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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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Constructing a degeneration (as a group scheme) of G_m to G_a
One more point of view : if X is a 2x2 matrix, its centralizer in PGL(2) is -- for generic X -- isomorphic to Gm; however, it is isomorphic to Ga if X is nonzero and nilpotent. This gives a typical …