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128-433: K-theory involves the construction of families of K - functors that map from topological spaces or schemes, or to be even more general: any object of a homotopy category to associated rings; these rings reflect some aspects of the structure of the original spaces or schemes. As with functors to groups in algebraic topology, the reason for this functorial mapping is that it is easier to compute some topological properties from

256-514: A c ∈ A {\displaystyle c\in A} such that a 1 + ′ b 2 + ′ c = a 2 + ′ b 1 + ′ c . {\displaystyle a_{1}+'b_{2}+'c=a_{2}+'b_{1}+'c.} Then, the set G ( A ) = A 2 / ∼ {\displaystyle G(A)=A^{2}/\sim } has

384-442: A {\displaystyle a} and b {\displaystyle b} in ⁠ G {\displaystyle G} ⁠ . If this additional condition holds, then the operation is said to be commutative , and the group is called an abelian group . It is a common convention that for an abelian group either additive or multiplicative notation may be used, but for a nonabelian group only multiplicative notation

512-414: A {\displaystyle a} and b {\displaystyle b} of G {\displaystyle G} to form an element of ⁠ G {\displaystyle G} ⁠ , denoted ⁠ a ⋅ b {\displaystyle a\cdot b} ⁠ , such that the following three requirements, known as group axioms , are satisfied: Formally,

640-412: A {\displaystyle a} and b {\displaystyle b} of a group ⁠ G {\displaystyle G} ⁠ , there is a unique solution x {\displaystyle x} in G {\displaystyle G} to the equation ⁠ a ⋅ x = b {\displaystyle a\cdot x=b} ⁠ , namely ⁠

768-430: A {\displaystyle a} and ⁠ b {\displaystyle b} ⁠ , the sum a + b {\displaystyle a+b} is also an integer; this closure property says that + {\displaystyle +} is a binary operation on ⁠ Z {\displaystyle \mathbb {Z} } ⁠ . The following properties of integer addition serve as

896-429: A − 1 ⋅ b {\displaystyle a^{-1}\cdot b} ⁠ . It follows that for each a {\displaystyle a} in G {\displaystyle G} , the function G → G {\displaystyle G\to G} that maps each x {\displaystyle x} to a ⋅ x {\displaystyle a\cdot x}

1024-406: A − 1 ) = φ ( a ) − 1 {\displaystyle \varphi (a^{-1})=\varphi (a)^{-1}} for all a {\displaystyle a} in ⁠ G {\displaystyle G} ⁠ . However, these additional requirements need not be included in the definition of homomorphisms, because they are already implied by

1152-558: A ↦ [ ( a , 0 ) ] , {\displaystyle a\mapsto [(a,0)],} which has a certain universal property . To get a better understanding of this group, consider some equivalence classes of the abelian monoid ( A , + ) {\displaystyle (A,+)} . Here we will denote the identity element of A {\displaystyle A} by 0 {\displaystyle 0} so that [ ( 0 , 0 ) ] {\displaystyle [(0,0)]} will be

1280-399: A − b . {\displaystyle a-b.} Another useful observation is the invariance of equivalence classes under scaling: The Grothendieck completion can be viewed as a functor G : A b M o n → A b G r p , {\displaystyle G:\mathbf {AbMon} \to \mathbf {AbGrp} ,} and it has the property that it

1408-446: A , b ) {\displaystyle (a,b)} we can find a minimal representative ( a ′ , b ′ ) {\displaystyle (a',b')} by using the invariance under scaling. For example, we can see from the scaling invariance that In general, if k := min { a , b } {\displaystyle k:=\min\{a,b\}} then This shows that we should think of

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1536-653: A covariant functor on the opposite category C o p {\displaystyle C^{\mathrm {op} }} . Some authors prefer to write all expressions covariantly. That is, instead of saying F : C → D {\displaystyle F\colon C\to D} is a contravariant functor, they simply write F : C o p → D {\displaystyle F\colon C^{\mathrm {op} }\to D} (or sometimes F : C → D o p {\displaystyle F\colon C\to D^{\mathrm {op} }} ) and call it

1664-402: A multiplicative group whenever the group operation is notated as multiplication; in this case, the identity is typically denoted ⁠ 1 {\displaystyle 1} ⁠ , and the inverse of an element x {\displaystyle x} is denoted ⁠ x − 1 {\displaystyle x^{-1}} ⁠ . In a multiplicative group,

1792-541: A polynomial ring is free ; this assertion is correct, but was not settled until 20 years later. ( Swan's theorem is another aspect of this analogy.) The other historical origin of algebraic K-theory was the work of J. H. C. Whitehead and others on what later became known as Whitehead torsion . There followed a period in which there were various partial definitions of higher K-theory functors . Finally, two useful and equivalent definitions were given by Daniel Quillen using homotopy theory in 1969 and 1972. A variant

1920-626: A topological space X in 1959, and using the Bott periodicity theorem they made it the basis of an extraordinary cohomology theory . It played a major role in the second proof of the Atiyah–Singer index theorem (circa 1962). Furthermore, this approach led to a noncommutative K-theory for C*-algebras . Already in 1955, Jean-Pierre Serre had used the analogy of vector bundles with projective modules to formulate Serre's conjecture , which states that every finitely generated projective module over

2048-820: A category, and similarly for D {\displaystyle D} , F o p {\displaystyle F^{\mathrm {op} }} is distinguished from F {\displaystyle F} . For example, when composing F : C 0 → C 1 {\displaystyle F\colon C_{0}\to C_{1}} with G : C 1 o p → C 2 {\displaystyle G\colon C_{1}^{\mathrm {op} }\to C_{2}} , one should use either G ∘ F o p {\displaystyle G\circ F^{\mathrm {op} }} or G o p ∘ F {\displaystyle G^{\mathrm {op} }\circ F} . Note that, following

2176-553: A category: the functor category . Morphisms in this category are natural transformations between functors. Functors are often defined by universal properties ; examples are the tensor product , the direct sum and direct product of groups or vector spaces, construction of free groups and modules, direct and inverse limits. The concepts of limit and colimit generalize several of the above. Universal constructions often give rise to pairs of adjoint functors . Functors sometimes appear in functional programming . For instance,

2304-411: A functor. Contravariant functors are also occasionally called cofunctors . There is a convention which refers to "vectors"—i.e., vector fields , elements of the space of sections Γ ( T M ) {\displaystyle \Gamma (TM)} of a tangent bundle T M {\displaystyle TM} —as "contravariant" and to "covectors"—i.e., 1-forms , elements of

