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129-439: In this article, all rings will be assumed to be commutative and with identity. Let S {\displaystyle S} be a commutative graded ring , where S = ⨁ i ≥ 0 S i {\displaystyle S=\bigoplus _{i\geq 0}S_{i}} is the direct sum decomposition associated with the gradation. The irrelevant ideal of S {\displaystyle S}

258-528: A ) {\displaystyle V(a)} form the closed sets of a topology on X {\displaystyle X} . Indeed, if ( a i ) i ∈ I {\displaystyle (a_{i})_{i\in I}} are a family of ideals, then we have ⋂ V ( a i ) = V ( ∑ a i ) {\textstyle \bigcap V(a_{i})=V\left(\sum a_{i}\right)} and if

387-413: A + b ) {\displaystyle (a+b)} element. This means the k {\displaystyle k} -th graded piece of S ∙ {\displaystyle S_{\bullet }} is the module S k = ⨁ a + b = k S a , b {\displaystyle S_{k}=\bigoplus _{a+b=k}S_{a,b}} In addition,

516-410: A = a a for all m , n ≥ 0 . A left zero divisor of a ring R is an element a in the ring such that there exists a nonzero element b of R such that ab = 0 . A right zero divisor is defined similarly. A nilpotent element is an element a such that a = 0 for some n > 0 . One example of a nilpotent element is a nilpotent matrix . A nilpotent element in a nonzero ring

645-489: A 1 for a "ring". Starting in the 1960s, it became increasingly common to see books including the existence of 1 in the definition of "ring", especially in advanced books by notable authors such as Artin, Bourbaki, Eisenbud, and Lang. There are also books published as late as 2022 that use the term without the requirement for a 1 . Likewise, the Encyclopedia of Mathematics does not require unit elements in rings. In

774-430: A base scheme . Formally, let X be any scheme and S be a sheaf of graded O X {\displaystyle O_{X}} -algebras (the definition of which is similar to the definition of O X {\displaystyle O_{X}} -modules on a locally ringed space ): that is, a sheaf with a direct sum decomposition where each S i {\displaystyle S_{i}}

903-541: A geometric object that are preserved under continuous deformations , such as stretching , twisting , crumpling, and bending; that is, without closing holes, opening holes, tearing, gluing, or passing through itself. A topological space is a set endowed with a structure, called a topology , which allows defining continuous deformation of subspaces, and, more generally, all kinds of continuity . Euclidean spaces , and, more generally, metric spaces are examples of topological spaces, as any distance or metric defines

1032-493: A multiplicative inverse . In 1921, Emmy Noether gave a modern axiomatic definition of commutative rings (with and without 1) and developed the foundations of commutative ring theory in her paper Idealtheorie in Ringbereichen . Fraenkel's axioms for a "ring" included that of a multiplicative identity, whereas Noether's did not. Most or all books on algebra up to around 1960 followed Noether's convention of not requiring

1161-452: A ring is a set equipped with two binary operations satisfying properties analogous to those of addition and multiplication of integers . Ring elements may be numbers such as integers or complex numbers , but they may also be non-numerical objects such as polynomials , square matrices , functions , and power series . Formally, a ring is a set endowed with two binary operations called addition and multiplication such that

1290-416: A robot can be described by a manifold called configuration space . In the area of motion planning , one finds paths between two points in configuration space. These paths represent a motion of the robot's joints and other parts into the desired pose. Disentanglement puzzles are based on topological aspects of the puzzle's shapes and components. In order to create a continuous join of pieces in

1419-464: A scheme . As in the case of the Spec construction there are many ways to proceed: the most direct one, which is also highly suggestive of the construction of regular functions on a projective variety in classical algebraic geometry, is the following. For any open set U {\displaystyle U} of Proj ⁡ S {\displaystyle \operatorname {Proj} S} (which

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1548-497: A smooth structure on a manifold to be defined. Smooth manifolds are "softer" than manifolds with extra geometric structures, which can act as obstructions to certain types of equivalences and deformations that exist in differential topology. For instance, volume and Riemannian curvature are invariants that can distinguish different geometric structures on the same smooth manifold – that is, one can smoothly "flatten out" certain manifolds, but it might require distorting

1677-449: A topology , called the Zariski topology , on Proj ⁡ S {\displaystyle \operatorname {Proj} S} by defining the closed sets to be those of the form where a {\displaystyle a} is a homogeneous ideal of S {\displaystyle S} . As in the case of affine schemes it is quickly verified that the V (

1806-731: A canonical projective morphism to the affine line A λ 1 {\displaystyle \mathbb {A} _{\lambda }^{1}} whose fibers are elliptic curves except at the points λ = 0 , 1 {\displaystyle \lambda =0,1} where the curves degenerate into nodal curves. So there is a fibration E λ ⟶ X ↓ A λ 1 − { 0 , 1 } {\displaystyle {\begin{matrix}E_{\lambda }&\longrightarrow &X\\&&\downarrow \\&&\mathbb {A} _{\lambda }^{1}-\{0,1\}\end{matrix}}} which

1935-421: A circle from two non-intersecting circles. The ideas underlying topology go back to Gottfried Wilhelm Leibniz , who in the 17th century envisioned the geometria situs and analysis situs . Leonhard Euler 's Seven Bridges of Königsberg problem and polyhedron formula are arguably the field's first theorems. The term topology was introduced by Johann Benedict Listing in the 19th century, although, it

2064-412: A convenient proof that any subgroup of a free group is again a free group. Differential topology is the field dealing with differentiable functions on differentiable manifolds . It is closely related to differential geometry and together they make up the geometric theory of differentiable manifolds. More specifically, differential topology considers the properties and structures that require only

2193-509: A few authors who use the term "ring" to refer to structures in which there is no requirement for multiplication to be associative. For these authors, every algebra is a "ring". The most familiar example of a ring is the set of all integers ⁠ Z , {\displaystyle \mathbb {Z} ,} ⁠ consisting of the numbers The axioms of a ring were elaborated as a generalization of familiar properties of addition and multiplication of integers. Some basic properties of

2322-421: A fixed set of lower powers, and thus the powers "cycle back". For instance, if a − 4 a + 1 = 0 then: and so on; in general, a is going to be an integral linear combination of 1 , a , and a . The first axiomatic definition of a ring was given by Adolf Fraenkel in 1915, but his axioms were stricter than those in the modern definition. For instance, he required every non-zero-divisor to have

