Sheaf theory was developed in mathematics to study the relationships between local and global phenomena. It has been applied in algebraic geometry, differential geometry, analysis, and even logic.
Conditions: The context is the general mathematical purpose of sheaf theory.
Sheaf theory was developed in mathematics to study the relationships between local and global phenomena. It has been applied in algebraic geometry, differential geometry, analysis, and even logic.
Conditions: The context is the general mathematical purpose of sheaf theory.
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.