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<head><title>ChainComplex -- the class of all chain complexes</title>
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<div><h1>ChainComplex -- the class of all chain complexes</h1>
<div class="single"><h2>Description</h2>
<div>For an overview of creating and using chain complexes in Macaulay2, see <a href="_chain_spcomplexes.html" title="">chain complexes</a>.<p/>
Common ways to create a chain complex<ul><li><span><a href="_chain__Complex.html" title="make a chain complex">chainComplex</a> -- make a chain complex</span></li>
<li><span><a href="_resolution_lp__Module_rp.html" title="compute a projective resolution of a module">resolution(Module)</a> -- compute a projective resolution of a module</span></li>
</ul>
Information about a chain complex<ul><li><span><a href="_length_lp__Chain__Complex_rp.html" title="length of a chain complex or graded module">length(ChainComplex)</a> -- length of a chain complex or graded module</span></li>
<li><span><a href="_min_lp__Graded__Module_rp.html" title="minimum of elements of a list">min(GradedModule)</a> -- minimum of elements of a list</span></li>
<li><span><a href="_max_lp__Graded__Module_rp.html" title="maximum of elements of a list">max(GradedModule)</a> -- maximum of elements of a list</span></li>
<li><span><a href="_betti_lp__Graded__Module_rp.html" title="display of degrees in a graded module">betti(GradedModule)</a> -- display of degrees in a graded module</span></li>
<li><span><a href="_ring.html" title="get the associated ring of an object">ring(ChainComplex)</a> -- get the associated ring of an object</span></li>
</ul>
Operations on chain complexes<ul><li><span><a href="_dual_lp__Chain__Complex_rp.html" title="dual">dual(ChainComplex)</a> -- dual</span></li>
<li><span><a href="___Chain__Complex_sp_pl_pl_sp__Chain__Complex.html" title="direct sum">ChainComplex ++ ChainComplex</a> -- direct sum</span></li>
<li><span><a href="___Chain__Complex_sp_st_st_sp__Chain__Complex.html" title="tensor product">ChainComplex ** Module</a> -- tensor product</span></li>
<li><span><a href="___Chain__Complex_sp_st_st_sp__Chain__Complex.html" title="tensor product">ChainComplex ** ChainComplex</a> -- tensor product</span></li>
<li><span><a href="___Hom_lp__Module_cm__Chain__Complex_rp.html" title="">Hom(ChainComplex,Module)</a></span></li>
<li><span><a href="___Chain__Complex_sp__Array.html" title="degree shift">ChainComplex Array</a> -- degree shift</span></li>
</ul>
</div>
</div>
<div class="waystouse"><h2>Functions and methods returning a chain complex :</h2>
<ul><li><span><a href="_chain__Complex.html" title="make a chain complex">chainComplex</a> -- make a chain complex</span></li>
<li><span><a href="___Chain__Complex_sp_st_st_sp__Chain__Complex.html" title="tensor product">ChainComplex ** ChainComplex</a> -- tensor product</span></li>
<li><span>ChainComplex ** Module, see <span><a href="___Chain__Complex_sp_st_st_sp__Chain__Complex.html" title="tensor product">ChainComplex ** ChainComplex</a> -- tensor product</span></span></li>
<li><span>Module ** ChainComplex, see <span><a href="___Chain__Complex_sp_st_st_sp__Chain__Complex.html" title="tensor product">ChainComplex ** ChainComplex</a> -- tensor product</span></span></li>
<li><span><a href="___Chain__Complex_sp_st_st_sp__Graded__Module.html" title="tensor product">ChainComplex ** GradedModule</a> -- tensor product</span></li>
<li><span><a href="___Chain__Complex_sp_st_st_sp__Ring.html" title="a binary operator, usually used for tensor product or Cartesian product">ChainComplex ** Ring</a> -- a binary operator, usually used for tensor product or Cartesian product</span></li>
<li><span><a href="___Chain__Complex_sp_pl_pl_sp__Chain__Complex.html" title="direct sum">ChainComplex ++ ChainComplex</a> -- direct sum</span></li>
<li><span><a href="___Chain__Complex_sp__Array.html" title="degree shift">ChainComplex Array</a> -- degree shift</span></li>
<li><span><a href="_chain__Complex_lp__Graded__Module_rp.html" title="make a chain complex from a graded module">chainComplex(GradedModule)</a> -- make a chain complex from a graded module</span></li>
<li><span>coimage(ChainComplexMap), see <span><a href="_coimage.html" title="coimage of a map">coimage</a> -- coimage of a map</span></span></li>
<li><span>cokernel(ChainComplexMap), see <span><a href="_cokernel.html" title="cokernel of a map of modules, graded modules, or chaincomplexes">cokernel</a> -- cokernel of a map of modules, graded modules, or chaincomplexes</span></span></li>
<li><span><a href="_complete_lp__Chain__Complex_rp.html" title="complete the internal parts">complete(ChainComplex)</a> -- complete the internal parts</span></li>