2451-396: A further construction. Over each open affine U , Proj S ( U ) bears an invertible sheaf O(1) , and the assumption we have just made ensures that these sheaves may be glued just like the Y U {\displaystyle Y_{U}} above; the resulting sheaf on P r o j ⁡ S {\displaystyle \operatorname {\mathbf {Proj} } S}

2580-426: A given space. Changing a topology consists of changing the collection of open sets. This changes which functions are continuous and which subsets are compact or connected. Metric spaces are an important class of topological spaces where the distance between any two points is defined by a function called a metric . In a metric space, an open set is a union of open disks, where an open disk of radius r centered at x

2709-420: A homeomorphism and the domain of the function is said to be homeomorphic to the range. Another way of saying this is that the function has a natural extension to the topology. If two spaces are homeomorphic, they have identical topological properties, and are considered topologically the same. The cube and the sphere are homeomorphic, as are the coffee cup and the doughnut. However, the sphere is not homeomorphic to

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2838-503: A limited sense (for example, spy ring), so if that were the etymology then it would be similar to the way "group" entered mathematics by being a non-technical word for "collection of related things". According to Harvey Cohn, Hilbert used the term for a ring that had the property of "circling directly back" to an element of itself (in the sense of an equivalence ). Specifically, in a ring of algebraic integers, all high powers of an algebraic integer can be written as an integral combination of

2967-847: A polynomial ring whose variables have non-standard degrees. For example, the weighted projective space P ( 1 , 1 , 2 ) {\displaystyle \mathbb {P} (1,1,2)} corresponds to taking Proj {\displaystyle \operatorname {Proj} } of the ring A [ X 0 , X 1 , X 2 ] {\displaystyle A[X_{0},X_{1},X_{2}]} where X 0 , X 1 {\displaystyle X_{0},X_{1}} have weight 1 {\displaystyle 1} while X 2 {\displaystyle X_{2}} has weight 2. The proj construction extends to bigraded and multigraded rings. Geometrically, this corresponds to taking products of projective schemes. For example, given

3096-511: A projective scheme using the proj construction for the graded algebra k [ X 0 , … , X n ] ∙ ( f 1 , … , f k ) ∙ {\displaystyle {\frac {k[X_{0},\ldots ,X_{n}]_{\bullet }}{(f_{1},\ldots ,f_{k})_{\bullet }}}} giving an embedding of projective varieties into projective schemes. Weighted projective spaces can be constructed using

3225-438: A quasicoherent sheaf on Proj ⁡ S {\displaystyle \operatorname {Proj} S} , denoted O X ( 1 ) {\displaystyle O_{X}(1)} or simply O ( 1 ) {\displaystyle {\mathcal {O}}(1)} , called the twisting sheaf of Serre . It can be checked that O ( 1 ) {\displaystyle {\mathcal {O}}(1)}

3354-503: A research article, the authors often specify which definition of ring they use in the beginning of that article. Gardner and Wiegandt assert that, when dealing with several objects in the category of rings (as opposed to working with a fixed ring), if one requires all rings to have a 1 , then some consequences include the lack of existence of infinite direct sums of rings, and that proper direct summands of rings are not subrings. They conclude that "in many, maybe most, branches of ring theory

3483-627: A ring A , the global sections of the structure sheaf form A itself, whereas the global sections of O X {\displaystyle {\mathcal {O}}_{X}} here form only the degree-zero elements of S {\displaystyle S} . If we define then each O ( n ) {\displaystyle {\mathcal {O}}(n)} contains the degree- n {\displaystyle n} information about S {\displaystyle S} , denoted S n {\displaystyle S_{n}} , and taken together they contain all

3612-528: A ring follow immediately from the axioms: Equip the set Z / 4 Z = { 0 ¯ , 1 ¯ , 2 ¯ , 3 ¯ } {\displaystyle \mathbb {Z} /4\mathbb {Z} =\left\{{\overline {0}},{\overline {1}},{\overline {2}},{\overline {3}}\right\}} with the following operations: Then ⁠ Z / 4 Z {\displaystyle \mathbb {Z} /4\mathbb {Z} } ⁠

3741-434: A ring is commutative has profound implications on its behavior. Commutative algebra , the theory of commutative rings , is a major branch of ring theory . Its development has been greatly influenced by problems and ideas of algebraic number theory and algebraic geometry . The simplest commutative rings are those that admit division by non-zero elements; such rings are called fields . Examples of commutative rings include

3870-406: A scheme Y which we define to be P r o j ⁡ S {\displaystyle \operatorname {\mathbf {Proj} } S} . It is not hard to show that defining each p U {\displaystyle p_{U}} to be the map corresponding to the inclusion of O X ( U ) {\displaystyle O_{X}(U)} into S ( U ) as

3999-534: A set (for instance, determining if a cloud of points is spherical or toroidal ). The main method used by topological data analysis is to: Several branches of programming language semantics , such as domain theory , are formalized using topology. In this context, Steve Vickers , building on work by Samson Abramsky and Michael B. Smyth , characterizes topological spaces as Boolean or Heyting algebras over open sets, which are characterized as semidecidable (equivalently, finitely observable) properties. Topology

Proj construction - Misplaced Pages Continue

4128-425: A sheaf of rings O X {\displaystyle O_{X}} on Proj ⁡ S {\displaystyle \operatorname {Proj} S} , and it may be shown that the pair ( Proj ⁡ S {\displaystyle \operatorname {Proj} S} , O X {\displaystyle O_{X}} ) is in fact a scheme (this is accomplished by showing that each of

4257-405: A subring ⁠ Z / 2 Z {\displaystyle \mathbb {Z} /2\mathbb {Z} } ⁠ , and if p {\displaystyle p} is prime, then ⁠ Z / p Z {\displaystyle \mathbb {Z} /p\mathbb {Z} } ⁠ has no subrings. The set of 2-by-2 square matrices with entries in a field F is With

4386-403: A topology on Proj ⁡ S {\displaystyle \operatorname {Proj} S} . The advantage of this approach is that the sets D ( f ) {\displaystyle D(f)} , where f {\displaystyle f} ranges over all homogeneous elements of the ring S {\displaystyle S} , form a base for this topology, which

