geometry, the fiber product of schemes is a fundamental construction. It has many interpretations and special cases. For example, the fiber product describes...

of mathematics, a pullback (also called a fiber product, fibre product, fibered product or Cartesian square) is the limit of a diagram consisting of two...

transformations, for purposes of coding theory and cryptography. Construction of the fiber product of schemes, in algebraic geometry. Change of basis Base change...

f:X\to Y} is a morphism of schemes, the fiber of a point p {\displaystyle p} in Y {\displaystyle Y} is the fiber product of schemes X × Y Spec ⁡ k ( p )...

and take an appropriate fiber product of schemes, with the morphism e that serves as the identity. The resulting group scheme is written μn (or μ μ n...

as schemes over Spec(k). To every algebraic extension L of k, the L-algebraic set associated to a given k-algebraic set V is the fiber product of schemes...

generally, the fiber product of schemes is defined by gluing together affine fiber products of this form. The tensor product can be used as a means of taking...

geometry Fiber product of schemes in algebraic geometry Change of base (disambiguation) in linear algebra or numeral systems Base change lifting of automorphic...

of Y, such that f restricted to Y′ is a flat morphism (generic flatness). Here 'restriction' is interpreted by means of the fiber product of schemes,...

of points. The fiber product of schemes always exists. That is, for any schemes X and Z with morphisms to a scheme Y, the categorical fiber product X...

scheme Scheme Éléments de géométrie algébrique Grothendieck's Séminaire de géométrie algébrique Fiber product of schemes Flat morphism Smooth scheme Finite...

topos. This article uses terminology from category theory. Fiber product of schemes "Base change", Encyclopedia of Mathematics, EMS Press, 2001 [1994]...

coordinate-free definition. As an example of the second situation, a common way of constructing the fiber product of schemes involves gluing along affine patches...

adjoint to fiber product of schemes, so the above definition can be rephrased in much more generality. In particular, one can replace the extension of fields...

of schemes that has fiber products and some final object S. That is, it is an S-scheme G equipped with one of the equivalent sets of data a triple of...

light (for illumination or data) through the core of the fiber. Its chief advantage over the glass product, other aspect being equal, is its robustness under...

called the complete linear system of D. Projective space over any scheme S can be defined as a fiber product of schemes P S n = P Z n × Spec ⁡ Z S . {\displaystyle...

the extension from affine schemes, where it is just the tensor product of R-algebras, to all schemes of the fiber product operation a significant (if...

EIA/TIA-598, "Optical Fiber Cable Color Coding" which defines identification schemes for fibers, buffered fibers, fiber units, and groups of fiber units within...

constructed by gluing affine schemes. For affine schemes, fiber products and pushouts correspond to tensor products and fiber squares of algebras. Hartshorne...

algebro-geometric context comes from the topological structure of the fiber product (in the category of schemes) X × Spec ( Z ) X {\displaystyle X\times _{{\textrm...

A fiber laser (or fibre laser in Commonwealth English) is a laser in which the active gain medium is an optical fiber doped with rare-earth elements such...

Product Sans is a contemporary geometric sans-serif typeface created by Google for branding purposes. It replaced the old Google logo on September 1,...

the relative dimension of a morphism of schemes is also important. By definition, the dimension of a scheme X is the dimension of the underlying topological...

Younique is an American Pyramid Scheme that markets beauty products. The company was founded in 2012 by Derek Maxfield, who is CEO, and Melanie Huscroft...

asbestos-cement products that were reinforced with asbestos fibers have developed products incorporating organic fibers. One such product was known as "Eternit"...

A staple fiber is a textile fiber of discrete length. The opposite is a filament fiber, which comes in continuous lengths. Staple length is a characteristic...

the category of all schemes over some fixed scheme equipped with some Grothendieck topology. Let F be a category and suppose it is fibered over C through...

functor and thus, unless higher Tor vanish, the scheme-theoretic intersection (i.e., fiber product of immersions) does not yield the correct intersection...

ringed space isomorphic to one of this form is called an affine scheme. General schemes are obtained by gluing affine schemes together. Similarly, for a module...