2432-605: A group ( G , ⋅ ) {\displaystyle (G,\cdot )} to a group ( H , ∗ ) {\displaystyle (H,*)} is a function φ : G → H {\displaystyle \varphi :G\to H} such that It would be natural to require also that φ {\displaystyle \varphi } respect identities, ⁠ φ ( 1 G ) = 1 H {\displaystyle \varphi (1_{G})=1_{H}} ⁠ , and inverses, φ (

2560-724: A group arose in the study of polynomial equations , starting with Évariste Galois in the 1830s, who introduced the term group (French: groupe ) for the symmetry group of the roots of an equation, now called a Galois group . After contributions from other fields such as number theory and geometry, the group notion was generalized and firmly established around 1870. Modern group theory —an active mathematical discipline—studies groups in their own right. To explore groups, mathematicians have devised various notions to break groups into smaller, better-understandable pieces, such as subgroups , quotient groups and simple groups . In addition to their abstract properties, group theorists also study

2688-411: A group called the dihedral group of degree four, denoted ⁠ D 4 {\displaystyle \mathrm {D} _{4}} ⁠ . The underlying set of the group is the above set of symmetries, and the group operation is function composition. Two symmetries are combined by composing them as functions, that is, applying the first one to the square, and the second one to the result of

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2816-467: A group is an ordered pair of a set and a binary operation on this set that satisfies the group axioms . The set is called the underlying set of the group, and the operation is called the group operation or the group law . A group and its underlying set are thus two different mathematical objects . To avoid cumbersome notation, it is common to abuse notation by using the same symbol to denote both. This reflects also an informal way of thinking: that

2944-587: A homomorphism of rings from the topological K-theory of a space to (the completion of) its rational cohomology. For a line bundle L , the Chern character ch is defined by More generally, if V = L 1 ⊕ ⋯ ⊕ L n {\displaystyle V=L_{1}\oplus \dots \oplus L_{n}} is a direct sum of line bundles, with first Chern classes x i = c 1 ( L i ) , {\displaystyle x_{i}=c_{1}(L_{i}),}

3072-492: A left identity (namely, ⁠ e {\displaystyle e} ⁠ ), and each element has a right inverse (which is e {\displaystyle e} for both elements). Furthermore, this operation is associative (since the product of any number of elements is always equal to the rightmost element in that product, regardless of the order in which these operations are done). However, ( G , ⋅ ) {\displaystyle (G,\cdot )}

3200-729: A left inverse is also a right inverse for the same element. Since they define exactly the same structures as groups, collectively the axioms are not weaker. In particular, assuming associativity and the existence of a left identity e {\displaystyle e} (that is, ⁠ e ⋅ f = f {\displaystyle e\cdot f=f} ⁠ ) and a left inverse f − 1 {\displaystyle f^{-1}} for each element f {\displaystyle f} (that is, ⁠ f − 1 ⋅ f = e {\displaystyle f^{-1}\cdot f=e} ⁠ ), one can show that every left inverse

3328-573: A linear algebraic group G {\displaystyle G} , via Quillen's Q-construction ; thus, by definition, In particular, K 0 G ( C ) {\displaystyle K_{0}^{G}(C)} is the Grothendieck group of Coh G ⁡ ( X ) {\displaystyle \operatorname {Coh} ^{G}(X)} . The theory was developed by R. W. Thomason in 1980s. Specifically, he proved equivariant analogs of fundamental theorems such as

3456-441: A model for the group axioms in the definition below. The integers, together with the operation ⁠ + {\displaystyle +} ⁠ , form a mathematical object belonging to a broad class sharing similar structural aspects. To appropriately understand these structures as a collective, the following definition is developed. The axioms for a group are short and natural ... Yet somehow hidden behind these axioms

3584-457: A point in the square to the corresponding point under the symmetry. For example, r 1 {\displaystyle r_{1}} sends a point to its rotation 90° clockwise around the square's center, and f h {\displaystyle f_{\mathrm {h} }} sends a point to its reflection across the square's vertical middle line. Composing two of these symmetries gives another symmetry. These symmetries determine

3712-476: A reflection along the diagonal ( ⁠ f d {\displaystyle f_{\mathrm {d} }} ⁠ ). Using the above symbols, highlighted in blue in the Cayley table: f h ∘ r 3 = f d . {\displaystyle f_{\mathrm {h} }\circ r_{3}=f_{\mathrm {d} }.} Given this set of symmetries and the described operation,

3840-752: A rotation over 360° which leaves the square unchanged. This is easily verified on the table. In contrast to the group of integers above, where the order of the operation is immaterial, it does matter in ⁠ D 4 {\displaystyle \mathrm {D} _{4}} ⁠ , as, for example, f h ∘ r 1 = f c {\displaystyle f_{\mathrm {h} }\circ r_{1}=f_{\mathrm {c} }} but ⁠ r 1 ∘ f h = f d {\displaystyle r_{1}\circ f_{\mathrm {h} }=f_{\mathrm {d} }} ⁠ . In other words, D 4 {\displaystyle \mathrm {D} _{4}}

3968-453: A series of terms, parentheses are usually omitted. The group axioms imply that the identity element is unique; that is, there exists only one identity element: any two identity elements e {\displaystyle e} and f {\displaystyle f} of a group are equal, because the group axioms imply ⁠ e = e ⋅ f = f {\displaystyle e=e\cdot f=f} ⁠ . It

K-theory - Misplaced Pages Continue

4096-444: A symmetry of the square. One of these ways is to first compose a {\displaystyle a} and b {\displaystyle b} into a single symmetry, then to compose that symmetry with ⁠ c {\displaystyle c} ⁠ . The other way is to first compose b {\displaystyle b} and ⁠ c {\displaystyle c} ⁠ , then to compose

4224-402: A uniform theory of groups started with Camille Jordan 's Traité des substitutions et des équations algébriques (1870). Walther von Dyck (1882) introduced the idea of specifying a group by means of generators and relations, and was also the first to give an axiomatic definition of an "abstract group", in the terminology of the time. As of the 20th century, groups gained wide recognition by

4352-759: Is ⁠ i d {\displaystyle \mathrm {id} } ⁠ , as it does not change any symmetry a {\displaystyle a} when composed with it either on the left or on the right. Inverse element : Each symmetry has an inverse: ⁠ i d {\displaystyle \mathrm {id} } ⁠ , the reflections ⁠ f h {\displaystyle f_{\mathrm {h} }} ⁠ , ⁠ f v {\displaystyle f_{\mathrm {v} }} ⁠ , ⁠ f d {\displaystyle f_{\mathrm {d} }} ⁠ , ⁠ f c {\displaystyle f_{\mathrm {c} }} ⁠ and

