Scheme theory enlarges the notion of an algebraic variety by taking into account multiplicities and allowing varieties to be defined over any commutative ring, not just algebraically closed fields. For example, the equations x=0 and x2=0 define the same algebraic variety but different schemes.
Conditions: The context is the motivation for scheme theory over classical algebraic geometry.; Schemes account for multiplicities.; Schemes can be defined over any commutative ring.
Scheme theory enlarges the notion of an algebraic variety by taking into account multiplicities and allowing varieties to be defined over any commutative ring, not just algebraically closed fields. For example, the equations x=0 and x2=0 define the same algebraic variety but different schemes.
Conditions: The context is the motivation for scheme theory over classical algebraic geometry.; Schemes account for multiplicities.; Schemes can be defined over any commutative ring.
A scheme is defined as a topological space together with commutative rings for all of its open sets, which arises from gluing together spectra of commutative rings along their open subsets. Equivalently, a scheme is a ringed space that is locally a spectrum of a commutative ring, or locally isomorphic to an affine space.
Conditions: It involves a topological space.; It involves commutative rings assigned to open sets.; It is constructed by gluing spectra of commutative rings.; It is a ringed space locally isomorphic to an affine space.
A scheme is defined as a topological space together with commutative rings for all of its open sets, which arises from gluing together spectra of commutative rings along their open subsets. Equivalently, a scheme is a ringed space that is locally a spectrum of a commutative ring, or locally isomorphic to an affine space.
Conditions: It involves a topological space.; It involves commutative rings assigned to open sets.; It is constructed by gluing spectra of commutative rings.; It is a ringed space locally isomorphic to an affine space.
Scheme theory allows one to use sheaf theory to study algebraic varieties in a way which is very similar to its use in the study of differential and analytic manifolds. A scheme can be expressed in terms of a sheaf of rings on the space.
Conditions: The context is algebraic geometry.; A scheme is expressed in terms of a sheaf of rings on the space.
Scheme theory allows one to use sheaf theory to study algebraic varieties in a way which is very similar to its use in the study of differential and analytic manifolds. A scheme can be expressed in terms of a sheaf of rings on the space.
Conditions: The context is algebraic geometry.; A scheme is expressed in terms of a sheaf of rings on the space.
Alexander Grothendieck introduced schemes in 1960 with the objective of developing the formalism needed to solve deep problems of algebraic geometry, such as the Weil conjectures.
Conditions: The context is the history of algebraic geometry in the 1960s.; The motivation is tied to solving specific deep problems like the Weil conjectures.
Alexander Grothendieck introduced schemes in 1960 with the objective of developing the formalism needed to solve deep problems of algebraic geometry, such as the Weil conjectures.
Conditions: The context is the history of algebraic geometry in the 1960s.; The motivation is tied to solving specific deep problems like the Weil conjectures.