Definition 10.57.1. Let S be a graded ring. We define \text{Proj}(S) to be the set of homogeneous prime ideals \mathfrak p of S such that S_{+} \not\subset \mathfrak p. The set \text{Proj}(S) is a subset of \mathop{\mathrm{Spec}}(S) and we endow it with the induced topology. The topological space \text{Proj}(S) is called the homogeneous spectrum of the graded ring S.
10.57 Proj of a graded ring
Let S be a graded ring. A homogeneous ideal is simply an ideal I \subset S which is also a graded submodule of S. Equivalently, it is an ideal generated by homogeneous elements. Equivalently, if f \in I and
is the decomposition of f into homogeneous parts in S then f_ i \in I for each i. To check that a homogeneous ideal \mathfrak p is prime it suffices to check that if ab \in \mathfrak p with a, b homogeneous then either a \in \mathfrak p or b \in \mathfrak p.
Note that by construction there is a continuous map
Let S = \oplus _{d \geq 0} S_ d be a graded ring. Let f\in S_ d and assume that d \geq 1. We define S_{(f)} to be the subring of S_ f consisting of elements of the form r/f^ n with r homogeneous and \deg (r) = nd. If M is a graded S-module, then we define the S_{(f)}-module M_{(f)} as the sub module of M_ f consisting of elements of the form x/f^ n with x homogeneous of degree nd.
Lemma 10.57.2. Let S be a \mathbf{Z}-graded ring containing a homogeneous invertible element of positive degree. Then the set G \subset \mathop{\mathrm{Spec}}(S) of \mathbf{Z}-graded primes of S (with induced topology) maps homeomorphically to \mathop{\mathrm{Spec}}(S_0).
Proof. First we show that the map is a bijection by constructing an inverse. Let f \in S_ d, d > 0 be invertible in S. If \mathfrak p_0 is a prime of S_0, then \mathfrak p_0S is a \mathbf{Z}-graded ideal of S such that \mathfrak p_0S \cap S_0 = \mathfrak p_0. And if ab \in \mathfrak p_0S with a, b homogeneous, then a^ db^ d/f^{\deg (a) + \deg (b)} \in \mathfrak p_0. Thus either a^ d/f^{\deg (a)} \in \mathfrak p_0 or b^ d/f^{\deg (b)} \in \mathfrak p_0, in other words either a^ d \in \mathfrak p_0S or b^ d \in \mathfrak p_0S. It follows that \sqrt{\mathfrak p_0S} is a \mathbf{Z}-graded prime ideal of S whose intersection with S_0 is \mathfrak p_0.
To show that the map is a homeomorphism we show that the image of G \cap D(g) is open. If g = \sum g_ i with g_ i \in S_ i, then by the above G \cap D(g) maps onto the set \bigcup D(g_ i^ d/f^ i) which is open. \square
For f \in S homogeneous of degree > 0 we define
Finally, for a homogeneous ideal I \subset S we define
We will use more generally the notation V_{+}(E) for any set E of homogeneous elements E \subset S.
Lemma 10.57.3 (Topology on Proj). Let S = \oplus _{d \geq 0} S_ d be a graded ring.
The sets D_{+}(f) are open in \text{Proj}(S).
We have D_{+}(ff') = D_{+}(f) \cap D_{+}(f').
Let g = g_0 + \ldots + g_ m be an element of S with g_ i \in S_ i. Then
D(g) \cap \text{Proj}(S) = (D(g_0) \cap \text{Proj}(S)) \cup \bigcup \nolimits _{i \geq 1} D_{+}(g_ i).Let g_0\in S_0 be a homogeneous element of degree 0. Then
D(g_0) \cap \text{Proj}(S) = \bigcup \nolimits _{f \in S_ d, \ d\geq 1} D_{+}(g_0 f).The open sets D_{+}(f) form a basis for the topology of \text{Proj}(S).
Let f \in S be homogeneous of positive degree. The ring S_ f has a natural \mathbf{Z}-grading. The ring maps S \to S_ f \leftarrow S_{(f)} induce homeomorphisms
D_{+}(f) \leftarrow \{ \mathbf{Z}\text{-graded primes of }S_ f\} \to \mathop{\mathrm{Spec}}(S_{(f)}).There exists an S such that \text{Proj}(S) is not quasi-compact.