4515-437: A topology. The deformations that are considered in topology are homeomorphisms and homotopies . A property that is invariant under such deformations is a topological property . The following are basic examples of topological properties: the dimension , which allows distinguishing between a line and a surface ; compactness , which allows distinguishing between a line and a circle; connectedness , which allows distinguishing

4644-435: Is quasicoherent by construction. If S {\displaystyle S} is generated by finitely many elements of degree 1 {\displaystyle 1} (e.g. a polynomial ring or a homogenous quotient of it), all quasicoherent sheaves on Proj ⁡ S {\displaystyle \operatorname {Proj} S} arise from graded modules by this construction. The corresponding graded module

4773-457: Is a π -system . The members of τ are called open sets in X . A subset of X is said to be closed if its complement is in τ (that is, its complement is open). A subset of X may be open, closed, both (a clopen set ), or neither. The empty set and X itself are always both closed and open. An open subset of X which contains a point x is called an open neighborhood of x . A function or map from one topological space to another

4902-752: Is a projective morphism . For any x ∈ X {\displaystyle x\in X} , the fiber of the above morphism over x {\displaystyle x} is the projective space P ( E ( x ) ) {\displaystyle \mathbb {P} ({\mathcal {E}}(x))} associated to the dual of the vector space E ( x ) := E ⊗ O X k ( x ) {\displaystyle {\mathcal {E}}(x):={\mathcal {E}}\otimes _{O_{X}}k(x)} over k ( x ) {\displaystyle k(x)} . If S {\displaystyle {\mathcal {S}}}

5031-430: Is a set R equipped with two binary operations + (addition) and ⋅ (multiplication) satisfying the following three sets of axioms, called the ring axioms : In notation, the multiplication symbol · is often omitted, in which case a · b is written as ab . In the terminology of this article, a ring is defined to have a multiplicative identity, while a structure with the same axiomatic definition but without

5160-466: Is a closed subscheme of P ( S 1 ) {\displaystyle \mathbb {P} ({\mathcal {S}}_{1})} and is then projective over X {\displaystyle X} . In fact, every closed subscheme of a projective P ( E ) {\displaystyle \mathbb {P} ({\mathcal {E}})} is of this form. As a special case, when E {\displaystyle {\mathcal {E}}}

5289-420: Is a product of projective schemes. There is an embedding of such schemes into projective space by taking the total graded algebra S ∙ , ∙ → S ∙ {\displaystyle S_{\bullet ,\bullet }\to S_{\bullet }} where a degree ( a , b ) {\displaystyle (a,b)} element is considered as a degree (

Proj construction - Misplaced Pages Continue

5418-661: Is a quantum field theory that computes topological invariants . Although TQFTs were invented by physicists, they are also of mathematical interest, being related to, among other things, knot theory , the theory of four-manifolds in algebraic topology, and to the theory of moduli spaces in algebraic geometry. Donaldson , Jones , Witten , and Kontsevich have all won Fields Medals for work related to topological field theory. The topological classification of Calabi–Yau manifolds has important implications in string theory , as different manifolds can sustain different kinds of strings. In cosmology, topology can be used to describe

5547-424: Is a quasi-coherent sheaf of graded O X {\displaystyle O_{X}} -modules, generated by S 1 {\displaystyle {\mathcal {S}}_{1}} and such that S 1 {\displaystyle {\mathcal {S}}_{1}} is of finite type, then P r o j S {\displaystyle \mathbf {Proj} {\mathcal {S}}}

5676-510: Is a ring: each axiom follows from the corresponding axiom for ⁠ Z . {\displaystyle \mathbb {Z} .} ⁠ If x is an integer, the remainder of x when divided by 4 may be considered as an element of ⁠ Z / 4 Z , {\displaystyle \mathbb {Z} /4\mathbb {Z} ,} ⁠ and this element is often denoted by " x mod 4 " or x ¯ , {\displaystyle {\overline {x}},} which

5805-502: Is a subring of the field of real numbers and also a subring of the ring of polynomials ⁠ Z [ X ] {\displaystyle \mathbb {Z} [X]} ⁠ (in both cases, ⁠ Z {\displaystyle \mathbb {Z} } ⁠ contains 1, which is the multiplicative identity of the larger rings). On the other hand, the subset of even integers ⁠ 2 Z {\displaystyle 2\mathbb {Z} } ⁠ does not contain

5934-455: Is a subring of  R , called the centralizer (or commutant) of  X . The center is the centralizer of the entire ring  R . Elements or subsets of the center are said to be central in  R ; they (each individually) generate a subring of the center. Let R be a ring. A left ideal of R is a nonempty subset I of R such that for any x, y in I and r in R , the elements x + y and rx are in I . If R I denotes

6063-517: Is also a smooth morphism of schemes (which can be checked using the Jacobian criterion ). The projective hypersurface Proj ⁡ ( C [ X 0 , … , X 4 ] / ( X 0 5 + ⋯ + X 4 5 ) ) {\displaystyle \operatorname {Proj} \left(\mathbb {C} [X_{0},\ldots ,X_{4}]/(X_{0}^{5}+\cdots +X_{4}^{5})\right)}

6192-603: Is also denoted O (1) and serves much the same purpose for P r o j ⁡ S {\displaystyle \operatorname {\mathbf {Proj} } S} as the twisting sheaf on the Proj of a ring does. Let E {\displaystyle {\mathcal {E}}} be a quasi-coherent sheaf on a scheme X {\displaystyle X} . The sheaf of symmetric algebras S y m O X ( E ) {\displaystyle \mathbf {Sym} _{O_{X}}({\mathcal {E}})}

6321-564: Is also possessed by any graded module M {\displaystyle M} over S {\displaystyle S} , and therefore with the appropriate minor modifications the preceding section constructs for any such M {\displaystyle M} a sheaf, denoted M ~ {\displaystyle {\tilde {M}}} , of O X {\displaystyle O_{X}} -modules on Proj ⁡ S {\displaystyle \operatorname {Proj} S} . This sheaf

6450-431: Is an O X {\displaystyle O_{X}} -module such that for every open subset U of X , S ( U ) is an O X ( U ) {\displaystyle O_{X}(U)} -algebra and the resulting direct sum decomposition is a grading of this algebra as a ring. Here we assume that S 0 = O X {\displaystyle S_{0}=O_{X}} . We make