4480-402: Is ⁠ b ⋅ a − 1 {\displaystyle b\cdot a^{-1}} ⁠ . For each ⁠ a {\displaystyle a} ⁠ , the function G → G {\displaystyle G\to G} that maps each x {\displaystyle x} to x ⋅ a {\displaystyle x\cdot a}

4608-419: Is a bijection ; it is called left multiplication by a {\displaystyle a} or left translation by ⁠ a {\displaystyle a} ⁠ . Similarly, given a {\displaystyle a} and ⁠ b {\displaystyle b} ⁠ , the unique solution to x ⋅ a = b {\displaystyle x\cdot a=b}

4736-413: Is a bijection called right multiplication by a {\displaystyle a} or right translation by ⁠ a {\displaystyle a} ⁠ . The group axioms for identity and inverses may be "weakened" to assert only the existence of a left identity and left inverses . From these one-sided axioms , one can prove that the left identity is also a right identity and

4864-584: Is a convergent spectral sequence E 1 p , q = ∐ x ∈ X ( p ) K − p − q ( k ( x ) ) ⇒ K − p − q ( X ) {\displaystyle E_{1}^{p,q}=\coprod _{x\in X^{(p)}}K^{-p-q}(k(x))\Rightarrow K_{-p-q}(X)} for X ( p ) {\displaystyle X^{(p)}}

4992-520: Is a free K ( X ) {\displaystyle K(X)} -module of rank r with basis 1 , ξ , … , ξ n − 1 {\displaystyle 1,\xi ,\dots ,\xi ^{n-1}} . This formula allows one to compute the Grothendieck group of P F n {\displaystyle \mathbb {P} _{\mathbb {F} }^{n}} . This make it possible to compute

5120-407: Is a natural example; it is contravariant in one argument, covariant in the other. A multifunctor is a generalization of the functor concept to n variables. So, for example, a bifunctor is a multifunctor with n = 2 . Two important consequences of the functor axioms are: One can compose functors, i.e. if F is a functor from A to B and G is a functor from B to C then one can form

5248-530: Is a necessary ingredient for defining K-theory since all definitions start by constructing an abelian monoid from a suitable category and turning it into an abelian group through this universal construction. Given an abelian monoid ( A , + ′ ) {\displaystyle (A,+')} let ∼ {\displaystyle \sim } be the relation on A 2 = A × A {\displaystyle A^{2}=A\times A} defined by if there exists

K-theory - Misplaced Pages Continue

5376-448: Is a set, ( R , + ) {\displaystyle (\mathbb {R} ,+)} is a group, and ( R , + , ⋅ ) {\displaystyle (\mathbb {R} ,+,\cdot )} is a field . But it is common to write R {\displaystyle \mathbb {R} } to denote any of these three objects. The additive group of the field R {\displaystyle \mathbb {R} }

5504-472: Is a short exact sequence where C Y / X {\displaystyle C_{Y/X}} is the conormal bundle of Y {\displaystyle Y} in X {\displaystyle X} . If we have a singular space Y {\displaystyle Y} embedded into a smooth space X {\displaystyle X} we define the virtual conormal bundle as Another useful application of virtual bundles

5632-443: Is also a right inverse of the same element as follows. Indeed, one has Similarly, the left identity is also a right identity: These proofs require all three axioms (associativity, existence of left identity and existence of left inverse). For a structure with a looser definition (like a semigroup ) one may have, for example, that a left identity is not necessarily a right identity. The same result can be obtained by only assuming

5760-491: Is also called K 0 ( X ) {\displaystyle K^{0}(X)} . One of the main techniques for computing the Grothendieck group for topological spaces comes from the Atiyah–Hirzebruch spectral sequence , which makes it very accessible. The only required computations for understanding the spectral sequences are computing the group K 0 {\displaystyle K^{0}} for

5888-540: Is an abelian monoid where the unit is given by the trivial vector bundle R 0 × X → X {\displaystyle \mathbb {R} ^{0}\times X\to X} . We can then apply the Grothendieck completion to get an abelian group from this abelian monoid. This is called the K-theory of X {\displaystyle X} and is denoted K 0 ( X ) {\displaystyle K^{0}(X)} . We can use

6016-476: Is applied. The words category and functor were borrowed by mathematicians from the philosophers Aristotle and Rudolf Carnap , respectively. The latter used functor in a linguistic context; see function word . Let C and D be categories . A functor F from C to D is a mapping that That is, functors must preserve identity morphisms and composition of morphisms. There are many constructions in mathematics that would be functors but for

6144-427: Is just a finite dimensional vector space, which is a free object in the category of coherent sheaves, hence projective, the monoid of isomorphism classes is N {\displaystyle \mathbb {N} } corresponding to the dimension of the vector space. It is an easy exercise to show that the Grothendieck group is then Z {\displaystyle \mathbb {Z} } . One important property of

6272-436: Is left adjoint to the corresponding forgetful functor U : A b G r p → A b M o n . {\displaystyle U:\mathbf {AbGrp} \to \mathbf {AbMon} .} That means that, given a morphism ϕ : A → U ( B ) {\displaystyle \phi :A\to U(B)} of an abelian monoid A {\displaystyle A} to

6400-423: Is multiplication. More generally, one speaks of an additive group whenever the group operation is notated as addition; in this case, the identity is typically denoted ⁠ 0 {\displaystyle 0} ⁠ , and the inverse of an element x {\displaystyle x} is denoted ⁠ − x {\displaystyle -x} ⁠ . Similarly, one speaks of

6528-448: Is not a group, since it lacks a right identity. When studying sets, one uses concepts such as subset , function, and quotient by an equivalence relation . When studying groups, one uses instead subgroups , homomorphisms , and quotient groups . These are the analogues that take the group structure into account. Group homomorphisms are functions that respect group structure; they may be used to relate two groups. A homomorphism from

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6656-443: Is not abelian. The modern concept of an abstract group developed out of several fields of mathematics. The original motivation for group theory was the quest for solutions of polynomial equations of degree higher than 4. The 19th-century French mathematician Évariste Galois , extending prior work of Paolo Ruffini and Joseph-Louis Lagrange , gave a criterion for the solvability of a particular polynomial equation in terms of

