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.
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.