6579-440: Is an example of a Fermat quintic threefold which is also a Calabi–Yau manifold . In addition to projective hypersurfaces, any projective variety cut out by a system of homogeneous polynomials f 1 = 0 , … , f k = 0 {\displaystyle f_{1}=0,\ldots ,f_{k}=0} in ( n + 1 ) {\displaystyle (n+1)} -variables can be converted into

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6708-408: Is an indispensable tool for the analysis of Proj ⁡ S {\displaystyle \operatorname {Proj} S} , just as the analogous fact for the spectrum of a ring is likewise indispensable. We also construct a sheaf on Proj ⁡ S {\displaystyle \operatorname {Proj} S} , called the “structure sheaf” as in the affine case, which makes it into

6837-403: Is by definition a set of homogeneous prime ideals of S {\displaystyle S} not containing S + {\displaystyle S_{+}} ) we define the ring O X ( U ) {\displaystyle O_{X}(U)} to be the set of all functions (where S ( p ) {\displaystyle S_{(p)}} denotes

6966-406: Is called continuous if the inverse image of any open set is open. If the function maps the real numbers to the real numbers (both spaces with the standard topology), then this definition of continuous is equivalent to the definition of continuous in calculus . If a continuous function is one-to-one and onto , and if the inverse of the function is also continuous, then the function is called

7095-423: Is called the subring generated by  E . For a ring R , the smallest subring of R is called the characteristic subring of R . It can be generated through addition of copies of 1 and  −1 . It is possible that n · 1 = 1 + 1 + ... + 1 ( n times) can be zero. If n is the smallest positive integer such that this occurs, then n is called the characteristic of  R . In some rings, n · 1

7224-790: Is consistent with the notation for 0, 1, 2, 3 . The additive inverse of any x ¯ {\displaystyle {\overline {x}}} in ⁠ Z / 4 Z {\displaystyle \mathbb {Z} /4\mathbb {Z} } ⁠ is − x ¯ = − x ¯ . {\displaystyle -{\overline {x}}={\overline {-x}}.} For example, − 3 ¯ = − 3 ¯ = 1 ¯ . {\displaystyle -{\overline {3}}={\overline {-3}}={\overline {1}}.} ⁠ Z / 4 Z {\displaystyle \mathbb {Z} /4\mathbb {Z} } ⁠ has

7353-454: Is denoted by R or R * or U ( R ) . For example, if R is the ring of all square matrices of size n over a field, then R consists of the set of all invertible matrices of size n , and is called the general linear group . A subset S of R is called a subring if any one of the following equivalent conditions holds: For example, the ring ⁠ Z {\displaystyle \mathbb {Z} } ⁠ of integers

7482-838: Is free over A , then and hence P ( E ) {\displaystyle \mathbb {P} ({\mathcal {E}})} is a projective space bundle. Many families of varieties can be constructed as subschemes of these projective bundles, such as the Weierstrass family of elliptic curves. For more details, see the main article. Global proj can be used to construct Lefschetz pencils . For example, let X = P s , t 1 {\displaystyle X=\mathbb {P} _{s,t}^{1}} and take homogeneous polynomials f , g ∈ C [ x 0 , … , x n ] {\displaystyle f,g\in \mathbb {C} [x_{0},\ldots ,x_{n}]} of degree k. We can consider

7611-419: Is in fact an invertible sheaf . One reason for the utility of O ( 1 ) {\displaystyle {\mathcal {O}}(1)} is that it recovers the algebraic information of S {\displaystyle S} that was lost when, in the construction of O X {\displaystyle O_{X}} , we passed to fractions of degree zero. In the case Spec A for

7740-605: Is locally free of rank n + 1 {\displaystyle n+1} , we get a projective bundle P ( E ) {\displaystyle \mathbb {P} ({\mathcal {E}})} over X {\displaystyle X} of relative dimension n {\displaystyle n} . Indeed, if we take an open cover of X by open affines U = Spec ⁡ ( A ) {\displaystyle U=\operatorname {Spec} (A)} such that when restricted to each of these, E {\displaystyle {\mathcal {E}}}

7869-532: Is naturally a quasi-coherent sheaf of graded O X {\displaystyle O_{X}} -modules, generated by elements of degree 1. The resulting scheme is denoted by P ( E ) {\displaystyle \mathbb {P} ({\mathcal {E}})} . If E {\displaystyle {\mathcal {E}}} is of finite type, then its canonical morphism p : P ( E ) → X {\displaystyle p:\mathbb {P} ({\mathcal {E}})\to X}

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7998-410: Is necessarily a zero divisor. An idempotent e {\displaystyle e} is an element such that e = e . One example of an idempotent element is a projection in linear algebra. A unit is an element a having a multiplicative inverse ; in this case the inverse is unique, and is denoted by a . The set of units of a ring is a group under ring multiplication; this group

8127-479: Is never zero for any positive integer n , and those rings are said to have characteristic zero . Given a ring R , let Z( R ) denote the set of all elements x in R such that x commutes with every element in R : xy = yx for any y in  R . Then Z( R ) is a subring of  R , called the center of  R . More generally, given a subset X of  R , let S be the set of all elements in R that commute with every element in  X . Then S

8256-794: Is not unique. A special case of the sheaf associated to a graded module is when we take M {\displaystyle M} to be S {\displaystyle S} itself with a different grading: namely, we let the degree d {\displaystyle d} elements of M {\displaystyle M} be the degree ( d + 1 ) {\displaystyle (d+1)} elements of S {\displaystyle S} , so M d = S d + 1 {\displaystyle M_{d}=S_{d+1}} and denote M = S ( 1 ) {\displaystyle M=S(1)} . We then obtain M ~ {\displaystyle {\tilde {M}}} as

8385-776: Is point-set topology. The basic object of study is topological spaces , which are sets equipped with a topology , that is, a family of subsets , called open sets , which is closed under finite intersections and (finite or infinite) unions . The fundamental concepts of topology, such as continuity , compactness , and connectedness , can be defined in terms of open sets. Intuitively, continuous functions take nearby points to nearby points. Compact sets are those that can be covered by finitely many sets of arbitrarily small size. Connected sets are sets that cannot be divided into two pieces that are far apart. The words nearby , arbitrarily small , and far apart can all be made precise by using open sets. Several topologies can be defined on

