Affine Group Schemes

Abstract

If R is any ring (commutative with 1), the 2 × 2 matrices with entries in R and determinant 1 form a groupSL2(R) under matrix multiplication. This is a familiar process for constructing a group from a ring. Another such process is GL2, where GL2(R) is the group of all 2 × 2 matrices with invertible determinant. Similarly we can form SL n and GL n . In particular there is GL1, denoted by the special symbol G m ; this is the multiplicative group, with G m (R) the set of invertible elements of R. It suggests the still simpler example G a , the additive group: G a (R) is just R itself under addition. Orthogonal groups are another common type; we can, for instance, get a group by taking all 2 ×2 matrices M over R satisfying MMt = I. A little less familiar isµ n , the nth roots of unity: if we set µ n (R) = {x ∈ R|xn = 1 }, we get a group under multiplication. All these are examples of affinc group schcmcs.