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GAP 4.8.9 installation with standard packages -- copy to your CoCalc project to get it
Project: cocalc-sagemath-dev-slelievre
Views: 418346<?xml version="1.0" encoding="UTF-8"?>12<!-- This is an automatically generated file. -->3<Chapter Label="Chapter_Module_Presentations">4<Heading>Module Presentations</Heading>56<Section Label="Chapter_Module_Presentations_Section_Functors">7<Heading>Functors</Heading>89<ManSection>10<Attr Arg="R" Name="FunctorStandardModuleLeft" Label="for IsHomalgRing"/>11<Returns>a functor12</Returns>13<Description>14The argument is a homalg ring <Math>R</Math>.15The output is a functor which takes16a left presentation as input and computes17its standard presentation.18</Description>19</ManSection>202122<ManSection>23<Attr Arg="R" Name="FunctorStandardModuleRight" Label="for IsHomalgRing"/>24<Returns>a functor25</Returns>26<Description>27The argument is a homalg ring <Math>R</Math>.28The output is a functor which takes29a right presentation as input and computes30its standard presentation.31</Description>32</ManSection>333435<ManSection>36<Attr Arg="R" Name="FunctorGetRidOfZeroGeneratorsLeft" Label="for IsHomalgRing"/>37<Returns>a functor38</Returns>39<Description>40The argument is a homalg ring <Math>R</Math>.41The output is a functor which takes42a left presentation as input and gets43rid of the zero generators.44</Description>45</ManSection>464748<ManSection>49<Attr Arg="R" Name="FunctorGetRidOfZeroGeneratorsRight" Label="for IsHomalgRing"/>50<Returns>a functor51</Returns>52<Description>53The argument is a homalg ring <Math>R</Math>.54The output is a functor which takes55a right presentation as input and gets56rid of the zero generators.57</Description>58</ManSection>596061<ManSection>62<Attr Arg="R" Name="FunctorLessGeneratorsLeft" Label="for IsHomalgRing"/>63<Returns>a functor64</Returns>65<Description>66The argument is a homalg ring <Math>R</Math>.67The output is functor which takes68a left presentation as input and computes69a presentation having less generators.70</Description>71</ManSection>727374<ManSection>75<Attr Arg="R" Name="FunctorLessGeneratorsRight" Label="for IsHomalgRing"/>76<Returns>a functor77</Returns>78<Description>79The argument is a homalg ring <Math>R</Math>.80The output is functor which takes81a right presentation as input and computes82a presentation having less generators.83</Description>84</ManSection>858687<ManSection>88<Attr Arg="R" Name="FunctorDualLeft" Label="for IsHomalgRing"/>89<Returns>a functor90</Returns>91<Description>92The argument is a homalg ring <Math>R</Math> that has an involution function.93The output is functor which takes94a left presentation <A>M</A> as input and computes95its Hom(M, R) as a left presentation.96</Description>97</ManSection>9899100<ManSection>101<Attr Arg="R" Name="FunctorDualRight" Label="for IsHomalgRing"/>102<Returns>a functor103</Returns>104<Description>105The argument is a homalg ring <Math>R</Math> that has an involution function.106The output is functor which takes107a right presentation <A>M</A> as input and computes108its Hom(M, R) as a right presentation.109</Description>110</ManSection>111112113<ManSection>114<Attr Arg="R" Name="FunctorDoubleDualLeft" Label="for IsHomalgRing"/>115<Returns>a functor116</Returns>117<Description>118The argument is a homalg ring <Math>R</Math> that has an involution function.119The output is functor which takes120a left presentation <A>M</A> as input and computes121its <A>Hom( Hom(M, R), R )</A> as a left presentation.122</Description>123</ManSection>124125126<ManSection>127<Attr Arg="R" Name="FunctorDoubleDualRight" Label="for IsHomalgRing"/>128<Returns>a functor129</Returns>130<Description>131The argument is a homalg ring <Math>R</Math> that has an involution function.132The output is functor which takes133a right presentation <A>M</A> as input and computes134its <A>Hom( Hom(M, R), R )</A> as a right presentation.135</Description>136</ManSection>137138139</Section>140141142<Section Label="Chapter_Module_Presentations_Section_GAP_Categories">143<Heading>GAP Categories</Heading>144145<ManSection>146<Filt Arg="object" Name="IsLeftOrRightPresentationMorphism" Label="for IsCapCategoryMorphism"/>147<Returns><C>true</C> or <C>false</C>148</Returns>149<Description>150The GAP category of morphisms in the category151of left or right presentations.152</Description>153</ManSection>154155156<ManSection>157<Filt Arg="object" Name="IsLeftPresentationMorphism" Label="for