The sets V_{+}(I) are closed.
Any closed subset T \subset \text{Proj}(S) is of the form V_{+}(I) for some homogeneous ideal I \subset S.
For any graded ideal I \subset S we have V_{+}(I) = \emptyset if and only if S_{+} \subset \sqrt{I}.
Proof. Since D_{+}(f) = \text{Proj}(S) \cap D(f), these sets are open. This proves (1). Also (2) follows as D(ff') = D(f) \cap D(f'). Similarly the sets V_{+}(I) = \text{Proj}(S) \cap V(I) are closed. This proves (8).
Suppose that T \subset \text{Proj}(S) is closed. Then we can write T = \text{Proj}(S) \cap V(J) for some ideal J \subset S. By definition of a homogeneous ideal if g \in J, g = g_0 + \ldots + g_ m with g_ d \in S_ d then g_ d \in \mathfrak p for all \mathfrak p \in T. Thus, letting I \subset S be the ideal generated by the homogeneous parts of the elements of J we have T = V_{+}(I). This proves (9).
The formula for \text{Proj}(S) \cap D(g), with g \in S is direct from the definitions. This proves (3). Consider the formula for \text{Proj}(S) \cap D(g_0). The inclusion of the right hand side in the left hand side is obvious. For the other inclusion, suppose g_0 \not\in \mathfrak p with \mathfrak p \in \text{Proj}(S). If all g_0f \in \mathfrak p for all homogeneous f of positive degree, then we see that S_{+} \subset \mathfrak p which is a contradiction. This gives the other inclusion. This proves (4).
The collection of opens D(g) \cap \text{Proj}(S) forms a basis for the topology since the standard opens D(g) \subset \mathop{\mathrm{Spec}}(S) form a basis for the topology on \mathop{\mathrm{Spec}}(S). By the formulas above we can express D(g) \cap \text{Proj}(S) as a union of opens D_{+}(f). Hence the collection of opens D_{+}(f) forms a basis for the topology also. This proves (5).
Proof of (6). First we note that D_{+}(f) may be identified with a subset (with induced topology) of D(f) = \mathop{\mathrm{Spec}}(S_ f) via Lemma 10.17.6. Note that the ring S_ f has a \mathbf{Z}-grading. The homogeneous elements are of the form r/f^ n with r \in S homogeneous and have degree \deg (r/f^ n) = \deg (r) - n\deg (f). The subset D_{+}(f) corresponds exactly to those prime ideals \mathfrak p \subset S_ f which are \mathbf{Z}-graded ideals (i.e., generated by homogeneous elements). Hence we have to show that the set of \mathbf{Z}-graded prime ideals of S_ f maps homeomorphically to \mathop{\mathrm{Spec}}(S_{(f)}). This follows from Lemma 10.57.2.
Let S = \mathbf{Z}[X_1, X_2, X_3, \ldots ] with grading such that each X_ i has degree 1. Then it is easy to see that
does not have a finite refinement. This proves (7).
Let I \subset S be a graded ideal. If \sqrt{I} \supset S_{+} then V_{+}(I) = \emptyset since every prime \mathfrak p \in \text{Proj}(S) does not contain S_{+} by definition. Conversely, suppose that S_{+} \not\subset \sqrt{I}. Then we can find an element f \in S_{+} such that f is not nilpotent modulo I. Clearly this means that one of the homogeneous parts of f is not nilpotent modulo I, in other words we may (and do) assume that f is homogeneous. This implies that I S_ f \not= S_ f, in other words that (S/I)_ f is not zero. Hence (S/I)_{(f)} \not= 0 since it is a ring which maps into (S/I)_ f. Pick a prime \mathfrak q \subset (S/I)_{(f)}. This corresponds to a graded prime of S/I, not containing the irrelevant ideal (S/I)_{+}. And this in turn corresponds to a graded prime ideal \mathfrak p of S, containing I but not containing S_{+} as desired. This proves (10) and finishes the proof. \square
Example 10.57.4. Let R be a ring. If S = R[X] with \deg (X) = 1, then the natural map \text{Proj}(S) \to \mathop{\mathrm{Spec}}(R) is a bijection and in fact a homeomorphism. Namely, suppose \mathfrak p \in \text{Proj}(S). Since S_{+} \not\subset \mathfrak p we see that X \not\in \mathfrak p. Thus if aX^ n \in \mathfrak p with a \in R and n > 0, then a \in \mathfrak p. It follows that \mathfrak p = \mathfrak p_0S with \mathfrak p_0 = \mathfrak p \cap R.