<li><span><a href="_cone_lp__Chain__Complex__Map_rp.html" title="mapping cone of a chain map">cone(ChainComplexMap)</a> -- mapping cone of a chain map</span></li>
<li><span><a href="_dual_lp__Chain__Complex_rp.html" title="dual">dual(ChainComplex)</a> -- dual</span></li>
<li><span>eagonNorthcott, see <span><a href="_eagon__Northcott_lp__Matrix_rp.html" title="Eagon-Northcott complex of a matrix of linear forms">eagonNorthcott(Matrix)</a> -- Eagon-Northcott complex of a matrix of linear forms</span></span></li>
<li><span><a href="___Graded__Module_sp_st_st_sp__Chain__Complex.html" title="tensor product">GradedModule ** ChainComplex</a> -- tensor product</span></li>
<li><span>Hom(ChainComplex,Module), see <span><a href="___Hom_lp__Module_cm__Chain__Complex_rp.html" title="">Hom(Module,ChainComplex)</a></span></span></li>
<li><span><a href="___Hom_lp__Module_cm__Chain__Complex_rp.html" title="">Hom(Module,ChainComplex)</a></span></li>
<li><span>image(ChainComplexMap), see <span><a href="_image.html" title="image of a map">image</a> -- image of a map</span></span></li>
<li><span><a href="_kernel_lp__Chain__Complex__Map_rp.html" title="kernel of a chain complex map">kernel(ChainComplexMap)</a> -- kernel of a chain complex map</span></li>
<li><span><a href="_koszul_lp__Matrix_rp.html" title="the Koszul complex">koszul(Matrix)</a> -- the Koszul complex</span></li>
<li><span>minimalPresentation(ChainComplex), see <span><a href="_minimal__Presentation_lp__Module_rp.html" title="minimal presentation of a module">minimalPresentation(Module)</a> -- minimal presentation of a module</span></span></li>
<li><span>prune(ChainComplex), see <span><a href="_minimal__Presentation_lp__Module_rp.html" title="minimal presentation of a module">minimalPresentation(Module)</a> -- minimal presentation of a module</span></span></li>
<li><span><a href="___Module_sp__Array.html" title="make a chain complex from a module">Module Array</a> -- make a chain complex from a module</span></li>
<li><span><a href="_new_sp__Chain__Complex.html" title="make a new chain complex from scratch">new ChainComplex</a> -- make a new chain complex from scratch</span></li>
<li><span><a href="_resolution_lp__Ideal_rp.html" title="compute a projective resolution of (the quotient ring corresponding to) an ideal">resolution(Ideal)</a> -- compute a projective resolution of (the quotient ring corresponding to) an ideal</span></li>
<li><span>resolution(MonomialIdeal), see <span><a href="_resolution_lp__Ideal_rp.html" title="compute a projective resolution of (the quotient ring corresponding to) an ideal">resolution(Ideal)</a> -- compute a projective resolution of (the quotient ring corresponding to) an ideal</span></span></li>
<li><span><a href="_resolution_lp__Module_rp.html" title="compute a projective resolution of a module">resolution(Module)</a> -- compute a projective resolution of a module</span></li>
<li><span>RingMap ChainComplex, see <span><a href="___Ring__Map_sp__Ring__Element.html" title="apply a ring map">RingMap RingElement</a> -- apply a ring map</span></span></li>
</ul>
<h2>Methods that use a chain complex :</h2>
<ul><li><span>ChainComplex == ChainComplex, see <span><a href="__eq_eq.html" title="equality">==</a> -- equality</span></span></li>
<li><span>ChainComplex == ZZ, see <span><a href="__eq_eq.html" title="equality">==</a> -- equality</span></span></li>
<li><span>ZZ == ChainComplex, see <span><a href="__eq_eq.html" title="equality">==</a> -- equality</span></span></li>
<li><span><a href="___Chain__Complex_sp_st_st_sp__Chain__Complex__Map.html" title="tensor product">ChainComplex ** ChainComplexMap</a> -- tensor product</span></li>
<li><span><a href="___Chain__Complex_sp^_sp__Z__Z.html" title="access member, cohomological degree">ChainComplex ^ ZZ</a> -- access member, cohomological degree</span></li>
<li><span><a href="___Chain__Complex_sp_us_sp__Z__Z.html" title="component">ChainComplex _ ZZ</a> -- component</span></li>
<li><span><a href="___Chain__Complex_sp_us_sp__Z__Z_sp_eq_sp__Thing.html" title="install component of chain complex">ChainComplex _ ZZ = Thing</a> -- install component of chain complex</span></li>
<li><span><a href="___Chain__Complex__Map_sp_st_st_sp__Chain__Complex.html" title="tensor product">ChainComplexMap ** ChainComplex</a> -- tensor product</span></li>
<li><span><a href="_components_lp__Chain__Complex_rp.html" title="list the components of a direct sum">components(ChainComplex)</a> -- list the components of a direct sum</span></li>
