4.2. Regular Schemes 127 Proof It Suffices to Show That Φ : A/M → Am

4.2. Regular schemes 127

Proof It suffices to show that ϕ : A/m → Am/mAm is an isomorphism because we will then have the isomorphism of the lemma by taking the tensor product ⊗Am. As A/m is a field, it suffices to show that ϕ is surjective. Let s ∈ A \ m. There exist a ∈ A and m ∈ m such that 1 = as + m. It follows that s−1 = a modulo mAm. This immediately implies that ϕ is surjective. In what follows, we are going to study an explicit case, where the notion of the Zariski tangent space takes the classical sense, namely, that of differential calculus (Proposition 2.5). Let Y = Spec k[T1,...,Tn] be the affine space over a field k, and let y ∈ Y (k) be a rational point. Let E denote the vector space kn. ∨ Let Dy : k[T1,...,Tn] → E (the dual of E) be the map defined by X ∂P DyP :(t1, . . . , tn) 7→ (y)ti. ∂Ti 1≤i≤n

The linear form DyP is called the differential of P in y. It is clear that Dy is a k-linear map and that Dy(PQ) = P (y)DyQ + Q(y)DyP .

Lemma 2.4. Let m denote the maximal ideal of k[T1,...,Tn] corresponding to y. 2 ∨ (a) The restriction of Dy to m induces an isomorphism m/m ' E .

(b) We have a canonical isomorphism TY,y ' E.

Proof (a) We know that m is of the form (T1 −λ1,...,Tn −λn), λi ∈ k (Exam- ple 2.3.32). The assertion then follows immediately using the Taylor expansion of P in (λ1, . . . , λn). Property (b) results from (a) and Lemma 2.3. If y is any closed (not necessarily rational) point of Y , then we have simi- n ∨ larly a k-linear map Dy : k[T1, ..., Tn] → (k(y) ) (dual as k(y)-vector space) which induces a k(y)-linear map m/m2 → (k(y)n)∨. For the proof of the next proposition, recall that if F is a vector space and if H is a subvector space of F , then (F/H)∨ is canonically isomorphic to the orthogonal of H in F :

(F/H)∨ ' H⊥ := {ϕ ∈ F ∨ | ϕ(v) = 0, ∀v ∈ H}.

n Proposition 2.5. Let X = V (I) be a closed subvariety of Y = Ak , and let x ∈ X be closed point. ⊥ (a) Then dimk(x) TX,x ≤ dimk(x)(DxI) . (b) Suppose that x is a rational point. Let f : X → Y denote the closed immersion. Then, by identifying TY,x with E as in Lemma 2.4, the lin- ear map Tf,x : TX,x → TY,x induces an isomorphism from TX,x onto ⊥ (DxI) ⊆ E. Consequently, TX,x can be identified with    ¯  ¯ X ∂P (t1, . . . , tn) ∈ E ¯ (x)ti = 0, ∀P ∈ I .  ∂Ti  1≤i≤n

It is therefore the tangent space to X at x in the classical sense. 128 4. Some local properties

Proof We have an exact sequence of vector spaces over k(x): 0 → I/(I ∩ m2) → m/m2 → n/n2 → 0, (2.1) where m is the maximal ideal of k[T1,...,Tn] corresponding to x ∈ Y , and n = m/I is the maximal ideal of OX (X) corresponding to x ∈ X. By Lemma 2.3 2 ∨ 2 and Exercise 2.1.5, TX,x ' (n/n ) and dimk(x) m/m ≤ n. As Dx induces a 2 linear surjective map I/(I ∩ m ) → DxI, we have 2 2 ⊥ dimk(x) TX,x = dimk(x) m/m − dimk(x)(I/(I ∩ m )) ≤ n − dimk(x) DxI. This implies (a). 2 ∨ (b) By the isomorphism Dx : m/m → E of Lemma 2.4(a), we obtain an exact sequence ∨ 2 0 → DxI → E → n/n → 0. 2 ∨ ⊥ Therefore TX,x ' (n/n ) ' (DxI) . 2 3 2 Example 2.6. Let X = Spec k[T,S]/(T − S ), and let x0 ∈ X ⊂ Ak be the rational point corresponding to the maximal ideal generated by T and S. 2 3 The differential form Dx(T − S ) is zero if and only if x = x0. Consequently, dimk TX,x = 1 for any x ∈ X(k) \{x0}, and dimk TX,x0 = 2.

+ X s x TX, x x t x0

Figure 7. Tangent space to a curve X.

4.2.2 Regular schemes and the Jacobian criterion Let (A, m) be a Noetherian local ring with residue field k = A/m. We have seen 2 that dimk m/m ≥ dim A (Corollary 2.5.14(b)). Definition 2.7. Let (A, m) be a Noetherian local ring. We say that A is regular 2 if dimk m/m = dim A. By Nakayama’s lemma, A is regular if and only if m is generated by dim A elements. Definition 2.8. Let X be a locally Noetherian scheme, and let x ∈ X be a point. We say that X is regular at x ∈ X, or that x is a regular point of X, if OX,x is regular, that is, dim OX,x = dimk(x) TX,x. We say that X is regular if it is regular at all of its points. A point x ∈ X which is not regular is called a singular point of X. A scheme that is not regular is said to be singular. Example 2.9. Any discrete valuation ring is regular. Consequently, any normal locally Noetherian scheme of dimension 1 is regular (Proposition 1.12). Con- versely, let (A, m) be a Noetherian regular local ring of dimension 1; then by hypothesis m is generated by one element, and A is a principal domain (see the second part of the proof of Proposition 1.12). Thus a Noetherian connected scheme of dimension 1 is a Dedekind scheme if and only if it is regular.