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A homomorphism of algebraic groups Φ : G → G, such that d1 (Φ) : T1 (G) → T1 (G) is nilpotent, is called a Frobenius endomorphism on G. If H ≤ G is a closed subgroup which is Φ-invariant, i. e. we have Φ(V ) ⊆ V , then the restriction of Φ|H is a Frobenius endomorphism on H. b) Let char(K) = p > 0 and q := pf for some f ∈ N. Then Φq : Kn → Kn : [x1 , . . , xn ] → [xq1 , . . , xqn ] is called the associated geometric Frobenius morphism on Kn . Hence the set of fixed points (Kn )Φq := {x ∈ Kn ; Φq (x) = x} = Fnq coincides with the finite set of Fq -rational points of Kn .

B) Show that the localisation RU is Noetherian. c) Show that dim(RU ) = sup{di − di−1 ; i ∈ N} ∈ N ∪ {∞}. Proof. 6]. 15) Exercise: Dimension and height. Give an example of a finitely generated K-algebra, where K is a field, which is not a domain, possessing an ideal I R such that dim(I) + ht(I) = dim(R). 16) Exercise: Catenary rings. A finite dimensional Noetherian ring R is called catenary, if for any prime ideals P ⊆ Q R all maximal chains P = P0 ⊂ · · · ⊂ Pr = Q of prime ideals have length r = ht(Q) − ht(P ).

Let V ⊆ Kn be closed, let 0 = f ∈ K[V ] and let x ∈ Vf . Using the closed embedding Vf → Kn+1 : y → [y, f (y)−1 ] give a definition of a Zariski tangent space Tx (Vf ), and show that it can be naturally identified with Tx (V ). 30) Exercise: Regular points. Let V be an irreducible affine variety over K. a) Show that for any x ∈ V we have dimK (Tx (V )) ≥ dim(V ). b) Show that the set of regular points is an open subset of V . Hint for (a). Consider the local ring Ox associated to x, and by using the Nakayama Lemma show that any subset S ⊆ Px generates the maximal ideal Px as an Ox -module if and only if it generates Px /Px2 as a K-vector space.

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Algebraic Groups by Jürgen Müller


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