And today, we start intersection theory. So, to establish notation a bit, we’re only going to be talking about algebraic schemes. These are separated schemes of finite type over our base field 
. That is, they admit finite open affine covers such that each affine is the spectrum of a finitely generated -algebra. We’re already almost to varieties, the only thing left is to assume reduced and irreducible (equivalently, integral) to get there, but we won’t do that unless necessary. Nor will we assume smooth if we don’t have to, so we’re hoping to get a theory that will work, at least somewhat, for singular varieties.
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