<?xml version="1.0" encoding="UTF-8" ?>DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN" "http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd " >html xmlns="http://www.w3.org/1999/xhtml " xml:lang="en" >head >script type="text/javascript" "https://cdn.jsdelivr.net/npm/mathjax@2/MathJax.js?config=TeX-AMS-MML_HTMLorMML " >script >title >GAP (MatricesForHomalg) - Chapter 4: Ring Maps</title >meta http-equiv="content-type" content="text/html; charset=UTF-8" />meta name="generator" content="GAPDoc2HTML" />link rel="stylesheet" type="text/css" href="manual.css" />script src="manual.js" type="text/javascript" ></script >script type="text/javascript" >overwriteStyle();</script >head >body class="chap4" onload="jscontent()" >div class="chlinktop" ><span class="chlink1" >Goto Chapter: </span ><a href="chap0_mj.html" >Top</a> <a href="chap1_mj.html" >1</a> <a href="chap2_mj.html" >2</a> <a href="chap3_mj.html" >3</a> <a href="chap4_mj.html" >4</a> <a href="chap5_mj.html" >5</a> <a href="chap6_mj.html" >6</a> <a href="chapA_mj.html" >A</a> <a href="chapB_mj.html" >B</a> <a href="chapC_mj.html" >C</a> <a href="chapD_mj.html" >D</a> <a href="chapE_mj.html" >E</a> <a href="chapF_mj.html" >F</a> <a href="chapBib_mj.html" >Bib</a> <a href="chapInd_mj.html" div >div class="chlinkprevnexttop" > <a href="chap0_mj.html" >[Top of Book]</a> <a href="chap0_mj.html#contents" >[Contents]</a> <a href="chap3_mj.html" >[Previous Chapter]</a> <a href="chap5_mj.html" >[Next Chapter]</a> </div >"mathjaxlink" class="pcenter" ><a href="chap4.html" >[MathJax off]</a></p>"X7B222197819984A6" name="X7B222197819984A6" ></a></p>div class="ChapSects" ><a href="chap4_mj.html#X7B222197819984A6" >4 <span class="Heading" >Ring Maps</span ></a>div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap4_mj.html#X7B99B8F5780E84C3" >4.1 <span class="Heading" >Ring Maps: Category and Representations</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap4_mj.html#X7E084D947E3AEFE6" >4.1-1 IsHomalgRingMap</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap4_mj.html#X87DB79AF83F17FB6" >4.1-2 IsHomalgRingSelfMap</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap4_mj.html#X7DFD1CBA83E63737" >4.1-3 IsHomalgRingMapRep</a></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap4_mj.html#X8717AEFB7BAC63F7" >4.2 <span class="Heading" >Ring Maps: Constructors</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap4_mj.html#X7F21AB318507FF83" >4.2-1 RingMap</a></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap4_mj.html#X85DA972D8701BC7C" >4.3 <span class="Heading" >Ring Maps: Properties</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap4_mj.html#X8555A4DF84C9165B" >4.3-1 IsMorphism</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap4_mj.html#X832893897FD3744D" >4.3-2 IsIdentityMorphism</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap4_mj.html#X87F79EA381E3E34F" >4.3-3 IsMonomorphism</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap4_mj.html#X849F620C824F4078" >4.3-4 IsEpimorphism</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap4_mj.html#X82B9422D7B01BA4A" >4.3-5 IsIsomorphism</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap4_mj.html#X790E34C5802D0F54" >4.3-6 IsAutomorphism</a></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap4_mj.html#X7EBF1DD67BD0758F" >4.4 <span class="Heading" >Ring Maps: Attributes</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap4_mj.html#X83678DEC78394702" >4.4-1 Source </a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap4_mj.html#X7EBE68567900396A" >4.4-2 Range</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap4_mj.html#X7C4F3F0F82C6EB88" >4.4-3 DegreeOfMorphism</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap4_mj.html#X785155EE844A98BD" >4.4-4 CoordinateRingOfGraph</a></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap4_mj.html#X7C7401BA7E2221CB" >4.5 <span class="Heading" >Ring Maps: Operations and Functions</span ></a>span >div >div >span class="Heading" >Ring Maps</span ></h3>strong class="pkg" >homalg</strong > ring map is a data structure for maps between finitely generated rings. <strong class="pkg" >homalg</strong > more or less provides the basic declarations and installs the generic methods for ring maps, but it is up to other high level packages to install methods applicable to specific rings. For example, the package <strong class="pkg" >Sheaves</strong > provides methods for ring maps of (finitely generated) affine rings.