Part (ii) of Lemma 6.14 describes the spectra of the two localisations \(A_f\) and \(A_\mathfrak {p}\) of Lecture 18.
Corollary 6.15. (i) For \(f \in A\), the map \(\iota ^*\) identifies \(\mathrm {Spec}(A_f)\) with the basic open set \(D(f) = \{ \mathfrak {q} \in \mathrm {Spec}(A) : f \notin \mathfrak {q} \}\).
(ii) For a prime ideal \(\mathfrak {p} \subset A\), the map \(\iota ^*\) identifies \(\mathrm {Spec}(A_\mathfrak {p})\) with \(\{ \mathfrak {q} \in \mathrm {Spec}(A) : \mathfrak {q} \subset \mathfrak {p} \}\), preserving inclusions. In particular, \(\dim A_{\mathfrak {p}} = h(\mathfrak {p})\).
Proof. A prime ideal \(\mathfrak {q}\) is disjoint from \(\{1, f, f^2, \ldots \}\) if and only if \(f \notin \mathfrak {q}\), and disjoint from \(A \setminus \mathfrak {p}\) if and only if \(\mathfrak {q} \subset \mathfrak {p}\). For the last statement, every chain of prime ideals contained in \(\mathfrak {p}\) can be extended to end at \(\mathfrak {p}\), so the supremum of their lengths is \(h(\mathfrak {p})\). □
Thus the localisation \(A_f\) plays the role of the ring of “regular functions” on the open set \(D(f)\).
\(\blacktriangleright \) The passage from \(A\) to \(A_{\mathfrak {p}}\) cuts out all the prime ideals except those contained in \(\mathfrak {p}\). This is in contrast to the passage from \(A\) to \(A/\mathfrak {p}\), which cuts out all the prime ideals except those containing \(\mathfrak {p}\). In Figure 10, for \(\mathfrak {p} = (Y - X^2)\) the first keeps the generic points of the plane and of the parabola, the second keeps the parabola with all its points. Given prime ideals \(\mathfrak {q} \subset \mathfrak {p}\), we can also localise with respect to \(\mathfrak {p}\) and quotient with respect to \(\mathfrak {q}\): by Lemma 6.14(iii), the prime ideals of \(A_\mathfrak {p}/\mathfrak {q}A_\mathfrak {p} \simeq (A/\mathfrak {q})_{\mathfrak {p}/\mathfrak {q}}\) are those lying between \(\mathfrak {q}\) and \(\mathfrak {p}\).
\(\sphericalangle \) Let \(I\) be an ideal in \(A\). Show that \(\iota ^{-1}(S^{-1}I) = \{ a \in A : as \in I \text { for some } s \in S \}\).
The ring \(A_{\mathfrak {p}}\) is the algebraic analogue of the germs of functions defined near the point \(\mathfrak {p}\) of \(\mathrm {Spec}(A)\). The dimension of \(A\) can be computed from these local rings: by definition \(\dim A = \sup _{\mathfrak {p}} h(\mathfrak {p})\), and \(h(\mathfrak {p}) \leq h(\mathfrak {m})\) when \(\mathfrak {p} \subset \mathfrak {m}\), so \[ \dim A = \sup _{\mathfrak {p} \in \mathrm {Spec}(A)} \dim A_{\mathfrak {p}} = \sup _{\mathfrak {m} \in \mathrm {mSpec}(A)} \dim A_{\mathfrak {m}}. \]
Next, we come to a very useful lemma in the study of local rings, which will later on help us understand how to compute \(\dim A_{\mathfrak {p}}\).
Lemma 6.16. (Nakayama’s Lemma) Let \(A\) be a local ring with maximal ideal \(\mathfrak {m}\). Let \(M\) be a finitely generated \(A\)-module and \(N \subset M\) be a submodule with the property that \(M = N + \mathfrak {m}M\). Then \(M=N\).
Proof. The \(A\)-module \(\overline {M} = M/N\) is finitely generated and satisfies \(\overline {M} = \mathfrak {m} \overline {M}\). We want to show that \(\overline {M}=\{0\}\). If not, let \(x_1,\ldots , x_n \in \overline {M}\) be generators, \(\overline {M} = A x_1 + A x_2 + \ldots + A x_n\), with \(n \geq 1\) minimal. Since \(\overline {M} = \mathfrak {m} \overline {M}\), we have \(x_n = \sum _{i=1}^n a_i x_i\) with \(a_i \in \mathfrak {m}\). Then \((1-a_n)x_n= \sum _{i=1}^{n-1} a_i x_i\), and \(1-a_n \notin \mathfrak {m}\) is a unit, because \(A\) is local and every non-unit lies in a maximal ideal. It follows that \(x_n \in A x_1 + A x_2 + \ldots + A x_{n-1}\), contradicting the minimality of \(n\). □
This is a simple but important result that allows us to make deductions about properties of \(M\) as an \(A\)-module from the properties of \(M/\mathfrak {m}M\) as an \(A/\mathfrak {m}\)-vector space. Thus, many properties of the finitely generated modules over a local ring \(A\) can be studied via linear algebra over the field \(A/\mathfrak {m}\).
