# Clarify on Propositions 3.36 and 3.38 in Qing Liu's algebraic geometry book

I have two questions, that I put together as they are related to the same topic, the projective scheme of a graded $$A$$-algebra $$B$$.

Question 1. Do you confirm that points (a), (b) and (c) of Proposition 3.36 below would not hold for any homogeneous $$f$$, but only for those in $$B_+$$?

Lemma 3.36. Let $$f\in B_+$$ be a homogeneous element of degree $$r$$.

(a) There exists a canonical homeomorphism $$\theta:D_+(f)\to\operatorname{Spec} B_{(f)}$$.

(b) Let $$D_+(g)\subset D_+(f)$$ and $$\alpha = g^rf^{-\deg g}\in B_{(f)}$$. Then $$\theta(D_+(g)=D(\alpha$$.

(c) We have a canonical homomorphism $$B_{(f)}\to B_{(g)}$$ which induces an isomorphism $$(B_{(f)})_\alpha \cong B_{(g)}$$. I ask because on my notes the same facts are stated without saying $$f\in B_+$$, but if such facts were true so in general, I feel like the usual construction of the scheme $$\operatorname{Proj} B$$, that relies heavily on (a), (b) and (c), would give a scheme isomorphic to $$\operatorname{Spec}B_{(1)}$$, i.e. $$\operatorname{Spec}B_{0}$$; clearly this is not the case in general.

From now on, if I write $$D_+(f)$$ the $$f$$ is assumed to be homogeneous in $$B_+$$.

Question 2. Once the scheme $$\operatorname{Proj} B$$ is built, we see any $$D_+(f)$$ is isomorphic to $$\operatorname{Spec}B_{(f)}$$. Thus there are morphisms $$D_+(f)\to \operatorname{Spec} A$$ for all $$f$$, which glue to a unique $$\operatorname{Proj} B\to\operatorname{Spec} A$$.

Proposition 3.38 below says the standard scheme structure on the space $$\operatorname{Proj} B$$ is the only one that admits a morphism to $$\operatorname{Spec} A$$ and, restricted to any $$D_+(f)$$, is isomorphic to $$\operatorname{Spec} B_{(f)}$$ over $$A$$.

Proposition 3.38. Let $$A$$ be a ring. Let $$B$$ be a graded algebra over $$A$$. Then we can endow $$\operatorname{Proj} B$$ with a unique structure of an $$A$$-scheme such that for any homogeneous $$f\in B_+$$, the open set $$D_+(f)$$ is affine and isomorphic to $$\operatorname{Spec} B_{(f)}$$.

Now, to determine uniquely a scheme structure on the space $$\operatorname{Proj} B$$, it is not enough to say that any $$D_+(f)$$ is abstractly isomorphic to $$\operatorname{Spec} B_{(f)}$$.

We need also isomorphisms between the scheme structures on $$D_+(f)\cap D_+(g)$$ induced by $$\operatorname{Spec} B_{(f)}$$ and $$\operatorname{Spec} B_{(g)}$$, and that these isomorphisms satisfy the cocycle conditions. These structures are both isomorphic to $$\operatorname{Spec} B_{(fg)}$$, so they are surely isomorphic to each other; but the isomorphism between them has to be unique, so that these structures on the various $$D_+(f)$$ can glue in only one way (the one giving standard $$\operatorname{Proj}B$$).

Likely to get this uniqueness we need to consider the $$A$$-scheme structure: I think that, if I proved that the only $$A$$-algebra automorphism of $$B_{(fg)}$$ is the identity, I would solve my problems; however this fact doesn't seem true, unless $$A=B_0$$.

Do you think it is true instead? Or is there another argument for uniqueness in Proposition 3.38? Question 2 is really problematic for some reason, I have been thinking to it in the last few days but I can't get a hold of it.

