Here we talk about the subtle property of schemes called dimension. We first try to motivate the definition and then we make attempts to define the notion.

Our intuition is derived from the theory of Manifolds, where we define the dimension of a real manifolds at a point to be the number of free variables locally at , or equivalently the dimension of the tangent space at . Similarly given a complex manifold we can define the complex dimension at to be the dimension of the tangent space at . Therefore if we define the dimension of a complex variety to the dimension of the tangent space, it does not work out because generally the complex variety has lot of singularities and therefore the tangent space defined does not realize our intuition. This is the first difficulty one faces. So we cannot go to tangent spaces, therefore we look at the stalk at itself. Given a complex manifold we have a ring of (real or complex) functions on it. Therefore we can localize and define the local ring(or the stalk) at to the ring of germs of functions at . This does indeed seem to perform better than the tangent space. This can be shown by a few examples. Let be the curve in the complex plane(). This curve has a singularity at the origin and the tangent space is two dimensional at that point. However the local ring at is where . We can show that this ring is one dimensional and therefore captures our intuition better.