Modern generalizations of algebraic varieties Scheme (mathematics)

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alexander grothendieck gave decisive definition, bringing conclusion generation of experimental suggestions , partial developments. defined spectrum of commutative ring space of prime ideals zariski topology, augments sheaf of rings: every zariski-open set assigns commutative ring, thought of ring of polynomial functions defined on set. these objects affine schemes; general scheme obtained gluing several such affine schemes, in analogy fact general varieties can obtained gluing affine varieties.


the generality of scheme concept criticized: schemes removed having straightforward geometrical interpretation, made concept difficult grasp. however, admitting arbitrary schemes makes whole category of schemes better-behaved. moreover, natural considerations regarding, example, moduli spaces, lead schemes non-classical . occurrence of these schemes not varieties (nor built varieties) in problems posed in classical terms made gradual acceptance of new foundations of subject.


subsequent work on algebraic spaces , algebraic stacks deligne, mumford, , michael artin, in context of moduli problems, has further enhanced geometric flexibility of modern algebraic geometry. grothendieck advocated types of ringed toposes generalisations of schemes, , following proposals relative schemes on ringed toposes developed m. hakim. recent ideas higher algebraic stacks , homotopical or derived algebraic geometry have regard further expanding algebraic reach of geometric intuition, bringing algebraic geometry closer in spirit homotopy theory.







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