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I downloaded a model from blendswap.com and when I rendered it I got this - enter image description here

The scene is made up of: the model, plane, hdr map and hemi lamp that is lighting the object. As you can see, the ground plane is not effected by the light. Its node setup is this - enter image description here

Why is the plane not effected by the light? I even replaced it with other lamps but the effect is the same. Why? How is that done?

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This is simply done by setting the color to black. The light of a diffuse material with a roughness of 0 is calculated using the Lambert lighting model.
This model calculates the light that an object reflects as follows:

Lambertian lighting

where I_d is the intensity of the diffusely reflected light, C is the material's color and I_l is the intensity of the incoming light, N is the surface's normal vector and L is the normalized vector from the surface point to the light source.

For typical colors C this will give you light areas where the surface faces the light source and darker areas where the surface points away from the light source. But since in your material the color is a pure Black (RGB (0,0,0)) the red green an blue light component will always be multiplied by 0. Thus the resulting surface color will always be black as well, no matter how strong the lightsource is.

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