6784-553: Is observed that the coordinate transformation symbol Λ i j {\displaystyle \Lambda _{i}^{j}} (representing the matrix Λ T {\displaystyle {\boldsymbol {\Lambda }}^{\textsf {T}}} ) acts on the "covector coordinates" "in the same way" as on the basis vectors: e i = Λ i j e j {\displaystyle \mathbf {e} _{i}=\Lambda _{i}^{j}\mathbf {e} _{j}} —whereas it acts "in

6912-451: Is referred to as the Grothendieck group ; K ( X ) has cohomological behavior and G ( X ) has homological behavior. If X is a smooth variety , the two groups are the same. If it is a smooth affine variety , then all extensions of locally free sheaves split, so the group has an alternative definition. In topology , by applying the same construction to vector bundles , Michael Atiyah and Friedrich Hirzebruch defined K ( X ) for

7040-615: Is smooth. The group K 0 ( X ) {\displaystyle K_{0}(X)} is special because there is also a ring structure: we define it as Using the Grothendieck–Riemann–Roch theorem , we have that is an isomorphism of rings. Hence we can use K 0 ( X ) {\displaystyle K_{0}(X)} for intersection theory . The subject can be said to begin with Alexander Grothendieck (1957), who used it to formulate his Grothendieck–Riemann–Roch theorem . It takes its name from

7168-660: Is the monster simple group , a huge and extraordinary mathematical object, which appears to rely on numerous bizarre coincidences to exist. The axioms for groups give no obvious hint that anything like this exists. Richard Borcherds , Mathematicians: An Outer View of the Inner World A group is a non-empty set G {\displaystyle G} together with a binary operation on ⁠ G {\displaystyle G} ⁠ , here denoted " ⁠ ⋅ {\displaystyle \cdot } ⁠ ", that combines any two elements

7296-521: Is the group whose underlying set is R {\displaystyle \mathbb {R} } and whose operation is addition. The multiplicative group of the field R {\displaystyle \mathbb {R} } is the group R × {\displaystyle \mathbb {R} ^{\times }} whose underlying set is the set of nonzero real numbers R ∖ { 0 } {\displaystyle \mathbb {R} \smallsetminus \{0\}} and whose operation

7424-399: Is the usual notation for composition of functions. A Cayley table lists the results of all such compositions possible. For example, rotating by 270° clockwise ( ⁠ r 3 {\displaystyle r_{3}} ⁠ ) and then reflecting horizontally ( ⁠ f h {\displaystyle f_{\mathrm {h} }} ⁠ ) is the same as performing

7552-419: Is thus customary to speak of the identity element of the group. The group axioms also imply that the inverse of each element is unique. Let a group element a {\displaystyle a} have both b {\displaystyle b} and c {\displaystyle c} as inverses. Then Therefore, it is customary to speak of the inverse of an element. Given elements

7680-443: Is used. Several other notations are commonly used for groups whose elements are not numbers. For a group whose elements are functions , the operation is often function composition ⁠ f ∘ g {\displaystyle f\circ g} ⁠ ; then the identity may be denoted id. In the more specific cases of geometric transformation groups, symmetry groups, permutation groups , and automorphism groups ,

7808-551: Is with the definition of a virtual tangent bundle of an intersection of spaces: Let Y 1 , Y 2 ⊂ X {\displaystyle Y_{1},Y_{2}\subset X} be projective subvarieties of a smooth projective variety. Then, we can define the virtual tangent bundle of their intersection Z = Y 1 ∩ Y 2 {\displaystyle Z=Y_{1}\cap Y_{2}} as Kontsevich uses this construction in one of his papers. Chern classes can be used to construct

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7936-403: The K 0 {\displaystyle K_{0}} or Hirzebruch surfaces. In addition, this can be used to compute the Grothendieck group K ( P n ) {\displaystyle K(\mathbb {P} ^{n})} by observing it is a projective bundle over the field F {\displaystyle \mathbb {F} } . One recent technique for computing

8064-617: The ( a , 0 ) {\displaystyle (a,0)} as positive integers and the ( 0 , b ) {\displaystyle (0,b)} as negative integers. There are a number of basic definitions of K-theory: two coming from topology and two from algebraic geometry. Given a compact Hausdorff space X {\displaystyle X} consider the set of isomorphism classes of finite-dimensional vector bundles over X {\displaystyle X} , denoted Vect ( X ) {\displaystyle {\text{Vect}}(X)} and let

8192-786: The Serre–Swan theorem and some algebra to get an alternative description of vector bundles over the ring of continuous complex-valued functions C 0 ( X ; C ) {\displaystyle C^{0}(X;\mathbb {C} )} as projective modules . Then, these can be identified with idempotent matrices in some ring of matrices M n × n ( C 0 ( X ; C ) ) {\displaystyle M_{n\times n}(C^{0}(X;\mathbb {C} ))} . We can define equivalence classes of idempotent matrices and form an abelian monoid Idem ( X ) {\displaystyle {\textbf {Idem}}(X)} . Its Grothendieck completion

8320-535: The Singularity category D s g ( X ) {\displaystyle D_{sg}(X)} from derived noncommutative algebraic geometry . It gives a long exact sequence starting with ⋯ → K 0 ( X ) → K 0 ( X ) → K s g ( X ) → 0 {\displaystyle \cdots \to K^{0}(X)\to K_{0}(X)\to K_{sg}(X)\to 0} where

8448-490: The opposite categories to C {\displaystyle C} and D {\displaystyle D} . By definition, F o p {\displaystyle F^{\mathrm {op} }} maps objects and morphisms in the identical way as does F {\displaystyle F} . Since C o p {\displaystyle C^{\mathrm {op} }} does not coincide with C {\displaystyle C} as

8576-414: The opposite functor F o p : C o p → D o p {\displaystyle F^{\mathrm {op} }\colon C^{\mathrm {op} }\to D^{\mathrm {op} }} , where C o p {\displaystyle C^{\mathrm {op} }} and D o p {\displaystyle D^{\mathrm {op} }} are

8704-572: The symmetry group of its roots (solutions). The elements of such a Galois group correspond to certain permutations of the roots. At first, Galois's ideas were rejected by his contemporaries, and published only posthumously. More general permutation groups were investigated in particular by Augustin Louis Cauchy . Arthur Cayley 's On the theory of groups, as depending on the symbolic equation θ n = 1 {\displaystyle \theta ^{n}=1} (1854) gives