• In the future, please try not to insert pictures of text in your questions. They are not searchable, and they are not accessible to users who make use of screen readers or other assistive technologies. I've replaced the pictures with MathJax for you this time. Commented Jul 13 at 16:28

Question 1: yes, $$f$$ must be in $$B_+$$. See for instance Stacks 01M3, Vakil 4.5.7 (2024-02-21 edition), or Hartshorne proposition II.2.5(a). Your observation that if this were true for $$f$$ of degree zero leading to $$\operatorname{Proj} B \cong \operatorname{Spec} B_{(1)} \cong \operatorname{Spec} B_0$$ is correct, which would be a contradiction: if $$B=k[x_0,\cdots,x_n]$$, then we would be saying $$\Bbb P^n_k\cong \Bbb A^0_k$$, which cannot happen for dimension reasons as soon as $$n>0$$.

Question 2: Let's revisit Liu's proof real quick.

Proof. Let $$X=\operatorname{Proj} B$$ and let $$\mathcal{B}$$ be the base for $$X$$ made up of the principal open set $$D_+(f)$$ with $$f\in B_+$$. For any $$D_+(f)\in\mathcal{B}$$, let $$\mathcal{O}_X(D_+(f))=B_{(f)}$$. Using lemma 3.36(c), we see that $$B_(f)$$ is canonically isomorphic to $$B_{(f')}$$ if $$D_+(f)=D_+(f')$$ and that we have a canonical restriction homomorphism $$\mathcal{O}_X(D_+(f))\to \mathcal{O}_X(D_+(g))$$ if $$D_+(g)\subset D_+(f)$$. Hence $$\mathcal{O}_X$$ is a $$\mathcal{B}$$-presheaf. Using the homomorphism $$\theta$$ of lemma 3.36, we see moreover that $$\mathcal{O}_X$$ is a $$\mathcal{B}$$-sheaf. We can then extend it to a sheaf $$\mathcal{O}_X$$ on $$X$$. The proposition is now clear, because $$(D_+(f),\mathcal{O}_X|_{D_+(f)})$$ is isomorphic, via $$\theta$$, to the affine scheme $$\operatorname{Spec} B_{(f)}$$. Finally, $$B_{(f)}$$ is naturally an $$A$$-algebra (since, by hypothesis, the image of $$A$$ in $$B$$ is contained in $$B_0$$), which gives the structure of an $$A$$-scheme on $$X$$.

The canonical homomorphisms $$B_{(f)}\to B_{(fg)}$$ and $$B_{(g)}\to B_{(fg)}$$ from lemma 3.36(c) provide the gluing data: if $$\alpha = (fg)^{\deg f}f^{-\deg fg}$$ and $$\beta = (fg)^{\deg g}g^{-\deg fg}$$, the induced isomorphisms $$(B_{(f)})_\alpha\cong B_{(fg)}$$ and $$(B_{(g)})_\beta\cong B_{(fg)}$$ are exactly what you need to glue.

I think Liu may have underspecified the assumptions here which are supposed to provide unicity - probably what one wants is the following assumptions:

• for all homogeneous $$f\in B_+$$, there is an isomorphism $$D_+(f)\cong\operatorname{Spec} B_{(f)}$$,
• for any homogeneous $$f,g\in B_+$$ and $$\alpha,\beta$$ as in the above paragraph, there is an isomorphism between $$D(\alpha)\subset D_+(f)$$ and $$D(\beta)\subset D_+(g)$$ obtained from the canonical homomorphisms of lemma 3.36(c),
• all of these isomorphisms are compatible with the projection to $$\operatorname{Spec} A$$.

(The goal is to make sure the assumptions for the gluing lemma are met, of course.) Most books I've read or taught out of go in the other direction here - they define the $$\operatorname{Proj}$$ construction then show it satisfies the desired properties. Personally, that's what I'd prefer if I was teaching the material, but to each their own.

• Thanks, just a question for a reality check: if the conditions in your first and second points are met, is the third point granted? (As the projections on $\operatorname{Spec}A$ are given by canonical homomorphism which commute with those of 3.36(c)) Commented Jul 13 at 19:05