Injective morphisms

Abstract

Let k be any field. Then a well-known result from linear algebra asserts that if ϕ: V ← W is an injective linear map between two finite dimensional vector spaces of the same dimension, then ϕ is surjective and the inverse map is again k-linear, i.e. a morphism in the category of vector spaces over k. In this chapter we will investigate to what extent this result can be generalized to the category of affine algebraic sets.