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ringed space wikipedia jump to content main menu main menu move to sidebar hide navigation main page contents current events random article about wikipedia contact us contribute help learn to edit community portal recent changes upload file special pages search search appearance donate create account log in personal tools donate create account log in contents move to sidebar hide top 1 definitions 2 examples 3 morphisms 4 tangent spaces 5 modules over the structure sheaf 6 citations 7 references 8 external links toggle the table of contents ringed space 12 languages العربية deutsch ελληνικά فارسی עברית 한국어 македонски မြန်မာဘာသာ русский українська tiếng việt 中文 edit links article talk english read edit view history tools tools move to sidebar hide actions read edit view history general what links here related changes upload file permanent link page information cite this page get shortened url switch to legacy parser print export download as pdf printable version in other projects wikidata item appearance move to sidebar hide from wikipedia the free encyclopedia redirected from structure sheaf sheaf of rings in mathematics in mathematics a ringed space is a family of commutative rings parametrized by open subsets of a topological space together with ring homomorphisms that play roles of restrictions precisely it is a topological space equipped with a sheaf of rings called a structure sheaf it is an abstraction of the concept of the rings of continuous scalar valued functions on open subsets among ringed spaces especially important and prominent is a locally ringed space a ringed space in which the analogy between the stalk at a point and the ring of germs of functions at a point is valid ringed spaces appear in analysis as well as complex algebraic geometry and the scheme theory of algebraic geometry note in the definition of a ringed space most expositions tend to restrict the rings to be commutative rings including hartshorne and wikipedia éléments de géométrie algébrique on the other hand does not impose the commutativity assumption although the book mostly considers the commutative case 1 definitions edit a ringed space x o x displaystyle x mathcal o _ x is a topological space x displaystyle x together with a sheaf of rings o x displaystyle mathcal o _ x on x displaystyle x the sheaf o x displaystyle mathcal o _ x is called the structure sheaf of x displaystyle x a locally ringed space is a ringed space x o x displaystyle x mathcal o _ x such that all stalks of o x displaystyle mathcal o _ x are local rings i e they have unique maximal ideals note that it is not required that o x u displaystyle mathcal o _ x u be a local ring for every open set u displaystyle u in fact this is almost never the case examples edit an arbitrary topological space x displaystyle x can be considered a locally ringed space by taking o x displaystyle mathcal o _ x to be the sheaf of real valued or complex valued continuous functions on open subsets of x displaystyle x the stalk at a point x displaystyle x can be thought of as the set of all germs of continuous functions at x displaystyle x this is a local ring with the unique maximal ideal consisting of those germs whose value at x displaystyle x is 0 displaystyle 0 if x displaystyle x is a manifold with some extra structure we can also take the sheaf of differentiable or holomorphic functions both of these give rise to locally ringed spaces if x displaystyle x is an algebraic variety carrying the zariski topology we can define a locally ringed space by taking o x u displaystyle mathcal o _ x u to be the ring of rational mappings defined on the zariski open set u displaystyle u that do not blow up become infinite within u displaystyle u the important generalization of this example is that of the spectrum of any commutative ring these spectra are also locally ringed spaces schemes are locally ringed spaces obtained by gluing together spectra of commutative rings morphisms edit a morphism from x o x displaystyle x mathcal o _ x to y o y displaystyle y mathcal o _ y is a pair f φ displaystyle f varphi where f x y displaystyle f x to y is a continuous map between the underlying topological spaces and φ o y f o x displaystyle varphi mathcal o _ y to f_ mathcal o _ x is a morphism from the structure sheaf of y displaystyle y to the direct image of the structure sheaf of x in other words a morphism from x o x displaystyle x mathcal o _ x to y o y displaystyle y mathcal o _ y is given by the following data a continuous map f x y displaystyle f x to y a family of ring homomorphisms φ v o y v o x f 1 v displaystyle varphi _ v mathcal o _ y v to mathcal o _ x f 1 v for every open set v displaystyle v of y displaystyle y that commute with the restriction maps that is if v 1 v 2 displaystyle v_ 1 subseteq v_ 2 are two open subsets of y displaystyle y then the following diagram must commute the vertical maps are the restriction homomorphisms there is an additional