8514-453: Is relevant to physics in areas such as condensed matter physics , quantum field theory and physical cosmology . The topological dependence of mechanical properties in solids is of interest in disciplines of mechanical engineering and materials science . Electrical and mechanical properties depend on the arrangement and network structures of molecules and elementary units in materials. The compressive strength of crumpled topologies

8643-459: Is studied in attempts to understand the high strength to weight of such structures that are mostly empty space. Topology is of further significance in Contact mechanics where the dependence of stiffness and friction on the dimensionality of surface structures is the subject of interest with applications in multi-body physics. A topological quantum field theory (or topological field theory or TQFT)

8772-420: Is the ideal generated by f {\displaystyle f} . For any ideal a {\displaystyle a} , the sets D ( a ) {\displaystyle D(a)} and V ( a ) {\displaystyle V(a)} are complementary, and hence the same proof as before shows that the sets D ( a ) {\displaystyle D(a)} form

8901-766: Is the ideal of elements of positive degree S + = ⨁ i > 0 S i . {\displaystyle S_{+}=\bigoplus _{i>0}S_{i}.} We say an ideal is homogeneous if it is generated by homogeneous elements. Then, as a set, Proj ⁡ S = { P ⊆ S  homogeneous prime ideal,  S + ⊈ P } . {\displaystyle \operatorname {Proj} S=\{P\subseteq S{\text{ homogeneous prime ideal, }}S_{+}\not \subseteq P\}.} For brevity we will sometimes write X {\displaystyle X} for Proj ⁡ S {\displaystyle \operatorname {Proj} S} . We may define

9030-428: Is the set of all points whose distance to x is less than r . Many common spaces are topological spaces whose topology can be defined by a metric. This is the case of the real line , the complex plane , real and complex vector spaces and Euclidean spaces . Having a metric simplifies many proofs. Algebraic topology is a branch of mathematics that uses tools from algebra to study topological spaces. The basic goal

9159-612: Is the sheaf associated to a graded S {\displaystyle S} -module M {\displaystyle M} we likewise expect it to contain lost grading information about M {\displaystyle M} . This suggests, though erroneously, that S {\displaystyle S} can in fact be reconstructed from these sheaves; as ⨁ n ≥ 0 H 0 ( X , O X ( n ) ) {\displaystyle \bigoplus _{n\geq 0}H^{0}(X,{\mathcal {O}}_{X}(n))} however, this

9288-442: Is the underlying set equipped with only the operation of addition. Although the definition requires that the additive group be abelian, this can be inferred from the other ring axioms. The proof makes use of the " 1 ", and does not work in a rng. (For a rng, omitting the axiom of commutativity of addition leaves it inferable from the remaining rng assumptions only for elements that are products: ab + cd = cd + ab .) There are

9417-437: Is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify up to homotopy equivalence. The most important of these invariants are homotopy groups , homology, and cohomology . Although algebraic topology primarily uses algebra to study topological problems, using topology to solve algebraic problems is sometimes also possible. Algebraic topology, for example, allows for

9546-414: Is true in the case that S {\displaystyle S} is a polynomial ring, below. This situation is to be contrasted with the fact that the spec functor is adjoint to the global sections functor in the category of locally ringed spaces . If A {\displaystyle A} is a ring, we define projective n -space over A {\displaystyle A} to be

9675-652: The X i {\displaystyle X_{i}} have weight ( 1 , 0 ) {\displaystyle (1,0)} and the Y i {\displaystyle Y_{i}} have weight ( 0 , 1 ) {\displaystyle (0,1)} . Then the proj construction gives Proj ( S ∙ , ∙ ) = P 1 × Spec ( C ) P 1 {\displaystyle {\text{Proj}}(S_{\bullet ,\bullet })=\mathbb {P} ^{1}\times _{{\text{Spec}}(\mathbb {C} )}\mathbb {P} ^{1}} which

9804-592: The Bridges of Königsberg , the result does not depend on the shape of the sphere; it applies to any kind of smooth blob, as long as it has no holes. To deal with these problems that do not rely on the exact shape of the objects, one must be clear about just what properties these problems do rely on. From this need arises the notion of homeomorphism . The impossibility of crossing each bridge just once applies to any arrangement of bridges homeomorphic to those in Königsberg, and

9933-403: The R -span of I , that is, the set of finite sums then I is a left ideal if RI ⊆ I . Similarly, a right ideal is a subset I such that IR ⊆ I . A subset I is said to be a two-sided ideal or simply ideal if it is both a left ideal and right ideal. A one-sided or two-sided ideal is then an additive subgroup of R . If E is a subset of R , then RE is a left ideal, called

10062-429: The geometrization conjecture (now theorem) in 3 dimensions – every 3-manifold can be cut into pieces, each of which has one of eight possible geometries. 2-dimensional topology can be studied as complex geometry in one variable ( Riemann surfaces are complex curves) – by the uniformization theorem every conformal class of metrics is equivalent to a unique complex one, and 4-dimensional topology can be studied from

10191-595: The plane , the sphere, and the torus, which can all be realized without self-intersection in three dimensions, and the Klein bottle and real projective plane , which cannot (that is, all their realizations are surfaces that are not manifolds). General topology is the branch of topology dealing with the basic set-theoretic definitions and constructions used in topology. It is the foundation of most other branches of topology, including differential topology, geometric topology, and algebraic topology. Another name for general topology

10320-501: The real line , the complex plane , and the Cantor set can be thought of as the same set with different topologies. Formally, let X be a set and let τ be a family of subsets of X . Then τ is called a topology on X if: If τ is a topology on X , then the pair ( X , τ ) is called a topological space. The notation X τ may be used to denote a set X endowed with the particular topology τ . By definition, every topology

10449-404: The scheme The grading on the polynomial ring S = A [ x 0 , … , x n ] {\displaystyle S=A[x_{0},\ldots ,x_{n}]} is defined by letting each x i {\displaystyle x_{i}} have degree one and every element of A {\displaystyle A} , degree zero. Comparing this to

10578-403: The stalk of the sheaf S at a point x of X , which is a graded algebra whose degree-zero elements form the ring O X , x {\displaystyle O_{X,x}} then the degree-one elements form a finitely-generated module over O X , x {\displaystyle O_{X,x}} and also generate the stalk as an algebra over it) then we may make