</p>"X7B99B8F5780E84C3" name="X7B99B8F5780E84C3" ></a></p>span class="Heading" >Ring Maps: Category and Representations</span ></h4>"X7E084D947E3AEFE6" name="X7E084D947E3AEFE6" ></a></p>div class="func" ><table class="func" width="100%" ><tr ><td class="tdleft" ><code class="func" >‣ IsHomalgRingMap</code >( <var class="Arg" >phi</var > )</td ><td class="tdright" >( category )</td ></tr ></table ></div >code class="code" >true</code > or <code class="code" >false</code ></p>strong class="pkg" >GAP</strong > category of ring maps.</p>"X87DB79AF83F17FB6" name="X87DB79AF83F17FB6" ></a></p>div class="func" ><table class="func" width="100%" ><tr ><td class="tdleft" ><code class="func" >‣ IsHomalgRingSelfMap</code >( <var class="Arg" >phi</var > )</td ><td class="tdright" >( category )</td ></tr ></table ></div >code class="code" >true</code > or <code class="code" >false</code ></p>strong class="pkg" >GAP</strong > category of ring self-maps.</p>strong class="pkg" >GAP</strong > category <code class="code" >IsHomalgRingMap</code >.)</p>"X7DFD1CBA83E63737" name="X7DFD1CBA83E63737" ></a></p>div class="func" ><table class="func" width="100%" ><tr ><td class="tdleft" ><code class="func" >‣ IsHomalgRingMapRep</code >( <var class="Arg" >phi</var > )</td ><td class="tdright" >( representation )</td tr ></table ></div >code class="code" >true</code > or <code class="code" >false</code ></p>strong class="pkg" >GAP</strong > representation of <strong class="pkg" >homalg</strong > ring maps.</p>strong class="pkg" >GAP</strong > category <code class="func" >IsHomalgRingMap</code > (<a href="chap4_mj.html#X7E084D947E3AEFE6" ><span class="RefLink" >4.1-1</span ></a>).)</p>"X8717AEFB7BAC63F7" name="X8717AEFB7BAC63F7" ></a></p>span class="Heading" >Ring Maps: Constructors</span ></h4>"X7F21AB318507FF83" name="X7F21AB318507FF83" ></a></p>div class="func" ><table class="func" width="100%" ><tr ><td class="tdleft" ><code class="func" >‣ RingMap</code >( <var class="Arg" >images</var >, <var class="Arg" >S</var >, <var class="Arg" >T</var > )</td ><td class="tdright" >( operation )</td ></tr ></table ></div >strong class="pkg" >homalg</strong > ring map </p>map (homomorphism) of finitely generated rings/algebras. It is represented by the images <var class="Arg" >images</var > of the set of generators of the source <strong class="pkg" >homalg</strong > ring <var class="Arg" >S</var > in terms of the generators of the target ring <var class="Arg" >T</var > (--> <a href="chap3_mj.html#X7C7962B97E6CDFE2" ><span class="RefLink" >3.2</span ></a>). Unless the source ring is free <em >and</em > given on free ring/algebra generators the returned map will cautiously be indicated using parenthesis: <q>homomorphism</q>. If source and target are identical objects, and only then, the ring map is created as a selfmap.</p>"X85DA972D8701BC7C" name="X85DA972D8701BC7C" ></a></p>span class="Heading" >Ring Maps: Properties</span ></h4>"X8555A4DF84C9165B" name="X8555A4DF84C9165B" ></a></p>div class="func" ><table class="func" width="100%" ><tr ><td class="tdleft" ><code class="func" >‣ IsMorphism</code >( <var class="Arg" >phi</var > )</td ><td class="tdright" >( property )</td ></tr ></table ></div >code class="code" >true</code > or <code class="code" >false</code ></p>var class="Arg" >phi</var > is a well-defined map , i.e. independent of all involved presentations.</p>"X832893897FD3744D" name="X832893897FD3744D" ></a></p>div class="func" ><table class="func" width="100%" ><tr ><td class="tdleft" ><code class="func" >‣ IsIdentityMorphism</code >( <var class="Arg" >phi</var > )</td ><td class="tdright" >( property )</td ></tr ></table div >code class="code" >true</code > or <code class="code" >false</code ></p>strong class="pkg" >homalg</strong > ring map <var class="Arg" >phi</var > is the identity morphism.</p>"X87F79EA381E3E34F" name="X87F79EA381E3E34F" ></a></p>div class="func" ><table class="func" width="100%" ><tr ><td class="tdleft" ><code class="func" >‣ IsMonomorphism</code >( <var class="Arg" >phi</var > )</td ><td class="tdright" >( property )</td ></tr ></table ></div >code class="code" >true</code > or <code class="code" >false</code ></p>strong class="pkg" >homalg</strong > ring map <var class="Arg" >phi</var > is a monomorphism.