By H. A. Nielsen

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2. The determinantal variety is normal. If m, n ≥ r > 1 then 0 is a singular point. 21. 1. Assume char(k) = 0 and let f : An → An be a morphism with Jacobian determinant Jf = det ∂fi ∂Xj Is f an isomorphism? 2. The question is unanswered even in case f : C2 → C2 ∈ k∗ Bibliography A. Altman and S. Kleiman, Introduction to Grothendieck duality theory, Springer-Verlag 1970. M. Atiyah and I. Macdonald, An introduction to commutative algebra, Addison-Wesley 1969. N. Bourbaki, Algébre commutative, Hermann-Masson 1961-.

PROJECTIVE LINE 53 9. 1. A group G which is an algebraic variety such that composition G × G → G and inversion G → G are morphisms is called an algebraic group. 2. The connected component Ge containing the unit is irreducible and again an algebraic group. Let G1 . . Gs be multiplication of the irreducible components containing e. The image contains all Gi so s = 1 and G1 is a subgroup. The cosets xG is an irreducible component containing x, so there are only finitely many cosets. These being disjoint gives Ge = G1 .

5. Let X = V (I) ⊂ An be an affine set. For x ∈ X and f ∈ I the differential n df = 1 ∂f (x)(Xi − xi ) ∂Xi is a linear polynomial in k[X1 , . . , Xn ]. The (embedded) tangent space to X at x is the affine linear subspace Tx X = V ({df |f ∈ I}) Let X = V (I) ⊂ Pn be a projective set. For x ∈ X and F ∈ I homogeneous the differential n ∂f dF = (x)Xi ∂X i 0 is a homogeneous polynomial in k[X0 , . . , Xn ]. The projective (embedded) tangent space to X at x is the projective linear subspace Tx X = V ({dF |F ∈ I}) n ∂f 0 ∂Xi Xi = deg(F ) F it follows that the restriction of the projective By Euler’s formula tangent space to an open affine coordinate space in Pn is identified with the affine tangent space.