Here is a first application of Nakayama’s lemma. For a finitely generated module \(M\), write \[ \mu (M) \coloneqq \min \{ n \geq 0 : M = A x_1 + \ldots + A x_n \text { for some } x_1,\ldots , x_n \in M \} \] for the minimal number of generators of \(M\).
Proposition 6.17. Let \(A\) be a local ring with maximal ideal \(\mathfrak {m}\) and residue field \(k = A/ \mathfrak {m}\), and let \(M\) be a finitely generated \(A\)-module. Elements \(x_1, \ldots , x_n \in M\) generate \(M\) if and only if their images \(\overline {x}_1, \ldots , \overline {x}_n\) span the \(k\)-vector space \(M/\mathfrak {m}M\). In particular, \(\mu (M) = \dim _{k} M/\mathfrak {m}M\).
Proof. If \(M = A x_1 + \ldots + A x_n\), then \(M/\mathfrak {m}M\) is spanned by \(\overline {x}_1, \ldots , \overline {x}_n\). Conversely, suppose that \(\overline {x}_1,\ldots , \overline {x}_n\) span \(M/\mathfrak {m}M\), and let \(N= A x_1 + \ldots + Ax_n \subset M\). By Nakayama’s lemma, it suffices to show that \(M = N + \mathfrak {m}M\). Let \(x \in M\). Then \(\overline {x} = \sum _{i=1}^n \overline {a}_i \overline {x}_i\) for some \(a_i \in A\), so \(x - \sum _{i=1}^n a_i x_i \in \mathfrak {m}M\), and \(x \in N + \mathfrak {m}M\). Finally, a \(k\)-vector space spanned by \(n\) elements has dimension at most \(n\), and a basis of \(M/\mathfrak {m}M\) lifts to a generating set of \(M\); hence \(\mu (M) = \dim _k M/\mathfrak {m}M\). □
Now let \(A\) be a Noetherian local ring with maximal ideal \(\mathfrak {m}\) and residue field \(k\). Then \(\mathfrak {m}\) is a finitely generated \(A\)-module, and we write \[ \mu (A) \coloneqq \mu (\mathfrak {m}) = \dim _k \mathfrak {m}/\mathfrak {m}^2 \] for the minimal number of generators of \(\mathfrak {m}\).
\(\sphericalangle \) Let \(\mathfrak {m}\) be a maximal ideal of a ring \(A\). Show that \(A_\mathfrak {m}/\mathfrak {m}^2 A_\mathfrak {m} \simeq A/\mathfrak {m}^2\), and deduce that \(\mathfrak {m}A_\mathfrak {m}/(\mathfrak {m}A_\mathfrak {m})^2 \simeq \mathfrak {m}/\mathfrak {m}^2\) (use Lemma 6.14(iii)).