<li><span>directSum(ChainComplex), see <span><a href="_direct__Sum.html" title="direct sum of modules or maps">directSum</a> -- direct sum of modules or maps</span></span></li>
<li><span><a href="_extend_lp__Chain__Complex_cm__Chain__Complex_cm__Matrix_rp.html" title="extend a module map to a chain map, if possible">extend(ChainComplex,ChainComplex,Matrix)</a> -- extend a module map to a chain map, if possible</span></li>
<li><span>gradedModule(ChainComplex), see <span><a href="_graded__Module.html" title="make a graded module">gradedModule</a> -- make a graded module</span></span></li>
<li><span><a href="___H__H_sp__Chain__Complex.html" title="homology of a chain complex">HH ChainComplex</a> -- homology of a chain complex</span></li>
<li><span><a href="___H__H^__Z__Z_sp__Chain__Complex.html" title="cohomology of a chain complex">HH^ZZ ChainComplex</a> -- cohomology of a chain complex</span></li>
<li><span><a href="___H__H_us__Z__Z_sp__Chain__Complex.html" title="homology of a chain complex">HH_ZZ ChainComplex</a> -- homology of a chain complex</span></li>
<li><span>inducedMap(ChainComplex,ChainComplex), see <span><a href="_induced__Map_lp__Module_cm__Module_rp.html" title="compute the map induced by the identity">inducedMap(Module,Module)</a> -- compute the map induced by the identity</span></span></li>
<li><span>isDirectSum(ChainComplex), see <span><a href="_is__Direct__Sum.html" title="whether something is a direct sum">isDirectSum</a> -- whether something is a direct sum</span></span></li>
<li><span>isHomogeneous(ChainComplex), see <span><a href="_is__Homogeneous.html" title="whether something is homogeneous (graded)">isHomogeneous</a> -- whether something is homogeneous (graded)</span></span></li>
<li><span><a href="_length_lp__Chain__Complex_rp.html" title="length of a chain complex or graded module">length(ChainComplex)</a> -- length of a chain complex or graded module</span></li>
<li><span><a href="_map_lp__Chain__Complex_cm__Chain__Complex_cm__Chain__Complex__Map_rp.html" title="">map(ChainComplex,ChainComplex,ChainComplexMap)</a></span></li>
<li><span><a href="_map_lp__Chain__Complex_cm__Chain__Complex_cm__Function_rp.html" title="make a map of chain complexes">map(ChainComplex,ChainComplex,Function)</a> -- make a map of chain complexes</span></li>
<li><span>ChainComplex ^ Array, see <span><a href="___Module_sp^_sp__Array.html" title="projection onto summand">Module ^ Array</a> -- projection onto summand</span></span></li>
<li><span>ChainComplex _ Array, see <span><a href="___Module_sp_us_sp__Array.html" title="inclusion from summand">Module _ Array</a> -- inclusion from summand</span></span></li>
<li><span><a href="_poincare_lp__Chain__Complex_rp.html" title="assemble degrees of a chain complex into a polynomial">poincare(ChainComplex)</a> -- assemble degrees of a chain complex into a polynomial</span></li>
<li><span>poincareN(ChainComplex), see <span><a href="_poincare__N.html" title="assemble degrees into polynomial">poincareN</a> -- assemble degrees into polynomial</span></span></li>
<li><span>regularity(ChainComplex), see <span><a href="_regularity.html" title="compute the Castelnuovo-Mumford regularity">regularity</a> -- compute the Castelnuovo-Mumford regularity</span></span></li>
<li><span>ring(ChainComplex), see <span><a href="_ring.html" title="get the associated ring of an object">ring</a> -- get the associated ring of an object</span></span></li>
<li><span>status(ChainComplex), see <span><a href="_status.html" title="status of a resolution computation">status</a> -- status of a resolution computation</span></span></li>
<li><span><a href="_sum_lp__Chain__Complex_rp.html" title="direct sum of the components of a chain complex">sum(ChainComplex)</a> -- direct sum of the components of a chain complex</span></li>
<li><span>tensorAssociativity(ChainComplex,ChainComplex,ChainComplex), see <span><a href="_tensor__Associativity.html" title="associativity isomorphisms for tensor products">tensorAssociativity</a> -- associativity isomorphisms for tensor products</span></span></li>
</ul>
</div>
<div class="waystouse"><h2>For the programmer</h2>
<p>The object <a href="___Chain__Complex.html" title="the class of all chain complexes">ChainComplex</a> is <span>a <a href="___Type.html">type</a></span>, with ancestor classes <a href="___Graded__Module.html" title="the class of all graded modules">GradedModule</a> < <a href="___Mutable__Hash__Table.html" title="the class of all mutable hash tables">MutableHashTable</a> < <a href="___Hash__Table.html" title="the class of all hash tables">HashTable</a> < <a href="___Thing.html" title="the class of all things">Thing</a>.</p>
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