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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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Can we classify reductive group schemes over curves

Can one classify all reductive group schemes over $C$? Certainly, you have the trivial ones (coming from pulling-back those living over the complex numbers). … I would already be very happy to see some "non-trivial" examples of reductive group schemes over the affine line. …