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2 made local representability theorem more precise

Stalk-local detection of irreducibility on locally Noetherian schemes, which I prove directly here with no primary decomposition tricks. It helps with a lot of exercises, and intuition.

Sheafification of base-presheaves (presheaves defined only on a base of open sets). I see from your TOC that you cover the unique extension of base sheaves to sheaves as per Kevin Lin's answer (E-H's Proposition I-12).

When I took Arthur Ogus' algebraic geometry class, he was very insistent about teaching us this, and it really paid off for the remainder of the course, particularly in exercises. It categorically exclaims (pun intended) the credo always start with the affine opens, so one sees explicitly how special and critical they are to the theory.

The sheaf of meromorphic functions $\mathcal{K}_X$ on $X$ can be defined by sheafifying the naive base-presheaf $\mathcal{K'}(U)=Frac(\mathcal{O}(U))$ on the base of open affines. This formula doesn't define a base-sheaf on affines, and as Georges Elencwajg and BCnrd explain here, it doesn't even define a presheaf when applied to arbitrary opens. I suggest at least mentioning these three facts, to save people from re-wasting the time that I and many others have in wondering what the resulting sheaf looks like.

Locally representable means representable, i.e. if $F:Sch^{op}\to Set$ is a functor with an open sheaf when restricted to a base of (Zariski) opens on every scheme, and $F$ has a covering by representable open subfunctors $F_i$, then $F$ is representable (very much along the lines of EGA 1 (1971), Chapter 0, Proposition 4.5.4). I advocate this because the work that goes into the proof is essentially the same work we inevitably do to prove fibered products of schemes exist, so it gives fibre products as a special case, but also offers up a rigorous-but-quick route to other constructions like global Spec and global Proj.

The general definition of quasicoherence and coherence for modules on local ringed spaces / non- locally Noetherian schemes... not as a gratuitous generality, but as a foreshadowing/reminder that presentations, not just surjections, are what make coherence work.

Basic Dedekind domain theory, along the lines of Lang's Algebraic Number Theory, chapter 1. I found curves and their divisors — even in characteristic 0 — impossible to understand until I read that.

Quasiseparatedness is something I'm glad to see you including, because using it explicitly is the key to a lot of proofs, so having it in mind as a word helps me remember how to do them.

Your affine communication lemma is a must-have, for anyone else reading this answer!

1

Stalk-local detection of irreducibility on locally Noetherian schemes, which I prove directly here with no primary decomposition tricks. It helps with a lot of exercises, and intuition.

Sheafification of base-presheaves (presheaves defined only on a base of open sets). I see from your TOC that you cover the unique extension of base sheaves to sheaves as per Kevin Lin's answer (E-H's Proposition I-12).

When I took Arthur Ogus' algebraic geometry class, he was very insistent about teaching us this, and it really paid off for the remainder of the course, particularly in exercises. It categorically exclaims (pun intended) the credo always start with the affine opens, so one sees explicitly how special and critical they are to the theory.

The sheaf of meromorphic functions $\mathcal{K}_X$ on $X$ can be defined by sheafifying the naive base-presheaf $\mathcal{K'}(U)=Frac(\mathcal{O}(U))$ on the base of open affines. This formula doesn't define a base-sheaf on affines, and as Georges Elencwajg and BCnrd explain here, it doesn't even define a presheaf when applied to arbitrary opens. I suggest at least mentioning these three facts, to save people from re-wasting the time that I and many others have in wondering what the resulting sheaf looks like.

Locally representable means representable, i.e. a functor with an open covering by representable subfunctors is representable. I advocate this because the work that goes into the proof is essentially the same work we inevitably do to prove fibered products exist, so it gives fibre products as a special case, but also offers up a rigorous-but-quick route to other constructions like global Spec and global Proj.

The general definition of quasicoherence and coherence for modules on local ringed spaces / non- locally Noetherian schemes... not as a gratuitous generality, but as a foreshadowing/reminder that presentations, not just surjections, are what make coherence work.

Basic Dedekind domain theory, along the lines of Lang's Algebraic Number Theory, chapter 1. I found curves and their divisors — even in characteristic 0 — impossible to understand until I read that.

Quasiseparatedness is something I'm glad to see you including, because using it explicitly is the key to a lot of proofs, so having it in mind as a word helps me remember how to do them.

Your affine communication lemma is a must-have, for anyone else reading this answer!