requirement for morphisms between locally ringed spaces the ring homomorphisms induced by φ displaystyle varphi between the stalks of y displaystyle y and the stalks of x displaystyle x must be local homomorphisms i e for every x x displaystyle x in x the maximal ideal of the local ring stalk at f x y displaystyle f x in y is mapped into the maximal ideal of the local ring at x x displaystyle x in x two morphisms can be composed to form a new morphism and we obtain the category of ringed spaces and the category of locally ringed spaces isomorphisms in these categories are defined as usual tangent spaces edit see also zariski tangent space locally ringed spaces have just enough structure to allow the meaningful definition of tangent spaces let x displaystyle x be a locally ringed space with structure sheaf o x displaystyle mathcal o _ x we want to define the tangent space t x x displaystyle t_ x x at the point x x displaystyle x in x take the local ring stalk r x displaystyle r_ x at the point x displaystyle x with maximal ideal m x displaystyle mathfrak m _ x then k x r x m x displaystyle k_ x r_ x mathfrak m _ x is a field and m x m x 2 displaystyle mathfrak m _ x mathfrak m _ x 2 is a vector space over that field the cotangent space the tangent space t x x displaystyle t_ x x is defined as the dual of this vector space the idea is the following a tangent vector at x displaystyle x should tell you how to differentiate functions at x displaystyle x i e the elements of r x displaystyle r_ x now it is enough to know how to differentiate functions whose value at x displaystyle x is zero since all other functions differ from these only by a constant and we know how to differentiate constants so we only need to consider m x displaystyle mathfrak m _ x furthermore if two functions are given with value zero at x displaystyle x then their product has derivative 0 at x displaystyle x by the product rule so we only need to know how to assign numbers to the elements of m x m x 2 displaystyle mathfrak m _ x mathfrak m _ x 2 and this is what the dual space does modules over the structure sheaf edit main article sheaf of modules given a locally ringed space x o x displaystyle x mathcal o _ x certain sheaves of modules on x displaystyle x occur in the applications the o x displaystyle mathcal o _ x modules to define them consider a sheaf f displaystyle mathcal f of abelian groups on x displaystyle x if f u displaystyle mathcal f u is a module over the ring o x u displaystyle mathcal o _ x u for every open set u displaystyle u in x displaystyle x and the restriction maps are compatible with the module structure then we call f displaystyle mathcal f an o x displaystyle mathcal o _ x module in this case the stalk of f displaystyle mathcal f at x displaystyle x will be a module over the local ring stalk r x displaystyle r_ x for every x x displaystyle x in x a morphism between two such o x displaystyle mathcal o _ x modules is a morphism of sheaves that is compatible with the given module structures the category of o x displaystyle mathcal o _ x modules over a fixed locally ringed space x o x displaystyle x mathcal o _ x is an abelian category an important subcategory of the category of o x displaystyle mathcal o _ x modules is the category of quasi coherent sheaves on x displaystyle x a sheaf of o x displaystyle mathcal o _ x modules is called quasi coherent if it is locally isomorphic to the cokernel of a map between free o x displaystyle mathcal o _ x modules a coherent sheaf f displaystyle f is a quasi coherent sheaf that is locally of finite type and for every open subset u displaystyle u of x displaystyle x the kernel of any morphism from a free o u displaystyle mathcal o _ u module of finite rank to f u displaystyle mathcal f _ u is also of finite type citations edit éléments de géométrie algébrique ch 0 4 1 1 references edit section 0 4 of grothendieck alexandre dieudonné jean 1960 éléments de géométrie algébrique i le langage des schémas publications mathématiques de l ihés 4 doi 10 1007 bf02684778 mr 0217083 hartshorne robin 1977 algebraic geometry graduate texts in mathematics vol 52 new york springer verlag isbn 978 0 387 90244 9 mr 0463157 external links edit onishchik a l 2001 1994 ringed space encyclopedia of mathematics ems press retrieved from https en wikipedia org w index php title ringed_space oldid 1255289727 categories sheaf theory scheme theory hidden categories articles with short description short description is different from wikidata this page was last edited on 4 november 2024 at 03 46 utc page was rendered with parsoid text is available under the creative commons attribution sharealike 4 0 license additional terms may apply by using this site you agree to the terms of use and privacy policy wikipedia is a registered trademark of the wikimedia foundation inc a non profit organization privacy policy about wikipedia disclaimers contact wikipedia legal safety contacts code of conduct 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