</p>"X849F620C824F4078" name="X849F620C824F4078" ></a></p>div class="func" ><table class="func" width="100%" ><tr ><td class="tdleft" ><code class="func" >‣ IsEpimorphism</code >( <var class="Arg" >phi</var > )</td ><td class="tdright" >( property )</td ></tr ></table ></div >code class="code" >true</code > or <code class="code" >false</code ></p>strong class="pkg" >homalg</strong > ring map <var class="Arg" >phi</var > is an epimorphism.</p>"X82B9422D7B01BA4A" name="X82B9422D7B01BA4A" ></a></p>div class="func" ><table class="func" width="100%" ><tr ><td class="tdleft" ><code class="func" >‣ IsIsomorphism</code >( <var class="Arg" >phi</var > )</td ><td class="tdright" >( property )</td ></tr ></table ></div >code class="code" >true</code > or <code class="code" >false</code ></p>strong class="pkg" >homalg</strong > ring map <var class="Arg" >phi</var > is an isomorphism.</p>"X790E34C5802D0F54" name="X790E34C5802D0F54" ></a></p>div class="func" ><table class="func" width="100%" ><tr ><td class="tdleft" ><code class="func" >‣ IsAutomorphism</code >( <var class="Arg" >phi</var > )</td ><td class="tdright" >( property )</td ></tr ></table ></div >code class="code" >true</code > or <code class="code" >false</code ></p>strong class="pkg" >homalg</strong > ring map <var class="Arg" >phi</var > is an automorphism.</p>"X7EBF1DD67BD0758F" name="X7EBF1DD67BD0758F" ></a></p>span class="Heading" >Ring Maps: Attributes</span ></h4>"X83678DEC78394702" name="X83678DEC78394702" ></a></p>Source </h5>div class="func" ><table class="func" width="100%" ><tr ><td class="tdleft" ><code class="func" >‣ Source </code >( <var class="Arg" >phi</var > )</td ><td class="tdright" >( attribute )</td ></tr ></table ></div >strong class="pkg" >homalg</strong > ring</p>source of the <strong class="pkg" >homalg</strong > ring map <var class="Arg" >phi</var >.</p>"X7EBE68567900396A" name="X7EBE68567900396A" ></a></p>div class="func" ><table class="func" width="100%" ><tr ><td class="tdleft" ><code class="func" >‣ Range</code >( <var class="Arg" >phi</var > )</td ><td class="tdright" >( attribute )</td ></tr ></table ></div >strong class="pkg" >homalg</strong > ring</p>strong class="pkg" >homalg</strong > ring map <var class="Arg" >phi</var >.</p>"X7C4F3F0F82C6EB88" name="X7C4F3F0F82C6EB88" ></a></p>div class="func" ><table class="func" width="100%" ><tr ><td class="tdleft" ><code class="func" >‣ DegreeOfMorphism</code >( <var class="Arg" >phi</var > )</td ><td class="tdright" >( attribute )</td ></tr ></table ></div >var class="Arg" >phi</var > of graded rings. <br /> (no method installed)</p>"X785155EE844A98BD" name="X785155EE844A98BD" ></a></p>div class="func" ><table class="func" width="100%" ><tr ><td class="tdleft" ><code class="func" >‣ CoordinateRingOfGraph</code >( <var class="Arg" >phi</var > )</td ><td class="tdright" >( attribute )</td ></tr ></table ></div >strong class="pkg" >homalg</strong > ring</p>map <var class="Arg" >phi</var >.</p>"X7C7401BA7E2221CB" name="X7C7401BA7E2221CB" ></a></p>span class="Heading" >Ring Maps: Operations and Functions</span ></h4>div class="chlinkprevnextbot" > <a href="chap0_mj.html" >[Top of Book]</a> <a href="chap0_mj.html#contents" >[Contents]</a> <a href="chap3_mj.html" >[Previous Chapter]</a> <a href="chap5_mj.html" >[Next Chapter]</a> </div >div class="chlinkbot" ><span class="chlink1" >Goto Chapter: </span ><a href="chap0_mj.html" >Top</a> <a href="chap1_mj.html" >1</a> <a href="chap2_mj.html" >2</a> <a href="chap3_mj.html" >3</a> <a href="chap4_mj.html" >4</a> <a href="chap5_mj.html" >5</a> <a href="chap6_mj.html" >6</a> <a href="chapA_mj.html" >A</a> <a href="chapB_mj.html" >B</a> <a href="chapC_mj.html" >C</a> <a href="chapD_mj.html" >D</a> <a href="chapE_mj.html" >E</a> <a href="chapF_mj.html" >F</a> <a href="chapBib_mj.html" >Bib</a> <a href="chapInd_mj.html" div >hr />"foot" >generated by <a href="https://www.math.rwth-aachen.de/~Frank.Luebeck/GAPDoc " >GAPDoc2HTML</a></p>body >html >quality 100%
¤ Dauer der Verarbeitung: 0.19 Sekunden
(vorverarbeitet)
¤ *© Formatika GbR, Deutschland