10707-412: The 1870s to the 1920s, with key contributions by Dedekind , Hilbert , Fraenkel , and Noether . Rings were first formalized as a generalization of Dedekind domains that occur in number theory , and of polynomial rings and rings of invariants that occur in algebraic geometry and invariant theory . They later proved useful in other branches of mathematics such as geometry and analysis . A ring

10836-984: The additional assumption that S is a quasi-coherent sheaf ; this is a “consistency” assumption on the sections over different open sets that is necessary for the construction to proceed. In this setup we may construct a scheme P r o j ⁡ S {\displaystyle \operatorname {\mathbf {Proj} } S} and a “projection” map p onto X such that for every open affine U of X , This definition suggests that we construct P r o j ⁡ S {\displaystyle \operatorname {\mathbf {Proj} } S} by first defining schemes Y U {\displaystyle Y_{U}} for each open affine U , by setting and maps p U : Y U → U {\displaystyle p_{U}\colon Y_{U}\to U} , and then showing that these data can be glued together “over” each intersection of two open affines U and V to form

10965-416: The branch of mathematics known as graph theory . Similarly, the hairy ball theorem of algebraic topology says that "one cannot comb the hair flat on a hairy ball without creating a cowlick ." This fact is immediately convincing to most people, even though they might not recognize the more formal statement of the theorem, that there is no nonvanishing continuous tangent vector field on the sphere. As with

11094-411: The canonical projections coming from the injections of these algebras from the tensor product diagram of commutative algebras. A generalization of the Proj construction replaces the ring S with a sheaf of algebras and produces, as the result, a scheme which might be thought of as a fibration of Proj's of rings. This construction is often used, for example, to construct projective space bundles over

11223-509: The concepts now known as homotopy and homology , which are now considered part of algebraic topology . Unifying the work on function spaces of Georg Cantor , Vito Volterra , Cesare Arzelà , Jacques Hadamard , Giulio Ascoli and others, Maurice Fréchet introduced the metric space in 1906. A metric space is now considered a special case of a general topological space, with any given topological space potentially giving rise to many distinct metric spaces. In 1914, Felix Hausdorff coined

11352-495: The definition of O ( 1 ) {\displaystyle {\mathcal {O}}(1)} , above, we see that the sections of O ( 1 ) {\displaystyle {\mathcal {O}}(1)} are in fact linear homogeneous polynomials, generated by the x i {\displaystyle x_{i}} themselves. This suggests another interpretation of O ( 1 ) {\displaystyle {\mathcal {O}}(1)} , namely as

11481-458: The definition of sheaves on those categories, and with that the definition of general cohomology theories. Topology has been used to study various biological systems including molecules and nanostructure (e.g., membraneous objects). In particular, circuit topology and knot theory have been extensively applied to classify and compare the topology of folded proteins and nucleic acids. Circuit topology classifies folded molecular chains based on

11610-638: The doughnut. While topological spaces can be extremely varied and exotic, many areas of topology focus on the more familiar class of spaces known as manifolds. A manifold is a topological space that resembles Euclidean space near each point. More precisely, each point of an n -dimensional manifold has a neighborhood that is homeomorphic to the Euclidean space of dimension n . Lines and circles , but not figure eights , are one-dimensional manifolds. Two-dimensional manifolds are also called surfaces , although not all surfaces are manifolds. Examples include

11739-533: The elements of degree zero yields the necessary consistency of the p U {\displaystyle p_{U}} , while the consistency of the Y U {\displaystyle Y_{U}} themselves follows from the quasi-coherence assumption on S . If S has the additional property that S 1 {\displaystyle S_{1}} is a coherent sheaf and locally generates S over S 0 {\displaystyle S_{0}} (that is, when we pass to

11868-400: The empty sequence. Authors who follow either convention for the use of the term "ring" may use one of the following terms to refer to objects satisfying the other convention: For each nonnegative integer n , given a sequence ( a 1 , … , a n ) {\displaystyle (a_{1},\dots ,a_{n})} of n elements of R , one can define

11997-429: The graded rings A ∙ = C [ X 0 , X 1 ] ,   B ∙ = C [ Y 0 , Y 1 ] {\displaystyle A_{\bullet }=\mathbb {C} [X_{0},X_{1}],{\text{ }}B_{\bullet }=\mathbb {C} [Y_{0},Y_{1}]} with the degree of each generator 1 {\displaystyle 1} . Then,

12126-397: The grading information that was lost. Likewise, for any sheaf of graded O X {\displaystyle {\mathcal {O}}_{X}} -modules N {\displaystyle N} we define and expect this “twisted” sheaf to contain grading information about N {\displaystyle N} . In particular, if N {\displaystyle N}

12255-407: The hairy ball theorem applies to any space homeomorphic to a sphere. Intuitively, two spaces are homeomorphic if one can be deformed into the other without cutting or gluing. A traditional joke is that a topologist cannot distinguish a coffee mug from a doughnut, since a sufficiently pliable doughnut could be reshaped to a coffee cup by creating a dimple and progressively enlarging it, while shrinking

12384-402: The hole into a handle. Homeomorphism can be considered the most basic topological equivalence . Another is homotopy equivalence . This is harder to describe without getting technical, but the essential notion is that two objects are homotopy equivalent if they both result from "squishing" some larger object. Topology, as a well-defined mathematical discipline, originates in the early part of

12513-616: The ideal sheaf I = ( s f + t g ) {\displaystyle {\mathcal {I}}=(sf+tg)} of O X [ x 0 , … , x n ] {\displaystyle {\mathcal {O}}_{X}[x_{0},\ldots ,x_{n}]} and construct global proj of this quotient sheaf of algebras O X [ x 0 , … , x n ] / I {\displaystyle {\mathcal {O}}_{X}[x_{0},\ldots ,x_{n}]/{\mathcal {I}}} . This can be described explicitly as

12642-458: The identity element 1 and thus does not qualify as a subring of  ⁠ Z ; {\displaystyle \mathbb {Z} ;} ⁠ one could call ⁠ 2 Z {\displaystyle 2\mathbb {Z} } ⁠ a subrng , however. An intersection of subrings is a subring. Given a subset E of R , the smallest subring of R containing E is the intersection of all subrings of R containing  E , and it

