The kernel of the canonical algebra map R → R/I is exactly I.
The cotangent module of ker(R → R/I) is the cotangent module I.Cotangent = I/I².
The canonical algebra map R → R/I is surjective.
Conormal exact sequence: for R → R/I over k, the maps
I/I² → (R/I) ⊗_R Ω_{R/k} → Ω_{(R/I)/k} form an exact sequence at the middle term.
The base-change map (R/I) ⊗_R Ω_{R/k} → Ω_{(R/I)/k} is surjective.
The induced map of Kähler differentials Ω_{R/k} → Ω_{(R/I)/(R/I)} is surjective.
First exact sequence of cotangent modules:
(R/I) ⊗_R Ω_{R/k} → Ω_{(R/I)/(R/I)} → 0 is exact, where the second map sends to zero.
Ω_{k[x₀,x₁]/k} is a free module of rank 2 over k[x₀,x₁].
Ω_{k[x₁,…,xₙ]/k} is a free module of rank n over k[x₁,…,xₙ].
Ω_{k[x₁,…,xₙ]/k} is a free module over k[x₁,…,xₙ].
Conormal exact sequence for a quotient of k[x₀,x₁] by an ideal I.
Surjectivity of the base-change map for a quotient of k[x₀,x₁].
Conormal exact sequence for a quotient of k[x₁,…,xₙ] by an ideal I.
If f : M → N is a surjective R-linear map and M is finite, then finrank N ≤ finrank M.
The base change (R/I) ⊗_R Ω_{R/k} of the rank-n free module of differentials of
R = k[x₁,…,xₙ] has (R/I)-rank n.
Rank bound: rank Ω_{(R/I)/k} ≤ n where R = k[x₁,…,xₙ].
Two-variable specialization: rank of Ω_{(R/I)/k} is at most 2 for R = k[x₀,x₁].
From the conormal exact sequence on a degree-d hypersurface in ℙⁿ, the degree of
ω on the hypersurface is d(d - (n+1)).
For a plane curve C ⊂ ℙ² of degree d, the adjunction formula gives
deg ω_C = d(d-3) = (d-1)(d-2) - 2.
Proposition 35: the conormal sequence I/I² → (R/I) ⊗_R Ω_{R/k} → Ω_{(R/I)/k} → 0
is exact at the middle term and the right map is surjective.
When R is formally smooth over k, injectivity of I/I² → (R/I) ⊗_R Ω_{R/k} is equivalent
to the vanishing of H¹ of the cotangent complex of R/I over k.
When both R and R/I are formally smooth over k, the conormal map I/I² → (R/I) ⊗ Ω_{R/k}
is injective.
When both R and R/I are formally smooth over k, the conormal map is split-injective.
Corollary 26: when both R and R/I are formally smooth over k, the conormal sequence
0 → I/I² → (R/I) ⊗ Ω_{R/k} → Ω_{(R/I)/k} → 0 is short exact and split.
Specialization of the adjunction-formula numerics to plane curves: the genus-degree formula
g = (d-1)(d-2)/2 emerges from d(d-3) = (d-1)(d-2) - 2.
Adjunction formula for a principal divisor: ω_D ≃ ω_X(-D)|_D. In algebraic terms,
⋀^{n-1} Ω_{(R/I)/k} ≃ ((R/I) ⊗ ⋀^n Ω_{R/k}) ⊗ (I/I²)* when I is principal and both
R and R/I are formally smooth (Cor 24, Lec 19).
Exterior powers commute with base change for free modules:
S ⊗_R ⋀^n M ≃ ⋀^n_S (S ⊗_R M).
Decomposition of the top exterior power of a direct sum:
⋀^n (A ⊕ C) ≃ (⋀^{n-1} C ⊗ A) × ⋀^n C (used in the proof of the adjunction formula).
Algebraic adjunction formula: when both R and R/I are formally smooth over k and
Ω_{R/k} is free, the restriction of the top exterior power of Ω_{R/k} to R/I is isomorphic
to ⋀^{n-1} Ω_{(R/I)/k} ⊗ (I/I²). This is the algebraic form of ω_X|_D ≃ ω_D ⊗ N_{D/X}.