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The first purpose of schemes theory is the geometrical study of solutions of algebraic systems of equations, not only over the real/complex numbers, but also over integer numbers (and more generally over any commutative ring with 1). It was finalized by Alexandre Grothendieck, during the 1950s and the 1960s.

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characterisation of regular morphisms

The theorem is actually true for morphisms between Noetherian schemes, no base field required. As you probably expect, this paper makes heavy use of simplicial module categories. …
Eric Canton's user avatar