12771-521: The indexing set I is finite, then ⋃ V ( a i ) = V ( ∏ a i ) . {\textstyle \bigcup V(a_{i})=V\left(\prod a_{i}\right).} Equivalently, we may take the open sets as a starting point and define A common shorthand is to denote D ( S f ) {\displaystyle D(Sf)} by D ( f ) {\displaystyle D(f)} , where S f {\displaystyle Sf}

12900-409: The left ideal generated by E ; it is the smallest left ideal containing E . Similarly, one can consider the right ideal or the two-sided ideal generated by a subset of R . If x is in R , then Rx and xR are left ideals and right ideals, respectively; they are called the principal left ideals and right ideals generated by x . The principal ideal RxR is written as ( x ) . For example,

13029-476: The number of vertices, edges, and faces of the polyhedron). Some authorities regard this analysis as the first theorem, signaling the birth of topology. Further contributions were made by Augustin-Louis Cauchy , Ludwig Schläfli , Johann Benedict Listing , Bernhard Riemann and Enrico Betti . Listing introduced the term "Topologie" in Vorstudien zur Topologie , written in his native German, in 1847, having used

13158-458: The open subsets D ( f ) {\displaystyle D(f)} is in fact an affine scheme). The essential property of S {\displaystyle S} for the above construction was the ability to form localizations S ( p ) {\displaystyle S_{(p)}} for each prime ideal p {\displaystyle p} of S {\displaystyle S} . This property

13287-429: The operations of matrix addition and matrix multiplication , M 2 ⁡ ( F ) {\displaystyle \operatorname {M} _{2}(F)} satisfies the above ring axioms. The element ( 1 0 0 1 ) {\displaystyle \left({\begin{smallmatrix}1&0\\0&1\end{smallmatrix}}\right)} is the multiplicative identity of

13416-475: The overall shape of the universe . This area of research is commonly known as spacetime topology . In condensed matter a relevant application to topological physics comes from the possibility to obtain one-way current, which is a current protected from backscattering. It was first discovered in electronics with the famous quantum Hall effect , and then generalized in other areas of physics, for instance in photonics by F.D.M Haldane . The possible positions of

13545-462: The pairwise arrangement of their intra-chain contacts and chain crossings. Knot theory , a branch of topology, is used in biology to study the effects of certain enzymes on DNA. These enzymes cut, twist, and reconnect the DNA, causing knotting with observable effects such as slower electrophoresis . Topological data analysis uses techniques from algebraic topology to determine the large scale structure of

13674-481: The planar and higher-dimensional Schönflies theorem . In high-dimensional topology, characteristic classes are a basic invariant, and surgery theory is a key theory. Low-dimensional topology is strongly geometric, as reflected in the uniformization theorem in 2 dimensions – every surface admits a constant curvature metric; geometrically, it has one of 3 possible geometries: positive curvature /spherical, zero curvature/flat, and negative curvature/hyperbolic – and

13803-519: The plane into two parts, the part inside and the part outside. In one of the first papers in topology, Leonhard Euler demonstrated that it was impossible to find a route through the town of Königsberg (now Kaliningrad ) that would cross each of its seven bridges exactly once. This result did not depend on the lengths of the bridges or on their distance from one another, but only on connectivity properties: which bridges connect to which islands or riverbanks. This Seven Bridges of Königsberg problem led to

13932-426: The point of view of complex geometry in two variables (complex surfaces), though not every 4-manifold admits a complex structure. Occasionally, one needs to use the tools of topology but a "set of points" is not available. In pointless topology one considers instead the lattice of open sets as the basic notion of the theory, while Grothendieck topologies are structures defined on arbitrary categories that allow

14061-408: The product P n = ∏ i = 1 n a i {\displaystyle P_{n}=\prod _{i=1}^{n}a_{i}} recursively: let P 0 = 1 and let P m = P m −1 a m for 1 ≤ m ≤ n . As a special case, one can define nonnegative integer powers of an element a of a ring: a = 1 and a = a a for n ≥ 1 . Then

14190-787: The projective morphism Proj ⁡ ( C [ s , t ] [ x 0 , … , x n ] / ( s f + t g ) ) → P s , t 1 {\displaystyle \operatorname {Proj} (\mathbb {C} [s,t][x_{0},\ldots ,x_{n}]/(sf+tg))\to \mathbb {P} _{s,t}^{1}} . Ring (mathematics) Ring homomorphisms Algebraic structures Related structures Algebraic number theory Noncommutative algebraic geometry Free algebra Clifford algebra In mathematics , rings are algebraic structures that generalize fields : multiplication need not be commutative and multiplicative inverses need not exist. Informally,

14319-535: The requirement for a multiplicative identity is instead called a " rng " (IPA: / r ʊ ŋ / ) with a missing "i". For example, the set of even integers with the usual + and ⋅ is a rng, but not a ring. As explained in § History below, many authors apply the term "ring" without requiring a multiplicative identity. Although ring addition is commutative , ring multiplication is not required to be commutative: ab need not necessarily equal ba . Rings that also satisfy commutativity for multiplication (such as

14448-431: The requirement of the existence of a unity element is not sensible, and therefore unacceptable." Poonen makes the counterargument that the natural notion for rings would be the direct product rather than the direct sum. However, his main argument is that rings without a multiplicative identity are not totally associative, in the sense that they do not contain the product of any finite sequence of ring elements, including

14577-409: The ring is an abelian group with respect to the addition operator, and the multiplication operator is associative , is distributive over the addition operation, and has a multiplicative identity element . (Some authors define rings without requiring a multiplicative identity and instead call the structure defined above a ring with identity . See § Variations on the definition .) Whether

14706-441: The ring is noncommutative. More generally, for any ring R , commutative or not, and any nonnegative integer n , the square matrices of dimension n with entries in R form a ring; see Matrix ring . The study of rings originated from the theory of polynomial rings and the theory of algebraic integers . In 1871, Richard Dedekind defined the concept of the ring of integers of a number field. In this context, he introduced

14835-403: The ring of integers) are called commutative rings . Books on commutative algebra or algebraic geometry often adopt the convention that ring means commutative ring , to simplify terminology. In a ring, multiplicative inverses are not required to exist. A non zero commutative ring in which every nonzero element has a multiplicative inverse is called a field . The additive group of a ring

