An affine scheme is a locally ringed space that is isomorphic to the spectrum Spec(R) of a commutative ring R. General schemes are constructed by gluing together open sets of this simple kind, called affine schemes.
Conditions: The object is an affine scheme.; It is a locally ringed space.; It is isomorphic to the spectrum of a commutative ring.
An affine scheme is a locally ringed space that is isomorphic to the spectrum Spec(R) of a commutative ring R. General schemes are constructed by gluing together open sets of this simple kind, called affine schemes.
Conditions: The object is an affine scheme.; It is a locally ringed space.; It is isomorphic to the spectrum of a 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.
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.