IsLeftOrRightPresentationMorphism"/>158<Returns><C>true</C> or <C>false</C>159</Returns>160<Description>161The GAP category of morphisms in the category162of left presentations.163</Description>164</ManSection>165166167<ManSection>168<Filt Arg="object" Name="IsRightPresentationMorphism" Label="for IsLeftOrRightPresentationMorphism"/>169<Returns><C>true</C> or <C>false</C>170</Returns>171<Description>172The GAP category of morphisms in the category173of right presentations.174</Description>175</ManSection>176177178<ManSection>179<Filt Arg="object" Name="IsLeftOrRightPresentation" Label="for IsCapCategoryObject"/>180<Returns><C>true</C> or <C>false</C>181</Returns>182<Description>183The GAP category of objects in the category184of left presentations or right presentations.185</Description>186</ManSection>187188189<ManSection>190<Filt Arg="object" Name="IsLeftPresentation" Label="for IsLeftOrRightPresentation"/>191<Returns><C>true</C> or <C>false</C>192</Returns>193<Description>194The GAP category of objects in the category195of left presentations.196</Description>197</ManSection>198199200<ManSection>201<Filt Arg="object" Name="IsRightPresentation" Label="for IsLeftOrRightPresentation"/>202<Returns><C>true</C> or <C>false</C>203</Returns>204<Description>205The GAP category of objects in the category206of right presentations.207</Description>208</ManSection>209210211</Section>212213214<Section Label="Chapter_Module_Presentations_Section_Constructors">215<Heading>Constructors</Heading>216217<ManSection>218<Oper Arg="A, M, B" Name="PresentationMorphism" Label="for IsLeftOrRightPresentation, IsHomalgMatrix, IsLeftOrRightPresentation"/>219<Returns>a morphism in <Math>\mathrm{Hom}(A,B)</Math>220</Returns>221<Description>222The arguments are an object <Math>A</Math>, a homalg matrix <Math>M</Math>,223and another object <Math>B</Math>.224<Math>A</Math> and <Math>B</Math> shall either both be objects in the category225of left presentations or both be objects in the category226of right presentations.227The output is a morphism <Math>A \rightarrow B</Math> in the228the category of left or right presentations whose229underlying matrix is given by <Math>M</Math>.230</Description>231</ManSection>232233234<ManSection>235<Attr Arg="m" Name="AsMorphismBetweenFreeLeftPresentations" Label="for IsHomalgMatrix"/>236<Returns>a morphism in <Math>\mathrm{Hom}(F^r,F^c)</Math>237</Returns>238<Description>239The argument is a homalg matrix <Math>m</Math>.240The output is a morphism <Math>F^r \rightarrow F^c</Math> in the241the category of left presentations whose242underlying matrix is given by <Math>m</Math>,243where <Math>F^r</Math> and <Math>F^c</Math> are free left presentations of244ranks given by the number of rows and columns of <Math>m</Math>.245</Description>246</ManSection>247248249<ManSection>250<Attr Arg="m" Name="AsMorphismBetweenFreeRightPresentations" Label="for IsHomalgMatrix"/>251<Returns>a morphism in <Math>\mathrm{Hom}(F^c,F^r)</Math>252</Returns>253<Description>254The argument is a homalg matrix <Math>m</Math>.255The output is a morphism <Math>F^c \rightarrow F^r</Math> in the256the category of right presentations whose257underlying matrix is given by <Math>m</Math>,258where <Math>F^r</Math> and <Math>F^c</Math> are free right presentations of259ranks given by the number of rows and columns of <Math>m</Math>.260</Description>261</ManSection>262263264<ManSection>265<Oper Arg="M" Name="AsLeftPresentation" Label="for IsHomalgMatrix"/>266<Returns>an object267</Returns>268<Description>269The argument is a homalg matrix <Math>M</Math> over a ring <Math>R</Math>.270The output is an object in the category of left presentations271over <Math>R</Math>. This object has <Math>M</Math> as its underlying matrix.272</Description>273</ManSection>274275276<ManSection>277<Oper Arg="M" Name="AsRightPresentation" Label="for IsHomalgMatrix"/>278<Returns>an object279</Returns>280<Description>281The argument is a homalg matrix <Math>M</Math> over a ring <Math>R</Math>.282The output is an object in the category of right presentations283over <Math>R</Math>. This object has <Math>M</Math> as its underlying matrix.284</Description>285</ManSection>286287288<ManSection>289<Func Arg="M, l" Name="AsLeftOrRightPresentation" />290<Returns>an object291</Returns>292<Description>293The arguments are a homalg matrix <Math>M</Math> and a boolean <Math>l</Math>.294If <Math>l</Math> is <C>true</C>, the output is an object in the category295of