If \mathfrak p \in \text{Proj}(S), then we define S_{(\mathfrak p)} to be the ring whose elements are fractions r/f where r, f \in S are homogeneous elements of the same degree such that f \not\in \mathfrak p. As usual we say r/f = r'/f' if and only if there exists some f'' \in S homogeneous, f'' \not\in \mathfrak p such that f''(rf' - r'f) = 0. Given a graded S-module M we let M_{(\mathfrak p)} be the S_{(\mathfrak p)}-module whose elements are fractions x/f with x \in M and f \in S homogeneous of the same degree such that f \not\in \mathfrak p. We say x/f = x'/f' if and only if there exists some f'' \in S homogeneous, f'' \not\in \mathfrak p such that f''(xf' - x'f) = 0.
Lemma 10.57.5. Let S be a graded ring. Let M be a graded S-module. Let \mathfrak p be an element of \text{Proj}(S). Let f \in S be a homogeneous element of positive degree such that f \not\in \mathfrak p, i.e., \mathfrak p \in D_{+}(f). Let \mathfrak p' \subset S_{(f)} be the element of \mathop{\mathrm{Spec}}(S_{(f)}) corresponding to \mathfrak p as in Lemma 10.57.3. Then S_{(\mathfrak p)} = (S_{(f)})_{\mathfrak p'} and compatibly M_{(\mathfrak p)} = (M_{(f)})_{\mathfrak p'}.
Proof. We define a map \psi : M_{(\mathfrak p)} \to (M_{(f)})_{\mathfrak p'}. Let x/g \in M_{(\mathfrak p)}. We set
This makes sense since \deg (x) = \deg (g) and since g^{\deg (f)}/f^{\deg (g)} \not\in \mathfrak p'. We omit the verification that \psi is well defined, a module map and an isomorphism. Hint: the inverse sends (x/f^ n)/(g/f^ m) to (xf^ m)/(g f^ n). \square
Here is a graded variant of Lemma 10.15.2.
Lemma 10.57.6. Suppose S is a graded ring, \mathfrak p_ i, i = 1, \ldots , r homogeneous prime ideals and I \subset S_{+} a graded ideal. Assume I \not\subset \mathfrak p_ i for all i. Then there exists a homogeneous element x\in I of positive degree such that x\not\in \mathfrak p_ i for all i.
Proof. We may assume there are no inclusions among the \mathfrak p_ i. The result is true for r = 1. Suppose the result holds for r - 1. Pick x \in I homogeneous of positive degree such that x \not\in \mathfrak p_ i for all i = 1, \ldots , r - 1. If x \not\in \mathfrak p_ r we are done. So assume x \in \mathfrak p_ r. If I \mathfrak p_1 \ldots \mathfrak p_{r-1} \subset \mathfrak p_ r then I \subset \mathfrak p_ r a contradiction. Pick y \in I\mathfrak p_1 \ldots \mathfrak p_{r-1} homogeneous and y \not\in \mathfrak p_ r. Then x^{\deg (y)} + y^{\deg (x)} works. \square
Lemma 10.57.7. Let S be a graded ring. Let \mathfrak p \subset S be a prime. Let \mathfrak q be the homogeneous ideal of S generated by the homogeneous elements of \mathfrak p. Then \mathfrak q is a prime ideal of S.