14964-831: The ring. If A = ( 0 1 1 0 ) {\displaystyle A=\left({\begin{smallmatrix}0&1\\1&0\end{smallmatrix}}\right)} and B = ( 0 1 0 0 ) , {\displaystyle B=\left({\begin{smallmatrix}0&1\\0&0\end{smallmatrix}}\right),} then A B = ( 0 0 0 1 ) {\displaystyle AB=\left({\begin{smallmatrix}0&0\\0&1\end{smallmatrix}}\right)} while B A = ( 1 0 0 0 ) ; {\displaystyle BA=\left({\begin{smallmatrix}1&0\\0&0\end{smallmatrix}}\right);} this example shows that

15093-1145: The scheme Proj ( S ∙ , ∙ ) {\displaystyle {\text{Proj}}(S_{\bullet ,\bullet })} now comes with bigraded sheaves O ( a , b ) {\displaystyle {\mathcal {O}}(a,b)} which are the tensor product of the sheaves π 1 ∗ O ( a ) ⊗ π 2 ∗ O ( b ) {\displaystyle \pi _{1}^{*}{\mathcal {O}}(a)\otimes \pi _{2}^{*}{\mathcal {O}}(b)} where π 1 : Proj ( S ∙ , ∙ ) → Proj ( A ∙ ) {\displaystyle \pi _{1}:{\text{Proj}}(S_{\bullet ,\bullet })\to {\text{Proj}}(A_{\bullet })} and π 2 : Proj ( S ∙ , ∙ ) → Proj ( B ∙ ) {\displaystyle \pi _{2}:{\text{Proj}}(S_{\bullet ,\bullet })\to {\text{Proj}}(B_{\bullet })} are

15222-450: The set of all positive and negative multiples of 2 along with 0 form an ideal of the integers, and this ideal is generated by the integer  2 . In fact, every ideal of the ring of integers is principal. Topology Topology (from the Greek words τόπος , 'place, location', and λόγος , 'study') is the branch of mathematics concerned with the properties of

15351-539: The set of integers with their standard addition and multiplication, the set of polynomials with their addition and multiplication, the coordinate ring of an affine algebraic variety , and the ring of integers of a number field. Examples of noncommutative rings include the ring of n × n real square matrices with n ≥ 2 , group rings in representation theory , operator algebras in functional analysis , rings of differential operators , and cohomology rings in topology . The conceptualization of rings spanned

15480-852: The sheaf of “coordinates” for Proj ⁡ S {\displaystyle \operatorname {Proj} S} , since the x i {\displaystyle x_{i}} are literally the coordinates for projective n {\displaystyle n} -space. If we let the base ring be A = C [ λ ] {\displaystyle A=\mathbb {C} [\lambda ]} , then X = Proj ⁡ ( A [ X , Y , Z ] ∙ ( Z Y 2 − X ( X − Z ) ( X − λ Z ) ) ∙ ) {\displaystyle X=\operatorname {Proj} \left({\frac {A[X,Y,Z]_{\bullet }}{(ZY^{2}-X(X-Z)(X-\lambda Z))_{\bullet }}}\right)} has

15609-408: The space and affecting the curvature or volume. Geometric topology is a branch of topology that primarily focuses on low-dimensional manifolds (that is, spaces of dimensions 2, 3, and 4) and their interaction with geometry, but it also includes some higher-dimensional topology. Some examples of topics in geometric topology are orientability , handle decompositions , local flatness , crumpling and

15738-436: The subring of the ring of fractions S p {\displaystyle S_{p}} consisting of fractions of homogeneous elements of the same degree) such that for each prime ideal p {\displaystyle p} of U {\displaystyle U} : It follows immediately from the definition that the O X ( U ) {\displaystyle O_{X}(U)} form

15867-602: The tensor product of these algebras over C {\displaystyle \mathbb {C} } gives the bigraded algebra A ∙ ⊗ C B ∙ = S ∙ , ∙ = C [ X 0 , X 1 , Y 0 , Y 1 ] {\displaystyle {\begin{aligned}A_{\bullet }\otimes _{\mathbb {C} }B_{\bullet }&=S_{\bullet ,\bullet }\\&=\mathbb {C} [X_{0},X_{1},Y_{0},Y_{1}]\end{aligned}}} where

15996-646: The term "topological space" and gave the definition for what is now called a Hausdorff space . Currently, a topological space is a slight generalization of Hausdorff spaces, given in 1922 by Kazimierz Kuratowski . Modern topology depends strongly on the ideas of set theory, developed by Georg Cantor in the later part of the 19th century. In addition to establishing the basic ideas of set theory, Cantor considered point sets in Euclidean space as part of his study of Fourier series . For further developments, see point-set topology and algebraic topology. The 2022 Abel Prize

16125-468: The terms "ideal" (inspired by Ernst Kummer 's notion of ideal number) and "module" and studied their properties. Dedekind did not use the term "ring" and did not define the concept of a ring in a general setting. The term "Zahlring" (number ring) was coined by David Hilbert in 1892 and published in 1897. In 19th century German, the word "Ring" could mean "association", which is still used today in English in

16254-565: The twentieth century, but some isolated results can be traced back several centuries. Among these are certain questions in geometry investigated by Leonhard Euler . His 1736 paper on the Seven Bridges of Königsberg is regarded as one of the first practical applications of topology. On 14 November 1750, Euler wrote to a friend that he had realized the importance of the edges of a polyhedron . This led to his polyhedron formula , V − E + F = 2 (where V , E , and F respectively indicate

16383-512: The word for ten years in correspondence before its first appearance in print. The English form "topology" was used in 1883 in Listing's obituary in the journal Nature to distinguish "qualitative geometry from the ordinary geometry in which quantitative relations chiefly are treated". Their work was corrected, consolidated and greatly extended by Henri Poincaré . In 1895, he published his ground-breaking paper on Analysis Situs , which introduced

16512-436: Was awarded to Dennis Sullivan "for his groundbreaking contributions to topology in its broadest sense, and in particular its algebraic, geometric and dynamical aspects". The term topology also refers to a specific mathematical idea central to the area of mathematics called topology. Informally, a topology describes how elements of a set relate spatially to each other. The same set can have different topologies. For instance,

16641-438: Was not until the first decades of the 20th century that the idea of a topological space was developed. The motivating insight behind topology is that some geometric problems depend not on the exact shape of the objects involved, but rather on the way they are put together. For example, the square and the circle have many properties in common: they are both one dimensional objects (from a topological point of view) and both separate

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