8832-423: The 180° rotation r 2 {\displaystyle r_{2}} are their own inverse, because performing them twice brings the square back to its original orientation. The rotations r 3 {\displaystyle r_{3}} and r 1 {\displaystyle r_{1}} are each other's inverses, because rotating 90° and then rotation 270° (or vice versa) yields

8960-715: The Cayley table: ( f d ∘ f v ) ∘ r 2 = r 3 ∘ r 2 = r 1 f d ∘ ( f v ∘ r 2 ) = f d ∘ f h = r 1 . {\displaystyle {\begin{aligned}(f_{\mathrm {d} }\circ f_{\mathrm {v} })\circ r_{2}&=r_{3}\circ r_{2}=r_{1}\\f_{\mathrm {d} }\circ (f_{\mathrm {v} }\circ r_{2})&=f_{\mathrm {d} }\circ f_{\mathrm {h} }=r_{1}.\end{aligned}}} Identity element : The identity element

9088-622: The Chern character is defined additively The Chern character is useful in part because it facilitates the computation of the Chern class of a tensor product. The Chern character is used in the Hirzebruch–Riemann–Roch theorem . The equivariant algebraic K-theory is an algebraic K-theory associated to the category Coh G ⁡ ( X ) {\displaystyle \operatorname {Coh} ^{G}(X)} of equivariant coherent sheaves on an algebraic scheme X {\displaystyle X} with action of

9216-1272: The Chow ring of X {\displaystyle X} , essentially giving the computation of K 0 ( C ) {\displaystyle K_{0}(C)} . Note that because C {\displaystyle C} has no codimension 2 {\displaystyle 2} points, the only nontrivial parts of the spectral sequence are E 1 0 , q , E 1 1 , q {\displaystyle E_{1}^{0,q},E_{1}^{1,q}} , hence E ∞ 1 , − 1 ≅ E 2 1 , − 1 ≅ CH 1 ( C ) E ∞ 0 , 0 ≅ E 2 0 , 0 ≅ CH 0 ( C ) {\displaystyle {\begin{aligned}E_{\infty }^{1,-1}\cong E_{2}^{1,-1}&\cong {\text{CH}}^{1}(C)\\E_{\infty }^{0,0}\cong E_{2}^{0,0}&\cong {\text{CH}}^{0}(C)\end{aligned}}} The coniveau filtration can then be used to determine K 0 ( C ) {\displaystyle K_{0}(C)} as

9344-559: The German Klasse , meaning "class". Grothendieck needed to work with coherent sheaves on an algebraic variety X . Rather than working directly with the sheaves, he defined a group using isomorphism classes of sheaves as generators of the group, subject to a relation that identifies any extension of two sheaves with their sum. The resulting group is called K ( X ) when only locally free sheaves are used, or G ( X ) when all are coherent sheaves. Either of these two constructions

9472-482: The Grothendieck group is K 0 ( C ) = Z ⊕ Pic ( C ) {\displaystyle K_{0}(C)=\mathbb {Z} \oplus {\text{Pic}}(C)} for Picard group of C {\displaystyle C} . This follows from the Brown-Gersten-Quillen spectral sequence of algebraic K-theory . For a regular scheme of finite type over a field, there

9600-564: The Grothendieck group is the projective bundle formula: given a rank r vector bundle E {\displaystyle {\mathcal {E}}} over a Noetherian scheme X {\displaystyle X} , the Grothendieck group of the projective bundle P ( E ) = Proj ⁡ ( Sym ∙ ⁡ ( E ∨ ) ) {\displaystyle \mathbb {P} ({\mathcal {E}})=\operatorname {Proj} (\operatorname {Sym} ^{\bullet }({\mathcal {E}}^{\vee }))}

9728-442: The Grothendieck group of P n {\displaystyle \mathbb {P} ^{n}} comes from its stratification as P n = A n ∐ A n − 1 ∐ ⋯ ∐ A 0 {\displaystyle \mathbb {P} ^{n}=\mathbb {A} ^{n}\coprod \mathbb {A} ^{n-1}\coprod \cdots \coprod \mathbb {A} ^{0}} since

9856-902: The Grothendieck group of a Noetherian scheme X {\displaystyle X} is that it is invariant under reduction, hence K ( X ) = K ( X red ) {\displaystyle K(X)=K(X_{\text{red}})} . Hence the Grothendieck group of any Artinian F {\displaystyle \mathbb {F} } -algebra is a direct sum of copies of Z {\displaystyle \mathbb {Z} } , one for each connected component of its spectrum. For example, K 0 ( Spec ( F [ x ] ( x 9 ) × F ) ) = Z ⊕ Z {\displaystyle K_{0}\left({\text{Spec}}\left({\frac {\mathbb {F} [x]}{(x^{9})}}\times \mathbb {F} \right)\right)=\mathbb {Z} \oplus \mathbb {Z} } One of

9984-852: The Grothendieck group of coherent sheaves on affine spaces are isomorphic to Z {\displaystyle \mathbb {Z} } , and the intersection of A n − k 1 , A n − k 2 {\displaystyle \mathbb {A} ^{n-k_{1}},\mathbb {A} ^{n-k_{2}}} is generically A n − k 1 ∩ A n − k 2 = A n − k 1 − k 2 {\displaystyle \mathbb {A} ^{n-k_{1}}\cap \mathbb {A} ^{n-k_{2}}=\mathbb {A} ^{n-k_{1}-k_{2}}} for k 1 + k 2 ≤ n {\displaystyle k_{1}+k_{2}\leq n} . Another important formula for

10112-411: The Grothendieck group of spaces with minor singularities comes from evaluating the difference between K 0 ( X ) {\displaystyle K^{0}(X)} and K 0 ( X ) {\displaystyle K_{0}(X)} , which comes from the fact every vector bundle can be equivalently described as a coherent sheaf. This is done using the Grothendieck group of

10240-401: The Grothendieck group on weighted projective spaces since they typically have isolated quotient singularities. In particular, if these singularities have isotropy groups G i {\displaystyle G_{i}} then the map K 0 ( X ) → K 0 ( X ) {\displaystyle K^{0}(X)\to K_{0}(X)} is injective and

10368-422: The K-theory classification of Ramond–Ramond field strengths and the charges of stable D-branes was first proposed in 1997. The easiest example of the Grothendieck group is the Grothendieck group of a point Spec ( F ) {\displaystyle {\text{Spec}}(\mathbb {F} )} for a field F {\displaystyle \mathbb {F} } . Since a vector bundle over this space

