Workshop on group schemes and p-divisible groups: Homework 1. 1
... 8. Let k be a field, k/k an algebraic closure, and G a locally finite type k-group. (i) Prove that a group scheme is separated if and only if its identity section is a closed immersion (Hint: identity section is base change of the diagonal); deduce that G is k-separated. (ii) If Gk is smooth, prove ...
... 8. Let k be a field, k/k an algebraic closure, and G a locally finite type k-group. (i) Prove that a group scheme is separated if and only if its identity section is a closed immersion (Hint: identity section is base change of the diagonal); deduce that G is k-separated. (ii) If Gk is smooth, prove ...
Prime Spacing and the Hardy-Littlewood
... Proof of a): If p > 2 then of the integers 0,1,2, …, p a-1 exactly pa-1 integers are not relatively prime to pa, namely 0, p, 2p, … Each of these kills 2 arithmetic progressions, and so 2(pa) = pa – 2pa-1 = pa-1(p – 2). If p = 2, then each multiple of p only kills one pair of arithmetic progression ...
... Proof of a): If p > 2 then of the integers 0,1,2, …, p a-1 exactly pa-1 integers are not relatively prime to pa, namely 0, p, 2p, … Each of these kills 2 arithmetic progressions, and so 2(pa) = pa – 2pa-1 = pa-1(p – 2). If p = 2, then each multiple of p only kills one pair of arithmetic progression ...
