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At a point on a differentiable manifold, it is a vector space that consists of the tangent vectors to the manifold at that point.

A geometric object defined as the set of solutions to a system of polynomial equations.

A structure-preserving map between two algebraic objects, like varieties or schemes.

A map between varieties defined by ratios of polynomial functions.

An advanced mathematical structure that enlarges the notion of algebraic variety to include multiplicities and 'bad' intersection points.

A map between two algebraic structures of the same type that respects the structure.

A set of polynomials within a ring that is closed under addition and under multiplication by any polynomial in the ring.

A generalization of the concept of vector space over a field, where the field is replaced with a ring.

An algebraic structure consisting of a set equipped with two binary operations that generalize the arithmetic operations of addition and multiplication.