10496-475: The application of the Grothendieck construction on this abelian monoid. In algebraic geometry, the same construction can be applied to algebraic vector bundles over a smooth scheme. But, there is an alternative construction for any Noetherian scheme X {\displaystyle X} . If we look at the isomorphism classes of coherent sheaves Coh ⁡ ( X ) {\displaystyle \operatorname {Coh} (X)} we can mod out by

10624-418: The cokernel is annihilated by lcm ( | G 1 | , … , | G k | ) n − 1 {\displaystyle {\text{lcm}}(|G_{1}|,\ldots ,|G_{k}|)^{n-1}} for n = dim ⁡ X {\displaystyle n=\dim X} . For a smooth projective curve C {\displaystyle C}

10752-446: The composite functor G ∘ F from A to C . Composition of functors is associative where defined. Identity of composition of functors is the identity functor. This shows that functors can be considered as morphisms in categories of categories, for example in the category of small categories . A small category with a single object is the same thing as a monoid : the morphisms of a one-object category can be thought of as elements of

10880-728: The counter-diagonal ( ⁠ f c {\displaystyle f_{\mathrm {c} }} ⁠ ). Indeed, every other combination of two symmetries still gives a symmetry, as can be checked using the Cayley table. Associativity : The associativity axiom deals with composing more than two symmetries: Starting with three elements ⁠ a {\displaystyle a} ⁠ , ⁠ b {\displaystyle b} ⁠ and ⁠ c {\displaystyle c} ⁠ of ⁠ D 4 {\displaystyle \mathrm {D} _{4}} ⁠ , there are two possible ways of using these three symmetries in this order to determine

11008-433: The desired explicit direct sum since it gives an exact sequence 0 → F 1 ( K 0 ( X ) ) → K 0 ( X ) → K 0 ( X ) / F 1 ( K 0 ( X ) ) → 0 {\displaystyle 0\to F^{1}(K_{0}(X))\to K_{0}(X)\to K_{0}(X)/F^{1}(K_{0}(X))\to 0} where

11136-507: The different ways in which a group can be expressed concretely, both from a point of view of representation theory (that is, through the representations of the group ) and of computational group theory . A theory has been developed for finite groups , which culminated with the classification of finite simple groups , completed in 2004. Since the mid-1980s, geometric group theory , which studies finitely generated groups as geometric objects, has become an active area in group theory. One of

11264-404: The direct sum ⊕ {\displaystyle \oplus } of isomorphisms classes of vector bundles is well-defined, giving an abelian monoid ( Vect ( X ) , ⊕ ) {\displaystyle ({\text{Vect}}(X),\oplus )} . Then, the Grothendieck group K 0 ( X ) {\displaystyle K^{0}(X)} is defined by

11392-434: The equation from the equivalence relation to get n = n . {\displaystyle n=n.} This implies hence we have an additive inverse for each element in G ( A ) {\displaystyle G(A)} . This should give us the hint that we should be thinking of the equivalence classes [ ( a , b ) ] {\displaystyle [(a,b)]} as formal differences

11520-677: The existence of a right identity and a right inverse. However, only assuming the existence of a left identity and a right inverse (or vice versa) is not sufficient to define a group. For example, consider the set G = { e , f } {\displaystyle G=\{e,f\}} with the operator ⋅ {\displaystyle \cdot } satisfying e ⋅ e = f ⋅ e = e {\displaystyle e\cdot e=f\cdot e=e} and ⁠ e ⋅ f = f ⋅ f = f {\displaystyle e\cdot f=f\cdot f=f} ⁠ . This structure does have

11648-424: The fact that they "turn morphisms around" and "reverse composition". We then define a contravariant functor F from C to D as a mapping that Variance of functor (composite) Note that contravariant functors reverse the direction of composition. Ordinary functors are also called covariant functors in order to distinguish them from contravariant ones. Note that one can also define a contravariant functor as

11776-431: The final step taken by Aschbacher and Smith in 2004. This project exceeded previous mathematical endeavours by its sheer size, in both length of proof and number of researchers. Research concerning this classification proof is ongoing. Group theory remains a highly active mathematical branch, impacting many other fields, as the examples below illustrate. Basic facts about all groups that can be obtained directly from

11904-403: The first abstract definition of a finite group . Geometry was a second field in which groups were used systematically, especially symmetry groups as part of Felix Klein 's 1872 Erlangen program . After novel geometries such as hyperbolic and projective geometry had emerged, Klein used group theory to organize them in a more coherent way. Further advancing these ideas, Sophus Lie founded

12032-440: The first application. The result of performing first a {\displaystyle a} and then b {\displaystyle b} is written symbolically from right to left as b ∘ a {\displaystyle b\circ a} ("apply the symmetry b {\displaystyle b} after performing the symmetry ⁠ a {\displaystyle a} ⁠ "). This

12160-525: The group axioms are commonly subsumed under elementary group theory . For example, repeated applications of the associativity axiom show that the unambiguity of a ⋅ b ⋅ c = ( a ⋅ b ) ⋅ c = a ⋅ ( b ⋅ c ) {\displaystyle a\cdot b\cdot c=(a\cdot b)\cdot c=a\cdot (b\cdot c)} generalizes to more than three factors. Because this implies that parentheses can be inserted anywhere within such

12288-602: The group axioms can be understood as follows. Binary operation : Composition is a binary operation. That is, a ∘ b {\displaystyle a\circ b} is a symmetry for any two symmetries a {\displaystyle a} and ⁠ b {\displaystyle b} ⁠ . For example, r 3 ∘ f h = f c , {\displaystyle r_{3}\circ f_{\mathrm {h} }=f_{\mathrm {c} },} that is, rotating 270° clockwise after reflecting horizontally equals reflecting along

12416-492: The group is the same as the set except that it has been enriched by additional structure provided by the operation. For example, consider the set of real numbers ⁠ R {\displaystyle \mathbb {R} } ⁠ , which has the operations of addition a + b {\displaystyle a+b} and multiplication ⁠ a b {\displaystyle ab} ⁠ . Formally, R {\displaystyle \mathbb {R} }

12544-409: The higher terms come from higher K-theory . Note that vector bundles on a singular X {\displaystyle X} are given by vector bundles E → X s m {\displaystyle E\to X_{sm}} on the smooth locus X s m ↪ X {\displaystyle X_{sm}\hookrightarrow X} . This makes it possible to compute

