[{"@context":"http:\/\/schema.org\/","@type":"BlogPosting","@id":"https:\/\/wiki.edu.vn\/en\/wiki3\/mapping-cone-homological-algebra-wikipedia\/#BlogPosting","mainEntityOfPage":"https:\/\/wiki.edu.vn\/en\/wiki3\/mapping-cone-homological-algebra-wikipedia\/","headline":"Mapping cone (homological algebra) – Wikipedia","name":"Mapping cone (homological algebra) – Wikipedia","description":"before-content-x4 From Wikipedia, the free encyclopedia after-content-x4 Tool in homological algebra In homological algebra, the mapping cone is a construction","datePublished":"2014-09-22","dateModified":"2014-09-22","author":{"@type":"Person","@id":"https:\/\/wiki.edu.vn\/en\/wiki3\/author\/lordneo\/#Person","name":"lordneo","url":"https:\/\/wiki.edu.vn\/en\/wiki3\/author\/lordneo\/","image":{"@type":"ImageObject","@id":"https:\/\/secure.gravatar.com\/avatar\/c9645c498c9701c88b89b8537773dd7c?s=96&d=mm&r=g","url":"https:\/\/secure.gravatar.com\/avatar\/c9645c498c9701c88b89b8537773dd7c?s=96&d=mm&r=g","height":96,"width":96}},"publisher":{"@type":"Organization","name":"Enzyklop\u00e4die","logo":{"@type":"ImageObject","@id":"https:\/\/wiki.edu.vn\/wiki4\/wp-content\/uploads\/2023\/08\/download.jpg","url":"https:\/\/wiki.edu.vn\/wiki4\/wp-content\/uploads\/2023\/08\/download.jpg","width":600,"height":60}},"image":{"@type":"ImageObject","@id":"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/96c3298ea9aa77c226be56a7d8515baaa517b90b","url":"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/96c3298ea9aa77c226be56a7d8515baaa517b90b","height":"","width":""},"url":"https:\/\/wiki.edu.vn\/en\/wiki3\/mapping-cone-homological-algebra-wikipedia\/","wordCount":6714,"articleBody":"     (adsbygoogle = window.adsbygoogle || []).push({});before-content-x4From Wikipedia, the free encyclopedia       (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4Tool in homological algebraIn homological algebra, the mapping cone is a construction on a map of chain complexes inspired by the analogous construction in topology.  In the theory of triangulated categories it is a kind of combined kernel and cokernel: if the chain complexes take their terms in an abelian category, so that we can talk about cohomology, then the cone of a map f being acyclic means that the map is a quasi-isomorphism; if we pass to the derived category of complexes, this means that f is an isomorphism there, which recalls the familiar property of maps of groups, modules over a ring, or elements of an arbitrary abelian category that if the kernel and cokernel both vanish, then the map is an isomorphism.  If we are working in a t-category, then in fact the cone furnishes both the kernel and cokernel of maps between objects of its core.       (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4Table of ContentsDefinition[edit]Properties[edit]Mapping cylinder[edit]Topological inspiration[edit]References[edit]Definition[edit]The cone may be defined in the category of cochain complexes over any additive category (i.e., a category whose morphisms form abelian groups and in which we may construct a direct sum of any two objects).  Let A,B{displaystyle A,B} be two complexes, with differentials        (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4dA,dB;{displaystyle d_{A},d_{B};} i.e.,A=\u22ef\u2192An\u22121\u2192dAn\u22121An\u2192dAnAn+1\u2192\u22ef{displaystyle A=dots to A^{n-1}{xrightarrow {d_{A}^{n-1}}}A^{n}{xrightarrow {d_{A}^{n}}}A^{n+1}to cdots }and likewise for B.{displaystyle B.}For a map of complexes f:A\u2192B,{displaystyle f:Ato B,} we define the cone, often denoted by Cone\u2061(f){displaystyle operatorname {Cone} (f)} or C(f),{displaystyle C(f),} to be the following complex:C(f)=A[1]\u2295B=\u22ef\u2192An\u2295Bn\u22121\u2192An+1\u2295Bn\u2192An+2\u2295Bn+1\u2192\u22ef{displaystyle C(f)=A[1]oplus B=dots to A^{n}oplus B^{n-1}to A^{n+1}oplus B^{n}to A^{n+2}oplus B^{n+1}to cdots } on terms,with differentialdC(f)=(dA[1]0f[1]dB){displaystyle d_{C(f)}={begin{pmatrix}d_{A[1]}&0\\f[1]&d_{B}end{pmatrix}}} (acting as though on column vectors).Here A[1]{displaystyle A[1]} is the complex with A[1]n=An+1{displaystyle A[1]^{n}=A^{n+1}} and dA[1]n=\u2212dAn+1{displaystyle d_{A[1]}^{n}=-d_{A}^{n+1}}.Note that the differential on C(f){displaystyle C(f)} is different from the natural differential on A[1]\u2295B{displaystyle A[1]oplus B}, and that some authors use a different sign convention.Thus, if for example our complexes are of abelian groups, the differential would act asdC(f)n(an+1,bn)=(dA[1]n0f[1]ndBn)(an+1bn)=(\u2212dAn+10fn+1dBn)(an+1bn)=(\u2212dAn+1(an+1)fn+1(an+1)+dBn(bn))=(\u2212dAn+1(an+1),fn+1(an+1)+dBn(bn)).