\(\blacktriangleright \) Here is the geometric meaning of \(\mu (A)\). Let \(V \subset k^n\) be an affine variety with \(\mathcal {I}(V) = (f_1, \ldots , f_m)\), let \(\mathbf {v} \in V\), and let \(A = k[V]_{\mathfrak {m}_\mathbf {v}}\) be the local ring of \(V\) at \(\mathbf {v}\), so that \(\mathfrak {m}/\mathfrak {m}^2 \simeq \mathfrak {m}_\mathbf {v}/\mathfrak {m}_\mathbf {v}^2\) by the exercise. Every \(g \in k[X_1,\ldots ,X_n]\) can be written as \(g = g(\mathbf {v}) + d_\mathbf {v} g + r\), where \(d_\mathbf {v} g = \sum _i \frac {\partial g}{\partial X_i}(\mathbf {v}) (X_i - v_i)\) is the linear part of \(g\) at \(\mathbf {v}\), which we view as a linear form on \(k^n\), and \(r \in (X_1 - v_1, \ldots , X_n - v_n)^2\). Then \(g \mapsto d_\mathbf {v} g\) induces an isomorphism \(\mathfrak {m}_\mathbf {v}/\mathfrak {m}_\mathbf {v}^2 \simeq (k^n)^*/\langle d_\mathbf {v} f_1, \ldots , d_\mathbf {v} f_m \rangle \), and the right hand side is the dual of the tangent space \[ T_\mathbf {v} V = \{ \mathbf {w} \in k^n : d_\mathbf {v} f_j(\mathbf {w}) = 0 \text { for } j = 1, \ldots , m \}. \] So \(\mu (A) = \dim _k T_\mathbf {v}V\). For the parabola \(V = \mathcal {V}(Y - X^2)\), the tangent space at the origin is the line \(Y = 0\), so \(\mu (A) = 1 = \dim A\). For the cusp \(V = \mathcal {V}(Y^2 - X^3)\) of Figure 8, we have \(d_{\mathbf {0}}(Y^2 - X^3) = 0\), so the tangent space at the origin is all of \(k^2\) and \(\mu (A) = 2\), while \(\dim A = 1\). At every other point of the cusp the tangent space is a line, see Figure 14.
Question: What is the relation between \(\mu (A)\) and \(\dim A\)?
The answer will be given by the following important theorem of Krull, which we prove in Lectures 20 and 21. It provides a lower estimate for the number of generators of an ideal in a Noetherian ring (and for the number of equations needed to describe an affine variety).
Theorem 6.18. (Krull’s height theorem) Let \(A\) be a Noetherian ring, and let \(I =(a_1,\ldots , a_r)\) be an ideal generated by \(r\) elements. Let \(\mathfrak {p} \subset A\) be a prime ideal which is minimal (with respect to inclusion) among the prime ideals containing \(I\). Then \[ h(\mathfrak {p}) \leq r. \]
Corollary 6.19. Let \(A\) be a Noetherian local ring with maximal ideal \(\mathfrak {m}\). Then \(\dim A \leq \mu (A)\). In particular, \(\dim A\) is finite.
Proof. Every chain of prime ideals of \(A\) can be extended to end at \(\mathfrak {m}\), so \(\dim A = h(\mathfrak {m})\), and \(\mathfrak {m}\) is the only prime ideal containing \(\mathfrak {m}\). So Theorem 6.18 applied to \(I = \mathfrak {m} = (a_1, \ldots , a_r)\) with \(r = \mu (A)\) gives \(\dim A = h(\mathfrak {m}) \leq \mu (A)\). □
Theorem 6.18 makes precise the intuition that each additional equation can decrease the dimension of the space of solutions by at most one. Note that one can give a lower bound for \(\dim A\) by exhibiting a chain of prime ideals in \(A\), but an upper bound is not straightforward to obtain from the definition. This is another reason why Theorem 6.18 is important.
\(\blacktriangleright \) For a point \(\mathbf {v} \in k^n\), the local ring \(A = k[X_1,\ldots ,X_n]_{\mathfrak {m}_\mathbf {v}}\) is regular of dimension \(n\). Indeed, \(\mu (A) = n\), since \(\mathfrak {m}_\mathbf {v}/\mathfrak {m}_\mathbf {v}^2\) has basis \(X_1 - v_1, \ldots , X_n - v_n\), and the chain of prime ideals \((0) \subsetneq (X_1 - v_1) \subsetneq (X_1 - v_1, X_2 - v_2) \subsetneq \ldots \subsetneq \mathfrak {m}_\mathbf {v}\), whose quotients are polynomial rings, gives \(h(\mathfrak {m}_\mathbf {v}) \geq n\), so \(\dim A = n\) by Corollary 6.19. More generally, a point \(\mathbf {v}\) of an affine variety \(V\) is called a smooth point if \(\dim _k T_\mathbf {v} V = \dim k[V]_{\mathfrak {m}_\mathbf {v}}\), that is, if the local ring of \(V\) at \(\mathbf {v}\) is regular. By Corollary 6.19, \(\dim _k T_\mathbf {v} V \geq \dim k[V]_{\mathfrak {m}_\mathbf {v}}\) always, and the origin of the cusp is not a smooth point.
An important theorem due to Auslander and Buchsbaum states that regular local rings are UFDs. Unfortunately, we do not have time to prove this result in this course; see [4], where you can also find a proof that the localisation \(k[X_1,\ldots , X_n]_{\mathfrak {p}}\) at any prime ideal \(\mathfrak {p}\) is regular. (These are applications of the homological methods in algebra, which we do not get into in this course.)