12672-404: The identity element of ( G ( A ) , + ) . {\displaystyle (G(A),+).} First, ( 0 , 0 ) ∼ ( n , n ) {\displaystyle (0,0)\sim (n,n)} for any n ∈ A {\displaystyle n\in A} since we can set c = 0 {\displaystyle c=0} and apply

12800-424: The integers in a unique way). The concept of a group was elaborated for handling, in a unified way, many mathematical structures such as numbers, geometric shapes and polynomial roots . Because the concept of groups is ubiquitous in numerous areas both within and outside mathematics, some authors consider it as a central organizing principle of contemporary mathematics. In geometry , groups arise naturally in

12928-484: The isomorphism class of a vector bundle π : E → X {\displaystyle \pi :E\to X} be denoted [ E ] {\displaystyle [E]} . Since isomorphism classes of vector bundles behave well with respect to direct sums , we can write these operations on isomorphism classes by It should be clear that ( Vect ( X ) , ⊕ ) {\displaystyle ({\text{Vect}}(X),\oplus )}

13056-534: The left hand term is isomorphic to CH 1 ( C ) ≅ Pic ( C ) {\displaystyle {\text{CH}}^{1}(C)\cong {\text{Pic}}(C)} and the right hand term is isomorphic to C H 0 ( C ) ≅ Z {\displaystyle CH^{0}(C)\cong \mathbb {Z} } . Since Ext Ab 1 ( Z , G ) = 0 {\displaystyle {\text{Ext}}_{\text{Ab}}^{1}(\mathbb {Z} ,G)=0} , we have

13184-571: The localization theorem. Functor In mathematics , specifically category theory , a functor is a mapping between categories . Functors were first considered in algebraic topology , where algebraic objects (such as the fundamental group ) are associated to topological spaces , and maps between these algebraic objects are associated to continuous maps between spaces. Nowadays, functors are used throughout modern mathematics to relate various categories. Thus, functors are important in all areas within mathematics to which category theory

13312-878: The mapped rings than from the original spaces or schemes. Examples of results gleaned from the K-theory approach include the Grothendieck–Riemann–Roch theorem , Bott periodicity , the Atiyah–Singer index theorem , and the Adams operations . In high energy physics , K-theory and in particular twisted K-theory have appeared in Type II string theory where it has been conjectured that they classify D-branes , Ramond–Ramond field strengths and also certain spinors on generalized complex manifolds . In condensed matter physics K-theory has been used to classify topological insulators , superconductors and stable Fermi surfaces . For more details, see K-theory (physics) . The Grothendieck completion of an abelian monoid into an abelian group

13440-407: The monoid, and composition in the category is thought of as the monoid operation. Functors between one-object categories correspond to monoid homomorphisms . So in a sense, functors between arbitrary categories are a kind of generalization of monoid homomorphisms to categories with more than one object. Let C and D be categories. The collection of all functors from C to D forms the objects of

13568-388: The more familiar groups is the set of integers Z = { … , − 4 , − 3 , − 2 , − 1 , 0 , 1 , 2 , 3 , 4 , … } {\displaystyle \mathbb {Z} =\{\ldots ,-4,-3,-2,-1,0,1,2,3,4,\ldots \}} together with addition . For any two integers

13696-486: The most commonly used computations of the Grothendieck group is with the computation of K ( P n ) {\displaystyle K(\mathbb {P} ^{n})} for projective space over a field. This is because the intersection numbers of a projective X {\displaystyle X} can be computed by embedding i : X ↪ P n {\displaystyle i:X\hookrightarrow \mathbb {P} ^{n}} and using

13824-426: The operation symbol is usually omitted entirely, so that the operation is denoted by juxtaposition, a b {\displaystyle ab} instead of ⁠ a ⋅ b {\displaystyle a\cdot b} ⁠ . The definition of a group does not require that a ⋅ b = b ⋅ a {\displaystyle a\cdot b=b\cdot a} for all elements

13952-676: The opposite way" on the "vector coordinates" (but "in the same way" as on the basis covectors: e i = Λ j i e j {\displaystyle \mathbf {e} ^{i}=\Lambda _{j}^{i}\mathbf {e} ^{j}} ). This terminology is contrary to the one used in category theory because it is the covectors that have pullbacks in general and are thus contravariant , whereas vectors in general are covariant since they can be pushed forward . See also Covariance and contravariance of vectors . Every functor F : C → D {\displaystyle F\colon C\to D} induces

14080-407: The pioneering work of Ferdinand Georg Frobenius and William Burnside (who worked on representation theory of finite groups), Richard Brauer 's modular representation theory and Issai Schur 's papers. The theory of Lie groups, and more generally locally compact groups was studied by Hermann Weyl , Élie Cartan and many others. Its algebraic counterpart, the theory of algebraic groups ,

14208-406: The programming language Haskell has a class Functor where fmap is a polytypic function used to map functions ( morphisms on Hask , the category of Haskell types) between existing types to functions between some new types. Group (mathematics) In mathematics , a group is a set with an operation that associates an element of the set to every pair of elements of

14336-481: The property of opposite category , ( F o p ) o p = F {\displaystyle \left(F^{\mathrm {op} }\right)^{\mathrm {op} }=F} . A bifunctor (also known as a binary functor ) is a functor whose domain is a product category . For example, the Hom functor is of the type C × C → Set . It can be seen as a functor in two arguments. The Hom functor

14464-695: The push pull formula i ∗ ( [ i ∗ E ] ⋅ [ i ∗ F ] ) {\displaystyle i^{*}([i_{*}{\mathcal {E}}]\cdot [i_{*}{\mathcal {F}}])} . This makes it possible to do concrete calculations with elements in K ( X ) {\displaystyle K(X)} without having to explicitly know its structure since K ( P n ) = Z [ T ] ( T n + 1 ) {\displaystyle K(\mathbb {P} ^{n})={\frac {\mathbb {Z} [T]}{(T^{n+1})}}} One technique for determining

14592-584: The relation [ E ] = [ E ′ ] + [ E ″ ] {\displaystyle [{\mathcal {E}}]=[{\mathcal {E}}']+[{\mathcal {E}}'']} if there is a short exact sequence This gives the Grothendieck-group K 0 ( X ) {\displaystyle K_{0}(X)} which is isomorphic to K 0 ( X ) {\displaystyle K^{0}(X)} if X {\displaystyle X}

