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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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Making the étale topos construction a fully faithful 2-functor from schemes to Grothendieck ...

In particular the section "Exodromy for schemes & the Reconstruction Theorem". …
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Cartier and the continuity of the early history of schemes

I would say that the first written record of schemes (a la Grothendieck) was his talk at the ICM Edinburgh in 1958[3] (you can tell me a counterexample). … But my question is There is a published record of schemes (a la Cartier) of those times? …
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