Lecture 18: Kähler Differentials

Lecture  18.725 Algebraic Geometry I Lecture 18: Kähler Differentials Last time we proved that principal divisors on a complete normal curve has degree zero. This actually remains true for Cartier divisors on irreducible non-normal curves. To prove this, we show that the degree of a divisor is preserved under pull-back to normalization. Let D be a principal divisor on a non-normal irreducible curve X. We may assume that D = (f ) is supported at a point x, the curve is complete and normal away from x, so that f defines a map X → P1 . The total degree of the divisor of zeroes of f is the same on X and on the normalization N m(X), both are equal to the degree deg(f˜), where f˜ is the composition N m(X) → X → P1 . Today we begin the discussion of tangent and cotangent spaces and smoothness. The first step is to define (Kähler) differentials. Definition 1. Let A be a commutative k-algebra. ΩA is defined to be the A-module generated by expressions da, a ∈ A, modulo the following equations: • d(a + b) = da + db; • d(λa) = λda; • d(ab) = (da)b + a(db), where a, b ∈ A, λ ∈ k. Then ΩA is characterized by a universal property: Hom(ΩA , M ) = Der(A, M ) for any A-module M , where Der(A, M ) is the k-module of k-linear derivations from A to M . As an alternative way to define ΩA , suppose that A is generated by a1 , . . . , an . Let X = Spec A and Im be the ideal of X in the diagonal X ⊂ X × X. Then (ai ⊗ 1 − 1 ⊗ ai ) generate Im ⊂ A ⊗ A. Therefore ΩA is finitely generated. This approach also allows us to define a coherent sheaf ΩX on X, called the sheaf of differentials on X. Let f : A → B be a morphism of rings. Then there is a canonical morphism B ⊗A ΩA → ΩB given by da 7→ d(f a). Let Y = Spec B. Then this morphism of rings gives rise to the morphism of varieties Y → X, df : f ∗ ΩX → ΩY . Now for an arbitrary variety X over k, we may define the sheaf ΩX by gluing the above constructions on affine charts. Then it is straightforward to check that Hom(ΩX , F) = Der(OX , F), where F is a coherent sheaf on X, and Der(OX , F) is the set of k-linear derivations OX → F, i.e. sheaf morphisms satisfying Leibniz rule on each chart. Definition 2. Let X be a variety. The Zariski cotangent space of X at x ∈ X is defined to be the vector space {ξ : OX,x → k | ξ is linear and ξ(f g) = f (x)ξ(g) + g(x)ξ(f )}, i.e. it is the set of derivations at x, and it is denoted as Tx∗ X. One can check that (ΩX )x = Tx∗ X. Now we define the tangent sheaf TX on X as TX = Hom(ΩX , OX ). Note however that even though there is always a map ΩX → Hom(TX , OX ), it is not neccessarily an isomorphism. Lemma 1. dim(Tx∗ X) ≥ dimx (X). Proof. We may assume X = Spec A and m the maximal ideal corresponding to x. Let df1 , · · · , dfn be the generators of m/m2 , where each dfi is lifted to fi ∈ m. By Nakayama lemma, fi generate m. Now, as a consequence of the hypersurface theorem, dimx X ≤ n. Definition 3. Let x ∈ X. X is said to be smooth at x if dim(Tx∗ X) = dimx (X). Proposition 1. X is smooth at x ∈ X if and only if ΩX is locally free on a neighborhood of x. Proof. One direction (from right to left) will follow from the next proposition. For the other direction (from left to right), recall the lemma stated during the lecture on October 22th, asserting that if all fibers of a coherent sheaf have the same dimension, then the sheaf is locally free, combined with the fact (that we will prove next time) that smooth varieties are locally irreducible. 1 Lecture  18.725 Algebraic Geometry I Proposition 2. For a variety X, the set of smooth points in X is open and dense in X. Proof. It follows from the previous proposition(left to right) that the set of smooth points in X, which we denote by Xsm , is open in X. Now, to prove that Xsm is dense in X, we may assume X is affine and irreducible, and is actually embedded as a closed subset X ⊂ An . Let d = n − dim X. We proceed by induction on d. If d = 0 then X = An , which is smooth everywhere, and there is nothing to prove. Now for d > 0, we may find g ∈ k[An ] vanishing on X. choose g to have minimal degree among such functions. ∂g We claim that is not identically zero on X for at least one xi . To see this, suppose to the contrary that ∂xi ∂g is identically vanishing on X. If chark = 0, by the minimality of degree of g, this means g is a constant ∂xi function which is not zero. Then g cannot vanish on X, a contradiction. if chark = p, then replacing g with g 1/p gives a function identically vanishing on X with a smaller degree than g, a contradiction. Hence the ∂g claim holds. After a change of coordinate, we may assume that g is monic in xn and is not identically ∂xn zero on X. now, consider the projection π : An = Spec k[x1 , · · · , xn ] → An−1 = Spec k[x1 , · · · , xn−1 ]. Let Y be the image of X under this projection. Then since π is finite, dim Y = dim X. Since Y is a closed subset of An−1 we may apply the induction hypothesis on Y , so that the smooth points of Y consist an open and ∂g dense subset of Y . Now we claim that if x ∈ X is such that = 6 0 at x and π(x) is a smooth point of Y , ∂xn then X is smooth at x. Indeed, for such x, π : X → Y induces a surjection Tπ∗(x) Y ⊕ (gdxn |x )/dg|x → Tx∗ X. Therefore, dim Tx∗ X ≤ dim Ty∗ Y = dim Y = dim X. By a previous lemma, dim Tx∗ X = dim X. Hence x is a smooth point of X. The set of all such x is dense in X, hence Xsm is dense in X. Remark 1. A curve is defined to be a variety of dimension one. For a curve X, the following are equivalent: • X is smooth. • All the local rings of X are DVR(=discrete valuation ring)s. • X is normal. Remark 2. As a final remark, let X be a hypersurface in An with IX = (f ). Let x ∈ X. Then X is ∂fi smooth at x if and only if IX is locally generated by some f1 , · · · , fm such that rank( ) = m. This is also ∂xj n ∼ [ equivalent to saying that O X,x := lim OX,x /mx = k [[x1 , · · · xm ]]. ←− n 2