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The Stacks project

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

f = f_0 + f_1 + \ldots + f_ n

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.

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.

Note that by construction there is a continuous map

\text{Proj}(S) \longrightarrow \mathop{\mathrm{Spec}}(S_0).

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

D_{+}(f) = \{ \mathfrak p \in \text{Proj}(S) \mid f \not\in \mathfrak p \} .

Finally, for a homogeneous ideal I \subset S we define

V_{+}(I) = \{ \mathfrak p \in \text{Proj}(S) \mid I \subset \mathfrak p \} .

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.

  1. The sets D_{+}(f) are open in \text{Proj}(S).

  2. We have D_{+}(ff') = D_{+}(f) \cap D_{+}(f').

  3. 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).
  4. 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).
  5. The open sets D_{+}(f) form a basis for the topology of \text{Proj}(S).

  6. 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)}).
  7. There exists an S such that \text{Proj}(S) is not quasi-compact.

  8. The sets V_{+}(I) are closed.

  9. Any closed subset T \subset \text{Proj}(S) is of the form V_{+}(I) for some homogeneous ideal I \subset S.

  10. 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

\text{Proj}(S) = \bigcup \nolimits _{i = 1}^\infty D_{+}(X_ i)

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

\psi (x/g) = (x g^{\deg (f) - 1}/f^{\deg (x)})/(g^{\deg (f)}/f^{\deg (g)}).

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.

  1. Any minimal prime of S is a homogeneous ideal of S.

  2. 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

  1. the ring S_{(f)} is of finite type over R, and

  2. 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

\sum \nolimits _{e\deg (f) = \sum e_ i\deg (f_ i)} \lambda _{e_1 \ldots e_ n} f_1^{e_1} \ldots f_ n^{e_ n}/f^ e

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

f_1^{e_1} \ldots f_ n^{e_ n}/f^ e = f_ i^{\deg (f)}/f^{\deg (f_ i)} \cdot f_1^{e_1} \ldots f_ i^{e_ i - \deg (f)} \ldots f_ n^{e_ n}/f^{e - \deg (f_ i)}

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

  1. R' \cong S_{(f)} and M \cong N_{(f)} (as modules),

  2. S_0 = R and S is generated by finitely many elements of degree 1 over R, and

  3. 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

S_{(f)} \longrightarrow R', \quad X_ i/X_0 \longmapsto x_ i.

To do the same thing with the module M we choose a presentation

M = (R')^{\oplus r}/\sum \nolimits _{j \in J} R'k_ j

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

N = \mathop{\mathrm{Coker}}\Big( \bigoplus \nolimits _{j \in J} S(-d_ j) \xrightarrow {(K_{ij})} S^{\oplus r} \Big)

which works. Some details omitted. \square


Comments (7)

Comment #1373 by Keenan Kidwell on

In the previous section (Tag 00JL), the convention is made that a graded ring is a -graded ring. But Tag 00JO makes reference to -graded rings, the definition of which is obvious, but slightly clashes with the aforementioned convention, right? The remark at the beginning of this section that primality of a homogeneous ideal in a graded ring can be checked ``on homogeneous elements" in the natural sense is used implicitly for -graded rings in the proof of 00JO, as is the fact that the radical of a homogeneous ideal (in a -graded ring!) is itself homogeneous. The latter fact follows from the former via Tag 00JT, which shows that, while a priori the radical of an ideal is the intersection of the primes which contain it, if is homogeneous, the radical is already the intersection of the homogeneous primes which contain it. I'm just pointing a bunch of stuff out and not offering any constructive suggestions. I guess what I would propose is the following. Introduce the notion of a -graded ring (to be distinguished from a graded ring, which by convention is -graded). Give the all-important example of for graded and of positive degree (which is what we care about anyway). Perhaps the result about checking primality of homogeneous ideals (which holds in -graded rings) should get its own tag at the beginning of this section. One can then deduce 00JT from this result immediately (the proof of 00JT is basically the proof of the primality-checking thing). Now insert the corollary that radicals of homogeneous ideals (in -graded rings) are homogeneous. And cite this in the proof of 00JO. (The argument there proves that if and are homogeneous elements with in the radical of , then one of is in the radical of , so there is a direct but implicit application of the primality-checking result.) If this seems worthwhile, I could do it and submit it. I wrote this stuff up for myself at some point already anyway. But maybe I'm just being pedantic.

Comment #1374 by Keenan Kidwell on

Also, I think one should argue that for a homogeneous prime of (or it's obvious?). The forward containment is obvious. If is homogeneous, then , so . This is enough to conclude the reverse containment since is generated by its homogeneous elements.

Comment #1386 by on

You are right. In fact, we should introduce -graded rings, where is a commutative(?) monoid. Then we should introduce -graded modules. Except then there is the funny business with our graded modules over a graded ring not following this pattern. So I guess then we should introduce, given a homomorphism of monoids, -graded modules over a -graded ring. Sigh!

Yeah, some of this stuff has to go into Section 10.56 and note how that section does not have formal definitions. The initial write-up made the, reasonable I feel, assumption that the reader may already have an idea of what a -graded ring is. I've decided to leave it as is for now, but feel free to edit a bit and send in the result.

Comment #2803 by Saurav on

For part 10, lemma 10.56.3 : . One can then use the fact that if , the homogeneous part of a prime ideal is also a prime. Therefore if and only if for every which implies . (Shorter argument!)

Comment #2804 by Saurav on

For part 10, lemma 10.56.3 : . One can then use the fact that , the homogeneous part of a prime ideal is also a prime. Therefore if and only if for every which implies . (Shorter argument!)

Comment #2906 by on

@#2803 and #2804. Yes, I sort of agree. But not enough to replace the current one.

Comment #9879 by Yaowei Zhang on

I am confused about the proof for Lemma 10.57.7. Which states the following:\

Let be a graded ring and let be a prime ideal. Let be the homogeneous ideal generated by homogeneous elements of . Then is a prime. \

To check if a homogeneous ideal is prime, it is enough to check if then or for homogeneous elements . Let be homogeneous and . By definition of , it means for homogeneous. But that imply and thus or . Then are homogeneous elements in and hence generator of . Thus or . Then we are done, right? Not sure we need to assume non-homogeneous.


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