The maximal ideal m_x ⊂ k[x_1,…,x_n] of polynomials vanishing at the point x ∈ k^n,
defined as the kernel of evaluation at x.
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The vanishing ideal of a point x ∈ k^n is maximal, since the evaluation map is
surjective onto the field k.
The image of m_x in the quotient k[x_1,…,x_n]/I, well-defined whenever I ⊆ m_x;
this is the maximal ideal of x viewed as a point of the affine variety V(I).
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The image of m_x in k[x_1,…,x_n]/I is maximal.
The image of m_x in k[x_1,…,x_n]/I is prime (since maximal ideals are prime).
The local ring of the affine variety V(I) at the point x, obtained as the
localization of k[x_1,…,x_n]/I at the maximal ideal of x.
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Definition 34 (Lecture 18). The universal property characterising Kähler differentials:
A-linear maps Ω[A⁄k] → M correspond naturally to k-derivations A → M.
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Definition 35 (Lecture 18). The Zariski cotangent space of a local ring R, defined as
m/m² viewed as a vector space over the residue field.
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Lemma 30 (Lecture 18). For a Noetherian local ring, the Krull dimension is at most the dimension of the Zariski cotangent space.
Definition 36 (Lecture 18). A Noetherian local ring is smooth (regular) iff the Zariski cotangent space has dimension equal to the Krull dimension; equality in Lemma 30.
Proposition 29 (Lecture 18). A Noetherian local k-algebra is regular (smooth) iff its
module of Kähler differentials Ω[R⁄k] is free.
The regular locus is open: if a prime 𝔭 of a finite-type k-algebra A has regular
localization, then some f ∉ 𝔭 makes the localization regular at every prime not containing
f.
The generic point of an integral finite-type k-algebra is regular: localizing at the
zero ideal gives a field, hence trivially a regular local ring.
Proposition 30 (Lecture 18). For an integral finite-type k-algebra A, the smooth
(regular) locus is open and dense: there exists nonzero f ∈ A such that the localization
of A at every prime not containing f is regular.
Helper for Corollary 23: if the local ring at x of V(f_1,…,f_m) has Krull dimension
n - m, then m ≤ n (you cannot cut out a positive-dimensional variety with more equations
than ambient coordinates while preserving the expected dimension).
The cotangent space at a point of V(f_1,…,f_m) ⊂ 𝔸ⁿ has dimension equal to
n - rank(Jac(f)(x)): the differentials of the relations cut out a subspace of the ambient
cotangent space (m_x/m_x²).
Corollary 23 (Lecture 18, Jacobian criterion). For a complete intersection V(f_1,…,f_m)
of expected dimension n - m, the local ring at x is regular iff the Jacobian matrix has
full rank m at x.
Corollary 23, hypersurface case: for a hypersurface V(P) ⊂ 𝔸ⁿ, smoothness at x
amounts to some partial derivative ∂P/∂x_i being nonzero at x, i.e. the Jacobian
(1 × n) matrix has rank one.
Helper for Proposition 31: if I agrees with span(f) after inverting a single element
not vanishing at x, and span(f) is regular at x, then so is I. This is the local
descent step used to deduce regularity for I from regularity for a locally-generating
subfamily.
If the Jacobian rows (∂f_i/∂x_j)(x) are k-linearly independent, then the local ring
of V(f_1,…,f_m) at x has the expected Krull dimension n - m.
At a smooth (regular) point x of V(I), one can find polynomials f_1,…,f_m ∈ I
whose Jacobian rows at x are linearly independent — i.e. local generators for the
cotangent obstructions.
A smooth complete intersection is locally a domain: a regular local ring is in particular an integral domain.
If span(f) ⊂ I and the two ideals define local rings of the same Krull dimension at
x, with the smaller one a domain, then I and span(f) agree after multiplying by some
unit u (i.e. they agree locally near x).
If the local ring at x of V(I) is regular and f_1,…,f_m ∈ I have linearly
independent Jacobian rows at x, then the local rings of V(I) and V(span f) have the
same Krull dimension at x.
If each f_i vanishes at x, then span(f_1,…,f_m) ⊆ m_x.
Polynomials in an ideal I ⊆ m_x all vanish at the point x.
Proposition 31 (Lecture 18). Smoothness of V(I) at x is equivalent to the existence
of polynomials f_1,…,f_m ∈ I with linearly independent Jacobian rows at x whose span,
after inverting some u(x) ≠ 0, equals I locally.