left presentations.296If <Math>l</Math> is <C>false</C>, the output is an object in the category297of right presentations.298In both cases, the underlying matrix of the result is <Math>M</Math>.299</Description>300</ManSection>301302303<ManSection>304<Oper Arg="r, R" Name="FreeLeftPresentation" Label="for IsInt, IsHomalgRing"/>305<Returns>an object306</Returns>307<Description>308The arguments are a non-negative integer <Math>r</Math>309and a homalg ring <Math>R</Math>.310The output is an object in the category of left presentations311over <Math>R</Math>. It is represented by the <Math>0 \times r</Math> matrix and312thus it is free of rank <Math>r</Math>.313</Description>314</ManSection>315316317<ManSection>318<Oper Arg="r, R" Name="FreeRightPresentation" Label="for IsInt, IsHomalgRing"/>319<Returns>an object320</Returns>321<Description>322The arguments are a non-negative integer <Math>r</Math>323and a homalg ring <Math>R</Math>.324The output is an object in the category of right presentations325over <Math>R</Math>. It is represented by the <Math>r \times 0</Math> matrix and326thus it is free of rank <Math>r</Math>.327</Description>328</ManSection>329330331<ManSection>332<Attr Arg="A" Name="UnderlyingMatrix" Label="for IsLeftOrRightPresentation"/>333<Returns>a homalg matrix334</Returns>335<Description>336The argument is an object <Math>A</Math> in the category of left or right presentations337over a homalg ring <Math>R</Math>.338The output is the underlying matrix which presents <Math>A</Math>.339</Description>340</ManSection>341342343<ManSection>344<Attr Arg="A" Name="UnderlyingHomalgRing" Label="for IsLeftOrRightPresentation"/>345<Returns>a homalg ring346</Returns>347<Description>348The argument is an object <Math>A</Math> in the category of left or right presentations349over a homalg ring <Math>R</Math>.350The output is <Math>R</Math>.351</Description>352</ManSection>353354355<ManSection>356<Attr Arg="A" Name="Annihilator" Label="for IsLeftOrRightPresentation"/>357<Returns>a morphism in <Math>\mathrm{Hom}(I, F)</Math>358</Returns>359<Description>360The argument is an object <Math>A</Math> in the category of left or right presentations.361The output is the embedding of the annihilator <Math>I</Math> of <Math>A</Math>362into the free module <Math>F</Math> of rank <Math>1</Math>.363In particular, the annihilator itself is seen as a left or right presentation.364</Description>365</ManSection>366367368<ManSection>369<Attr Arg="R" Name="LeftPresentations" Label="for IsHomalgRing"/>370<Returns>a category371</Returns>372<Description>373The argument is a homalg ring <Math>R</Math>.374The output is the category of free left presentations375over <Math>R</Math>.376</Description>377</ManSection>378379380<ManSection>381<Attr Arg="R" Name="RightPresentations" Label="for IsHomalgRing"/>382<Returns>a category383</Returns>384<Description>385The argument is a homalg ring <Math>R</Math>.386The output is the category of free right presentations387over <Math>R</Math>.388</Description>389</ManSection>390391392</Section>393394395<Section Label="Chapter_Module_Presentations_Section_Attributes">396<Heading>Attributes</Heading>397398<ManSection>399<Attr Arg="R" Name="UnderlyingHomalgRing" Label="for IsLeftOrRightPresentationMorphism"/>400<Returns>a homalg ring401</Returns>402<Description>403The argument is a morphism <Math>\alpha</Math> in the category404of left or right presentations over a homalg ring <Math>R</Math>.405The output is <Math>R</Math>.406</Description>407</ManSection>408409410<ManSection>411<Attr Arg="alpha" Name="UnderlyingMatrix" Label="for IsLeftOrRightPresentationMorphism"/>412<Returns>a homalg matrix413</Returns>414<Description>415The argument is a morphism <Math>\alpha</Math> in the category416of left or right presentations.417The output is its underlying homalg matrix.418</Description>419</ManSection>420421422</Section>423424425<Section Label="Chapter_Module_Presentations_Section_Non-Categorical_Operations">426<Heading>Non-Categorical Operations</Heading>427428<ManSection>429<Oper Arg="A, i" Name="StandardGeneratorMorphism" Label="for IsLeftOrRightPresentation, IsInt"/>430<Returns>a morphism in <Math>\mathrm{Hom}(F, A)</Math>431</Returns>432<Description>433The argument is an object <Math>A</Math> in the category of434left or right presentations over a homalg ring <Math>R</Math>435with underlying matrix <Math>M</Math>436and an integer <Math>i</Math>.437The output is a morphism <Math>F \rightarrow A</Math> given438by