Proof. Suppose f, g \in S are such that fg \in \mathfrak q. Let f_ d (resp. g_ e) be the homogeneous part of f (resp. g) of degree d (resp. e). Assume d, e are maxima such that f_ d \not= 0 and g_ e \not= 0. By assumption we can write fg = \sum a_ i f_ i with f_ i \in \mathfrak p homogeneous. Say \deg (f_ i) = d_ i. Then f_ d g_ e = \sum a_ i' f_ i with a_ i' to homogeneous par of degree d + e - d_ i of a_ i (or 0 if d + e -d_ i < 0). Hence f_ d \in \mathfrak p or g_ e \in \mathfrak p. Hence f_ d \in \mathfrak q or g_ e \in \mathfrak q. In the first case replace f by f - f_ d, in the second case replace g by g - g_ e. Then still fg \in \mathfrak q but the discrete invariant d + e has been decreased. Thus we may continue in this fashion until either f or g is zero. This clearly shows that fg \in \mathfrak q implies either f \in \mathfrak q or g \in \mathfrak q as desired. \square
Lemma 10.57.8. Let S be a graded ring.
Any minimal prime of S is a homogeneous ideal of S.
Given a homogeneous ideal I \subset S any minimal prime over I is homogeneous.
Proof. The first assertion holds because the prime \mathfrak q constructed in Lemma 10.57.7 satisfies \mathfrak q \subset \mathfrak p. The second because we may consider S/I and apply the first part. \square
Lemma 10.57.9. Let R be a ring. Let S be a graded R-algebra. Let f \in S_{+} be homogeneous. Assume that S is of finite type over R. Then
the ring S_{(f)} is of finite type over R, and
for any finite graded S-module M the module M_{(f)} is a finite S_{(f)}-module.
Proof. Choose f_1, \ldots , f_ n \in S which generate S as an R-algebra. We may assume that each f_ i is homogeneous (by decomposing each f_ i into its homogeneous components). An element of S_{(f)} is a sum of the form
with \lambda _{e_1 \ldots e_ n} \in R. Thus S_{(f)} is generated as an R-algebra by the f_1^{e_1} \ldots f_ n^{e_ n} /f^ e with the property that e\deg (f) = \sum e_ i\deg (f_ i). If e_ i \geq \deg (f) then we can write this as
Thus we only need the elements f_ i^{\deg (f)}/f^{\deg (f_ i)} as well as the elements f_1^{e_1} \ldots f_ n^{e_ n} /f^ e with e \deg (f) = \sum e_ i \deg (f_ i) and e_ i < \deg (f). This is a finite list and we see that (1) is true.
To see (2) suppose that M is generated by homogeneous elements x_1, \ldots , x_ m. Then arguing as above we find that M_{(f)} is generated as an S_{(f)}-module by the finite list of elements of the form f_1^{e_1} \ldots f_ n^{e_ n} x_ j /f^ e with e \deg (f) = \sum e_ i \deg (f_ i) + \deg (x_ j) and e_ i < \deg (f). \square
Lemma 10.57.10. Let R be a ring. Let R' be a finite type R-algebra, and let M be a finite R'-module. There exists a graded R-algebra S, a graded S-module N and an element f \in S homogeneous of degree 1 such that
R' \cong S_{(f)} and M \cong N_{(f)} (as modules),
S_0 = R and S is generated by finitely many elements of degree 1 over R, and
N is a finite S-module.
Proof. We may write R' = R[x_1, \ldots , x_ n]/I for some ideal I. For an element g \in R[x_1, \ldots , x_ n] denote \tilde g \in R[X_0, \ldots , X_ n] the element homogeneous of minimal degree such that g = \tilde g(1, x_1, \ldots , x_ n). Let \tilde I \subset R[X_0, \ldots , X_ n] generated by all elements \tilde g, g \in I. Set S = R[X_0, \ldots , X_ n]/\tilde I and denote f the image of X_0 in S. By construction we have an isomorphism
To do the same thing with the module M we choose a presentation
with k_ j = (k_{1j}, \ldots , k_{rj}). Let d_{ij} = \deg (\tilde k_{ij}). Set d_ j = \max \{ d_{ij}\} . Set K_{ij} = X_0^{d_ j - d_{ij}}\tilde k_{ij} which is homogeneous of degree d_ j. With this notation we set
which works. Some details omitted. \square
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