In algebraic geometry and commutative algebra, the theorems of generic flatness and generic freeness state that under certain hypotheses, a sheaf of modules on a scheme is flat or free. They are due to Alexander Grothendieck.
Generic flatness states that if Y is an integral locally noetherian scheme, u : X → Y is a finite type morphism of schemes, and F is a coherent OX-module, then there is a non-empty open subset U of Y such that the restriction of F to u−1(U) is flat over U.[1]
Because Y is integral, U is a dense open subset of Y. This can be applied to deduce a variant of generic flatness which is true when the base is not integral.[2] Suppose that S is a noetherian scheme, u : X → S is a finite type morphism, and F is a coherent OX module. Then there exists a partition of S into locally closed subsets S1, ..., Sn with the following property: Give each Si its reduced scheme structure, denote by Xi the fiber product X ×SSi, and denote by Fi the restriction F ⊗OSOSi; then each Fi is flat.
called faithfully flat if it is both surjective and flat. Two basic intuitions regarding flat morphisms are: flatness is a generic property; and the failure...
study of flatness to the case of local rings. They are often expressed by saying that flatness is a local property. The definition of a flat morphism...
three lists of generic and genericized trademarks are: marks which were originally legally protected trademarks, but have been genericized and have lost...
Flagstone (flag) is a genericflat stone, sometimes cut in regular rectangular or square shape and usually used for paving slabs or walkways, patios, flooring...
Koszul complex Spectrum of a ring Zariski topology Kähler differential Genericflatness Irrelevant ideal Locally ringed space Coherent sheaf Invertible sheaf...
August 2018. {{cite web}}: |first= has generic name (help) Midgley, Carol (28 June 2018). "TV review: Stath Lets Flats; The People Vs The NHS: Who Gets the...
geometry and computational geometry, general position is a notion of genericity for a set of points, or other geometric objects. It means the general...
place of one, the singular impersonal pronoun, in colloquial speech. The generic you is primarily a colloquial substitute for one. For instance, "Brushing...
lights, but are not often eaten because of their green-colored flesh. The generic name Ablennes – formerly misspelled Athlennes – means ‘without mucosity’...
the foot (rolling inwards), although the term 'flat foot' is still applicable as it is a somewhat generic term. Muscular training of the feet is helpful...
Bumper cars or dodgems are the generic names for a type of flat amusement ride consisting of multiple small electrically powered cars which draw power...
Stock pot is a generic name for one of the most common types of cooking pot used worldwide. A stock pot is traditionally used to make stock or broth,...
represented on all continents except Antarctica, and is most diverse at both the generic and species levels in the Old World tropics. Silvanids generally are small...
type of container, while plain text refers to a type of content. At a generic level of description, there are two kinds of computer files: text files...
storeys tall, and with outdoor parking space, while apartment is more generic and may also include luxury condominiums. In Japanese English loanwords...
Filters may be contained in cartridges or canisters; there are also flexible flat and pancake-style filters (see illustrations). Filters are usually mounted...
discrete valuation ring whose spectrum has one closed point and one open (generic) point. We glue these spectra together by identifying their open points...
term "bubble wrap" is owned by Sealed Air Corporation, but has become a generic trademark. Similar product names include bubble pack, air bubble packing...
weakening an assumption to the finite case (e.g., the characterization of flatness with the Tor functor). An example of a link between finite generation and...
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