the <Math>i</Math>-th row or column of <Math>M</Math>, where <Math>F</Math>439is a free left or right presentation of rank <Math>1</Math>.440</Description>441</ManSection>442443444<ManSection>445<Attr Arg="A" Name="CoverByFreeModule" Label="for IsLeftOrRightPresentation"/>446<Returns>a morphism in <Math>\mathrm{Hom}(F,A)</Math>447</Returns>448<Description>449The argument is an object <Math>A</Math> in the category of450left or right presentations.451The output is a morphism from a free module <Math>F</Math>452to <Math>A</Math>, which maps the standard generators of453the free module to the generators of <Math>A</Math>.454</Description>455</ManSection>456457458</Section>459460461<Section Label="Chapter_Module_Presentations_Section_Natural_Transformations">462<Heading>Natural Transformations</Heading>463464<ManSection>465<Attr Arg="R" Name="NaturalIsomorphismFromIdentityToStandardModuleLeft" Label="for IsHomalgRing"/>466<Returns>a natural transformation <Math>\mathrm{Id} \rightarrow \mathrm{StandardModuleLeft}</Math>467</Returns>468<Description>469The argument is a homalg ring <Math>R</Math>.470The output is the natural isomorphism from the identity functor471to the left standard module functor.472</Description>473</ManSection>474475476<ManSection>477<Attr Arg="R" Name="NaturalIsomorphismFromIdentityToStandardModuleRight" Label="for IsHomalgRing"/>478<Returns>a natural transformation <Math>\mathrm{Id} \rightarrow \mathrm{StandardModuleRight}</Math>479</Returns>480<Description>481The argument is a homalg ring <Math>R</Math>.482The output is the natural isomorphism from the identity functor483to the right standard module functor.484</Description>485</ManSection>486487488<ManSection>489<Attr Arg="R" Name="NaturalIsomorphismFromIdentityToGetRidOfZeroGeneratorsLeft" Label="for IsHomalgRing"/>490<Returns>a natural transformation <Math>\mathrm{Id} \rightarrow \mathrm{GetRidOfZeroGeneratorsLeft}</Math>491</Returns>492<Description>493The argument is a homalg ring <Math>R</Math>.494The output is the natural isomorphism from the identity functor495to the functor that gets rid of zero generators of left modules.496</Description>497</ManSection>498499500<ManSection>501<Attr Arg="R" Name="NaturalIsomorphismFromIdentityToGetRidOfZeroGeneratorsRight" Label="for IsHomalgRing"/>502<Returns>a natural transformation <Math>\mathrm{Id} \rightarrow \mathrm{GetRidOfZeroGeneratorsRight}</Math>503</Returns>504<Description>505The argument is a homalg ring <Math>R</Math>.506The output is the natural isomorphism from the identity functor507to the functor that gets rid of zero generators of right modules.508</Description>509</ManSection>510511512<ManSection>513<Attr Arg="R" Name="NaturalIsomorphismFromIdentityToLessGeneratorsLeft" Label="for IsHomalgRing"/>514<Returns>a natural transformation <Math>\mathrm{Id} \rightarrow \mathrm{LessGeneratorsLeft}</Math>515</Returns>516<Description>517The argument is a homalg ring <Math>R</Math>.518The output is the natural morphism from the identity functor519to the left less generators functor.520</Description>521</ManSection>522523524<ManSection>525<Attr Arg="R" Name="NaturalIsomorphismFromIdentityToLessGeneratorsRight" Label="for IsHomalgRing"/>526<Returns>a natural transformation <Math>\mathrm{Id} \rightarrow \mathrm{LessGeneratorsRight}</Math>527</Returns>528<Description>529The argument is a homalg ring <Math>R</Math>.530The output is the natural morphism from the identity functor531to the right less generator functor.532</Description>533</ManSection>534535536<ManSection>537<Attr Arg="R" Name="NaturalTransformationFromIdentityToDoubleDualLeft" Label="for IsHomalgRing"/>538<Returns>a natural transformation <Math>\mathrm{Id} \rightarrow \mathrm{FunctorDoubleDualLeft}</Math>539</Returns>540<Description>541The argument is a homalg ring <Math>R</Math>.542The output is the natural morphism from the identity functor543to the double dual functor in left Presentations category.544</Description>545</ManSection>546547548<ManSection>549<Attr Arg="R" Name="NaturalTransformationFromIdentityToDoubleDualRight" Label="for IsHomalgRing"/>550<Returns>a natural transformation <Math>\mathrm{Id} \rightarrow \mathrm{FunctorDoubleDualRight}</Math>551</Returns>552<Description>553The argument is a homalg ring <Math>R</Math>.554The output is the natural morphism from the identity functor555to the double dual functor in right Presentations category.556</Description>557</ManSection>558559560</Section>561562563</Chapter>564565566567