fix: proofread Algebraic Geometry II: Affine Schemes part

[vEnhance]

Proofreading pass over the Algebraic Geometry II: Affine Schemes part (part of the #315 campaign), covering the six chapters: sheaves, localization, spec-zariski, spec-sheaf, spec-examples, mor-scheme.

Most fixes are minor typos and grammar. A few small mathematical/notational corrections are worth calling out:

Math / calculation fixes

• spec-sheaf.tex (gyrotop problem): the scheme was written Spec ℂ[x,y,z]/(xy,z), but that collapses z=0 everywhere and is inconsistent with the given solution (which uses A_z ≅ k[z,z⁻¹], a punctured line, and (xz,yz)). Corrected to (xz,yz) — the plane-plus-axis "top" shape the problem intends.
• spec-sheaf.tex (stalks-are-localizations proof): the injectivity step had h(g₁f₂ − f₂g₁) = 0, but g₁f₂ − f₂g₁ ≡ 0 identically. Corrected to h(f₁g₂ − f₂g₁) = 0 (the genuine cross-multiplication), and tightened "some nonzero h" to "some h ∉ 𝔭" so the subsequent D(h) step is valid.
• spec-zariski.tex (Krull dimension example): the length-1 chain for ℂ[x,y]/(y−x²) was written (x,y) ⊊ (y−x²), which is the reversed (and false) inclusion. Corrected to (y−x²) ⊊ (x,y).
• spec-examples.tex: evaluating 20/19 at (7) gives 4 (mod 7), not 3 (since 19≡5, 20≡6, 6·5⁻¹ ≡ 6·3 ≡ 4).
• localization.tex: A_(x) = {f/g : g(0)≠0} was described as "the Laurent series"; it is actually the rational functions with no pole at the origin (Laurent series/polynomials are the other localization). Reworded.
• mor-scheme.tex: general t↦t² remark said (x−a) maps to (y−f(y)); should be (y−f(a)).

Notation fixes

• spec-zariski.tex: f ∈ ℂ → f ∈ A for a polynomial; A = Spec ℤ → A = ℤ (with "function on Spec A").
• spec-sheaf.tex: f ∈ Spec A → f ∈ A; 𝒪_{Spec X,(x)} → 𝒪_{X,(x)} (double Spec); residue field A/𝔪 → R/𝔪; U ∋ m → U ∋ 𝔪.
• mor-scheme.tex: several 𝒪_{Spec X}/𝒪_{Spec Y} (double Spec on spaces X,Y) → 𝒪_X/𝒪_Y; k[y] → ℂ[y]; f^♯ → π^♯ and f^{pre} → π^{pre}.
• sheaves.tex: category-lovers aside wrote 𝒮ℱ_p(U); corrected to 𝒮ℱ(U) / 𝒮ℱ_p.

Typos / grammar

• sheaves.tex: "thrown way" → "thrown away"; "a open set" → "an open set".
• localization.tex: "also explain" → "explains"; "those elements of those elements" → "those elements".
• spec-zariski.tex: "as it always moving" → "as it is always moving"; "old geometric'' " → "old geometric'' points"; "is the called" → "is then called the"; "considering the space" → "consider the space".
• spec-examples.tex: "are don't look" → "don't look"; "only way" → "only one way"; "is the again" → "is again"; "How it does differ" → "How does it differ"; pmod{x-1,y-2} → pmod{x-1,y+2}; "un-bending the hyperbola" → "un-bending the parabola" (parabola section); "(0) is longer prime" → "no longer prime"; "deleted it" → "deleted"; "draw the visualize" → "visualize"; "this point has only one closed point" → "this space".
• spec-sheaf.tex: "if and only of" → "if and only if".
• mor-scheme.tex: "give a define" → "define"; "didn't to specify" → "didn't have to specify"; "two construction gives" → "two constructions give"; "well-definced" → "well-defined".

🤖 Generated with Claude Code

---

Generated by Claude Code