal of R is irrelevant since its image under all homomorphisms under considertation is 0. Let (0) = _iQ_i be a primary decomposition of the zero ideal in R. From this collection we drop all those primary ideals that contain a power of Irr(R) (note that it is enough that for each generator of Irr(R) there is some power which lies in an ideal for that ideal to contain some power of Irr(R)). Let N' be the intersection of the remaining ideals if that set is non-empty and N' = R otherwise. It is clear that N' N_R since N'Irr(R)^l _iQ_i = (0) for a suitable l. Conversely, let Q_i be a primary ideal in the decomposition above that does not contain any power Irr(R)^l, then there is an i such that Y _i^l does not lie in Q_i for any l. But Y _i^lN_R = 0 for l sufficiently large, so N_R Q_i. This proves that N_R N' as well.

We note that N_R = R if and only if Irr(R) is nilpotent which happens if and only if R is a finite R₀ module. Moreover, these conditions imply that Proj(R)(A) is empty for all A. One we prove Hilbert’s Nullstellensatz we will show that the converse is true as well.

1.6. Homogenisation. Let R be a finitely generated ring which we think of as a quotient of the polynomial ring with its induced filtration by degree. The associated graded ring R^h can be thought of as the subring of R[T] whose homogeneous elements of degree n have the form aTⁿ, where the degree of a is at most n. The natural homomorphism R[T] R which sends T to 1 induces a homomorphism R^h R as well; this is the dehomogenisation homomorphism. If I is an ideal in R, then its homogenisation is the ideal J = I^h in that consists of elements aTⁿ with a in I. If bTⁿ is an element of such that bTⁿT^m lies in J then clearly bTⁿ itself lies in J; in other words (J : T^m) = J for all m. Conversely, given a homogeneous ideal J in R^h we can take its dehomogenisation I and consider I^h. If bTⁿ is an element of J, then b is an element of R and so bTⁿ lies in I^h; thus J is contained in I^h. In fact (J : T^m) I^h for all m and I^h is the stable union of the ideals (J : T^m); the increasing sequence (J : T^m) stops after a finite stage by the Noetherian-ness of R^h.

From this description it is clear that the homogenisation of the primary decomposition of an ideal is a primary decomposition of the homogenisation.

Let f : R^h A be a morphism which maps the irrelevant ideal to the unit ideal. T is an element of degree 1 in the irrelevant ideal. If T is itself mapped to a unit in A^×, then we can take the equivalent homomorphism ^{-1 .} f which maps T to 1 and thus factors through R. In other words, the elements of Proj(R)(A) that are represented by homomorphisms R^h A that map T to a unit correspond naturally to elements of Spec(R)(A). Thus Spec(R) can be identifed as a subscheme of Proj(R).

At the other extreme we can consider a homomorphism f : R^h A that sends T to 0 and yet maps the irrelevant ideal to the unit ideal. The multiples of T in R^h have the form aTⁿ where the degree of a is strictly smaller than n (so that aTⁿ = T(aT^n-1)). All such elements must also go to 0. It follows that f actually gives a homomorphism (R_n/R_n-1) A. When R is considered as a quotient of a polynomial ring by an ideal I and R_n is the induced filtration, this amounts to considering the quotient of the same polynomial ring by the ideal generated by the leading homogeneous components of elements of I. These leading homogeneous components define the asymptotes at infinity of Spec(R). Thus, we can think of Proj(R) as the union of Spec(R) with its asymptotes at infinity.

1.7. Blow-up. Let R be a finitely generated ring and I an ideal. Let us write R as a quotient of the polynomial ring k[X₁,...,X_n,] where the X_i generate R as a k-algebra and X_k, ..., X_n generate the ideal I. The graded ring _I = Iⁿ (where I⁰ = R) can then be naturally thought of as a quotient of the graded polynomial ring k[X₁,...,X_n,Y _k,...,Y _n] where deg(X_i) = 0 and deg(Y _i) = 1. Moreover, we have the obvious relations X_iY _j = X_jY _i. Let K be the ideal generated by these relations and B be Proj(k[X₁,...,X_n,Y _k,...,Y _k]/K); the blow-up scheme Proj(_I) is a closed subscheme of B. Note that B is itself the blow-up of the polynomial ring with respect to the ideal generated by X_k, ..., X_n. An element B(A) is given by taking elements a₁,..., a_n and b_k, ...b_n of A such that (b_k,...,b_n) generate the unit ideal in A and a_ib_j = a_jb_i; in other words (a_k,...,a_n) and (b_k,...,b_n) are proportional. When A has only one maximal ideal this amounts to saying that (a_k,...,a_n) is a multiple of (b_k,...,b_n). Now, if (a_k,...,a_n) generate the unit ideal then multiplying factor is a unit in A and so unto equivalence we can determine the b’s uniquely. In other words, each element of B(A) gives an element of ⁿ and is point is “outside the linear subspace defined by X_k = = X_n = 0” then the element of B(A) is uniquely determined. Here we interpret the complement of the vanishing of the ideal as the locus where the image of the ideal generates the unit ideal; this is not the set-theoretic complement. Similarly, the blow-up Proj(_I) of R along I has a natural map to Spec(R) which is a bijection on the locus of maps R A under which the image of I generates the unit ideal.

The points of Proj(_I) that map to points of Spec(R/I) Spec(R) correspond to the asymptotes of Spec(R) along Spec(R/I). In terms of rings we see that these are points of Proj((Iⁿ/I^n-1)).