What would you like to understand?

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 $x^2 = 0$ 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 $x^2 = 0$ 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.