14720-713: The resulting symmetry with ⁠ a {\displaystyle a} ⁠ . These two ways must give always the same result, that is, ( a ∘ b ) ∘ c = a ∘ ( b ∘ c ) , {\displaystyle (a\circ b)\circ c=a\circ (b\circ c),} For example, ( f d ∘ f v ) ∘ r 2 = f d ∘ ( f v ∘ r 2 ) {\displaystyle (f_{\mathrm {d} }\circ f_{\mathrm {v} })\circ r_{2}=f_{\mathrm {d} }\circ (f_{\mathrm {v} }\circ r_{2})} can be checked using

14848-537: The sequence of abelian groups above splits, giving the isomorphism. Note that if C {\displaystyle C} is a smooth projective curve of genus g {\displaystyle g} over C {\displaystyle \mathbb {C} } , then K 0 ( C ) ≅ Z ⊕ ( C g / Z 2 g ) {\displaystyle K_{0}(C)\cong \mathbb {Z} \oplus (\mathbb {C} ^{g}/\mathbb {Z} ^{2g})} Moreover,

14976-499: The set (as does every binary operation) and satisfies the following constraints: the operation is associative , it has an identity element , and every element of the set has an inverse element . Many mathematical structures are groups endowed with other properties. For example, the integers with the addition operation form an infinite group, which is generated by a single element called ⁠ 1 {\displaystyle 1} ⁠ (these properties characterize

15104-584: The set of codimension p {\displaystyle p} points, meaning the set of subschemes x : Y → X {\displaystyle x:Y\to X} of codimension p {\displaystyle p} , and k ( x ) {\displaystyle k(x)} the algebraic function field of the subscheme. This spectral sequence has the property E 2 p , − p ≅ CH p ( X ) {\displaystyle E_{2}^{p,-p}\cong {\text{CH}}^{p}(X)} for

15232-1196: The space of sections Γ ( T ∗ M ) {\displaystyle \Gamma {\mathord {\left(T^{*}M\right)}}} of a cotangent bundle T ∗ M {\displaystyle T^{*}M} —as "covariant". This terminology originates in physics, and its rationale has to do with the position of the indices ("upstairs" and "downstairs") in expressions such as x ′ i = Λ j i x j {\displaystyle {x'}^{\,i}=\Lambda _{j}^{i}x^{j}} for x ′ = Λ x {\displaystyle \mathbf {x} '={\boldsymbol {\Lambda }}\mathbf {x} } or ω i ′ = Λ i j ω j {\displaystyle \omega '_{i}=\Lambda _{i}^{j}\omega _{j}} for ω ′ = ω Λ T . {\displaystyle {\boldsymbol {\omega }}'={\boldsymbol {\omega }}{\boldsymbol {\Lambda }}^{\textsf {T}}.} In this formalism it

15360-461: The spheres S n {\displaystyle S^{n}} . There is an analogous construction by considering vector bundles in algebraic geometry . For a Noetherian scheme X {\displaystyle X} there is a set Vect ( X ) {\displaystyle {\text{Vect}}(X)} of all isomorphism classes of algebraic vector bundles on X {\displaystyle X} . Then, as before,

15488-488: The structure of a group ( G ( A ) , + ) {\displaystyle (G(A),+)} where: Equivalence classes in this group should be thought of as formal differences of elements in the abelian monoid. This group ( G ( A ) , + ) {\displaystyle (G(A),+)} is also associated with a monoid homomorphism i : A → G ( A ) {\displaystyle i:A\to G(A)} given by

15616-566: The study of Lie groups in 1884. The third field contributing to group theory was number theory . Certain abelian group structures had been used implicitly in Carl Friedrich Gauss 's number-theoretical work Disquisitiones Arithmeticae (1798), and more explicitly by Leopold Kronecker . In 1847, Ernst Kummer made early attempts to prove Fermat's Last Theorem by developing groups describing factorization into prime numbers . The convergence of these various sources into

15744-558: The study of symmetries and geometric transformations : The symmetries of an object form a group, called the symmetry group of the object, and the transformations of a given type form a general group. Lie groups appear in symmetry groups in geometry, and also in the Standard Model of particle physics . The Poincaré group is a Lie group consisting of the symmetries of spacetime in special relativity . Point groups describe symmetry in molecular chemistry . The concept of

15872-946: The symbol ∘ {\displaystyle \circ } is often omitted, as for multiplicative groups. Many other variants of notation may be encountered. Two figures in the plane are congruent if one can be changed into the other using a combination of rotations , reflections , and translations . Any figure is congruent to itself. However, some figures are congruent to themselves in more than one way, and these extra congruences are called symmetries . A square has eight symmetries. These are: [REDACTED] f h {\displaystyle f_{\mathrm {h} }} (horizontal reflection) [REDACTED] f d {\displaystyle f_{\mathrm {d} }} (diagonal reflection) [REDACTED] f c {\displaystyle f_{\mathrm {c} }} (counter-diagonal reflection) These symmetries are functions. Each sends

16000-575: The techniques above using the derived category of singularities for isolated singularities can be extended to isolated Cohen-Macaulay singularities, giving techniques for computing the Grothendieck group of any singular algebraic curve. This is because reduction gives a generically smooth curve, and all singularities are Cohen-Macaulay. One useful application of the Grothendieck-group is to define virtual vector bundles. For example, if we have an embedding of smooth spaces Y ↪ X {\displaystyle Y\hookrightarrow X} then there

16128-574: The underlying abelian monoid of an abelian group B , {\displaystyle B,} there exists a unique abelian group morphism G ( A ) → B . {\displaystyle G(A)\to B.} An illustrative example to look at is the Grothendieck completion of N {\displaystyle \mathbb {N} } . We can see that G ( ( N , + ) ) = ( Z , + ) . {\displaystyle G((\mathbb {N} ,+))=(\mathbb {Z} ,+).} For any pair (

16256-432: Was also given by Friedhelm Waldhausen in order to study the algebraic K-theory of spaces, which is related to the study of pseudo-isotopies. Much modern research on higher K-theory is related to algebraic geometry and the study of motivic cohomology . The corresponding constructions involving an auxiliary quadratic form received the general name L-theory . It is a major tool of surgery theory . In string theory ,

16384-431: Was first shaped by Claude Chevalley (from the late 1930s) and later by the work of Armand Borel and Jacques Tits . The University of Chicago 's 1960–61 Group Theory Year brought together group theorists such as Daniel Gorenstein , John G. Thompson and Walter Feit , laying the foundation of a collaboration that, with input from numerous other mathematicians, led to the classification of finite simple groups , with

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