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 MM^t = 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.