§1.0. Background: Affine Varieties 7 f1,..., fs . Using Lemma 1.0.6, it is straightforward to show that V is smooth at p if and only if the Jacobian matrix (1.0.3) Jp( f1,..., fs) = fi ∂x j (p) 1≤i≤s,1≤ j≤n has rank n d (Exercise 1.0.9). Here is a simple example. Example 1.0.8. As noted in Example 1.0.4, the plane curve C defined by x3 = y2 has I(C) = x3 y2⊆ C[x,y]. A point p = (a,b) C has Jacobian Jp = (3a2,−2b), so the origin is the only singular point of C. Since Tp(V) = HomC(mV ,p /m2 V,p ,C), we see that V is smooth at p when dim V equals the dimension of mV ,p /m2 V ,p as a vector space over OV ,p /mV ,p . In terms of commutative algebra, this means that p V is smooth if and only if OV,p is a regular local ring. See [10, p. 123] or [89, 10.3]. We can relate smoothness and normality as follows. Proposition 1.0.9. A smooth irreducible affine variety V is normal. Proof. Recall that C[X] C(V) and OV,p C(V) since V is irreducible. In §3.0 we will see that C[V] = p∈V OV,p. By Exercise 1.0.7, C[V] is normal once we prove that OV ,p is normal for all p V. Hence it suffices to show that OV,p is normal whenever p is smooth. This follows from some powerful results in commutative algebra: OV ,p is a regular local ring when p is a smooth point of V (see above), and every regular local ring is a UFD (see [89, Thm. 19.19]). Then we are done since every UFD is normal. A direct proof that OV,p is normal at a smooth point p V is sketched in Exercise 1.0.10. The converse of Propostion 1.0.9 can fail. We will see in §1.3 that the affine variety V(xy zw) C4 is normal, yet V(xy zw) is singular at the origin. Products of Affine Varieties. Given affine varieties V1 and V2, there are several ways to show that the cartesian product V1 ×V2 is an affine variety. The most direct way is to proceed as follows. Let V1 Cm = Spec(C[x1,...,xm]) and V2 Cn = Spec(C[y1,...,yn]). Take I(V1) = f1,..., fs and I(V2) = g1,...,gt . Since the fi and g j depend on separate sets of variables, it follows that V1 ×V2 = V( f1,..., fs,g1,...,gt) Cm+n is an affine variety. A fancier method is to use the mapping properties of the product. This will also give an intrinsic description of its coordinate ring. Given V1 and V2 as above,
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