{displaystyle {begin{array}{ccl}d_{C(f)}^{n}(a^{n+1},b^{n})&=&{begin{pmatrix}d_{A[1]}^{n}&0\\f[1]^{n}&d_{B}^{n}end{pmatrix}}{begin{pmatrix}a^{n+1}\\b^{n}end{pmatrix}}\\&=&{begin{pmatrix}-d_{A}^{n+1}&0\\f^{n+1}&d_{B}^{n}end{pmatrix}}{begin{pmatrix}a^{n+1}\\b^{n}end{pmatrix}}\\&=&{begin{pmatrix}-d_{A}^{n+1}(a^{n+1})\\f^{n+1}(a^{n+1})+d_{B}^{n}(b^{n})end{pmatrix}}\\&=&left(-d_{A}^{n+1}(a^{n+1}),f^{n+1}(a^{n+1})+d_{B}^{n}(b^{n})right).end{array}}}Properties[edit]Suppose now that we are working over an abelian category, so that the homology of a complex is defined. The main use of the cone is to identify quasi-isomorphisms: if the cone is acyclic, then the map is a quasi-isomorphism.  To see this, we use the existence of a triangleA\u2192fB\u2192C(f)\u2192A[1]{displaystyle A{xrightarrow {f}}Bto C(f)to A[1]}where the maps B\u2192C(f),C(f)\u2192A[1]{displaystyle Bto C(f),C(f)to A[1]} are given by the direct summands (see Homotopy category of chain complexes).  Since this is a triangle, it gives rise to a long exact sequence on homology groups:\u22ef\u2192Hi\u22121(C(f))\u2192Hi(A)\u2192f\u2217Hi(B)\u2192Hi(C(f))\u2192\u22ef{displaystyle dots to H_{i-1}(C(f))to H_{i}(A){xrightarrow {f^{*}}}H_{i}(B)to H_{i}(C(f))to cdots }and if C(f){displaystyle C(f)} is acyclic then by definition, the outer terms above are zero.  Since the sequence is exact, this means that f\u2217{displaystyle f^{*}} induces an isomorphism on all homology groups, and hence (again by definition) is a quasi-isomorphism.This fact recalls the usual alternative characterization of isomorphisms in an abelian category as those maps whose kernel and cokernel both vanish.  This appearance of a cone as a combined kernel and cokernel is not accidental; in fact, under certain circumstances the cone literally embodies both.  Say for example that we are working over an abelian category and A,B{displaystyle A,B} have only one nonzero term in degree 0:A=\u22ef\u21920\u2192A0\u21920\u2192\u22ef,{displaystyle A=dots to 0to A_{0}to 0to cdots ,}B=\u22ef\u21920\u2192B0\u21920\u2192\u22ef,{displaystyle B=dots to 0to B_{0}to 0to cdots ,}and therefore f:A\u2192B{displaystyle fcolon Ato B} is just f0:A0\u2192B0{displaystyle f_{0}colon A_{0}to B_{0}} (as a map of objects of the underlying abelian category).  Then the cone is justC(f)=\u22ef\u21920\u2192A0[\u22121]\u2192f0B0[0]\u21920\u2192\u22ef.{displaystyle C(f)=dots to 0to {underset {[-1]}{A_{0}}}{xrightarrow {f_{0}}}{underset {[0]}{B_{0}}}to 0to cdots .}(Underset text indicates the degree of each term.) The homology of this complex is thenH\u22121(C(f))=ker\u2061(f0),{displaystyle H_{-1}(C(f))=operatorname {ker} (f_{0}),}H0(C(f))=coker\u2061(f0),{displaystyle H_{0}(C(f))=operatorname {coker} (f_{0}),}Hi(C(f))=0\u00a0for\u00a0i\u2260\u22121,0.\u00a0{displaystyle H_{i}(C(f))=0{text{ for }}ineq -1,0. }This is not an accident and in fact occurs in every t-category.Mapping cylinder[edit]A related notion is the mapping cylinder: let f:A\u2192B{displaystyle fcolon Ato B} be a morphism of chain complexes, let further g:Cone\u2061(f)[\u22121]\u2192A{displaystyle gcolon operatorname {Cone} (f)[-1]to A} be the natural map. The mapping cylinder of f is by definition the mapping cone of g.Topological inspiration[edit]This complex is called the cone in analogy to the mapping cone (topology) of a continuous map of topological spaces \u03d5:X\u2192Y{displaystyle phi :Xrightarrow Y}: the complex of singular chains of the topological cone cone(\u03d5){displaystyle cone(phi )} is homotopy equivalent to the cone (in the chain-complex-sense) of the induced map of singular chains of X to Y. The mapping cylinder of a map of complexes is similarly related to the mapping cylinder of continuous maps.References[edit]     (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4"},{"@context":"http:\/\/schema.org\/","@type":"BreadcrumbList","itemListElement":[{"@type":"ListItem","position":1,"item":{"@id":"https:\/\/wiki.edu.vn\/en\/wiki3\/#breadcrumbitem","name":"Enzyklop\u00e4die"}},{"@type":"ListItem","position":2,"item":{"@id":"https:\/\/wiki.edu.vn\/en\/wiki3\/mapping-cone-homological-algebra-wikipedia\/#breadcrumbitem","name":"Mapping cone (homological algebra) – Wikipedia"}}]}]