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<div><h1>BoijSoederberg -- Betti diagram routines</h1>
<div class="single"><h2>Description</h2>
<div><em>BoijSoederberg</em> is a package designed to help with the investigation of the Boij-Soederberg conjectures and theorems. For the definitions and conjectures, see math.AC/0611081, "Graded Betti numbers of Cohen-Macaulay modules and the Multiplicity conjecture", by Mats Boij, Jonas Soederberg.<p/>
<h2>Manipulation of Betti diagrams</h2>
<ul><li><span><a href="_mat2betti_lp__Matrix_cm__Z__Z_rp.html" title="matrix to Betti diagram">mat2betti</a> -- matrix to Betti diagram</span></li>
<li><span><a href="_matrix_lp__Betti__Tally_cm__Z__Z_cm__Z__Z_rp.html" title="Betti diagram to matrix">matrix(BettiTally,ZZ,ZZ)</a> -- Betti diagram to matrix</span></li>
<li><span><a href="_lowest__Degrees_lp__Betti__Tally_rp.html" title="list of lowest degree shifts">lowestDegrees</a> -- list of lowest degree shifts</span></li>
<li><span><a href="_highest__Degrees_lp__Betti__Tally_rp.html" title="list of highest degree shifts">highestDegrees</a> -- list of highest degree shifts</span></li>
<li><span><a href="../../Macaulay2Doc/html/___Betti__Tally.html" title="the class of all Betti tallies">BettiTally</a> -- the class of all Betti tallies</span></li>
</ul>
<h2>Pure Betti diagrams</h2>
<ul><li><span><a href="_pure__Betti_lp__List_rp.html" title="list of smallest integral Betti numbers corresponding to a degree sequence">pureBetti</a> -- list of smallest integral Betti numbers corresponding to a degree sequence</span></li>
<li><span><a href="_pure__Betti__Diagram_lp__List_rp.html" title="pure Betti diagram given a list of degrees">pureBettiDiagram</a> -- pure Betti diagram given a list of degrees</span></li>
<li><span><a href="_is__Pure_lp__Betti__Tally_rp.html" title="is a Betti diagram pure?">isPure</a> -- is a Betti diagram pure?</span></li>
</ul>
<h2>Cohomology tables</h2>
<ul><li><span><a href="___Cohomology__Tally.html" title="cohomology table">CohomologyTally</a> -- cohomology table</span></li>
<li><span><a href="_pure__Cohomology__Table_lp__List_cm__Z__Z_cm__Z__Z_rp.html" title="pure cohomology table given zeros of Hilbert polynomial">pureCohomologyTable</a> -- pure cohomology table given zeros of Hilbert polynomial</span></li>
<li><span><a href="_bott.html" title="cohomology of Schur functors of tautological bundle on P^n">bott</a> -- cohomology of Schur functors of tautological bundle on P^n</span></li>
</ul>
<h2>Decomposition into pure diagrams</h2>
<ul><li><span><a href="_decompose_lp__Betti__Tally_rp.html" title="write a Betti diagram as a positive combination of pure integral diagrams">decompose(BettiTally)</a> -- write a Betti diagram as a positive combination of pure integral diagrams</span></li>
</ul>
<h2>Three constructions for pure resolutions. These routines provide the zero-th betti number given a degree sequence.</h2>
<ul><li><span><a href="_pure__Two__Invariant.html" title="first betti number of specific exact complex">pureTwoInvariant</a> -- first betti number of specific exact complex</span></li>
<li><span><a href="_pure__Weyman.html" title="first betti number of specific exact complex">pureWeyman</a> -- first betti number of specific exact complex</span></li>
<li><span><a href="_pure__Char__Free.html" title="first betti number of specific exact complex">pureCharFree</a> -- first betti number of specific exact complex</span></li>
<li><span><a href="_pure__All.html" title="Vector of first betti number of our three specific exact complexes">pureAll</a> -- Vector of first betti number of our three specific exact complexes</span></li>
</ul>
<h2>Constructions often leading to pure resolutions</h2>
<ul><li><span><a href="_random__Module_lp__List_cm__Z__Z_rp.html" title="module with random relations in prescribed degrees">randomModule</a> -- module with random relations in prescribed degrees</span></li>
<li><span><a href="_random__Socle__Module_lp__List_cm__Z__Z_rp.html" title="random finite length module with prescribed number of socle elements in single degree">randomSocleModule</a> -- random finite length module with prescribed number of socle elements in single degree</span></li>
</ul>
<h2>Facet equation and the dot product between Betti diagrams and cohomology tables</h2>
<ul><li><span><a href="_facet__Equation_lp__List_cm__Z__Z_cm__Z__Z_cm__Z__Z_rp.html" title="The upper facet equation corresponding to (L,i)">facetEquation</a> -- The upper facet equation corresponding to (L,i)</span></li>
<li><span><a href="_dot__Product.html" title="entry by entry dot product of two Betti diagrams">dotProduct</a> -- entry by entry dot product of two Betti diagrams</span></li>
<li><span><tt>supportFunctional</tt> (missing documentation<!-- tag: supportFunctional -->)</span></li>
<li><span><tt>BettiTally * CohomologyTally</tt> (missing documentation<!-- tag: (*,BettiTally,CohomologyTally) -->)</span></li>
</ul>
</div>
</div>
<div class="single"><h2>Authors</h2>
<ul><li><div class="single"><a href="http://www.msri.org/~de/">David Eisenbud</a><span> <<a href="mailto:de@msri.org">de@msri.org</a>></span></div>
</li>
<li><div class="single">Frank Schreyer<span> <<a href="mailto:schreyer@math.uni-sb.de">schreyer@math.uni-sb.de</a>></span></div>
</li>
<li><div class="single"><a href="http://www.math.cornell.edu/~mike">Mike Stillman</a><span> <<a href="mailto:mike@math.cornell.edu">mike@math.cornell.edu</a>></span></div>
</li>
</ul>
</div>
<div class="single"><h2>Version</h2>
This documentation describes version <b>1.2</b> of BoijSoederberg.</div>
<div class="single"><h2>Source code</h2>
The source code from which this documentation is derived is in the file <a href="../../../../Macaulay2/BoijSoederberg.m2">BoijSoederberg.m2</a>.</div>
<div class="single"><h2>Exports</h2>
<ul><li><div class="single">Types<ul><li><span><a href="___Cohomology__Tally.html" title="cohomology table">CohomologyTally</a> -- cohomology table</span></li>
</ul>
</div>
</li>
<li><div class="single">Functions<ul><li><span><a href="_bott.html" title="cohomology of Schur functors of tautological bundle on P^n">bott</a> -- cohomology of Schur functors of tautological bundle on P^n</span></li>
<li><span><a href="_dot__Product.html" title="entry by entry dot product of two Betti diagrams">dotProduct</a> -- entry by entry dot product of two Betti diagrams</span></li>
<li><span>facetEquation, see <span><a href="_facet__Equation_lp__List_cm__Z__Z_cm__Z__Z_cm__Z__Z_rp.html" title="The upper facet equation corresponding to (L,i)">facetEquation(List,ZZ,ZZ,ZZ)</a> -- The upper facet equation corresponding to (L,i)</span></span></li>
<li><span>highestDegrees, see <span><a href="_highest__Degrees_lp__Betti__Tally_rp.html" title="list of highest degree shifts">highestDegrees(BettiTally)</a> -- list of highest degree shifts</span></span></li>
<li><span>isPure, see <span><a href="_is__Pure_lp__Betti__Tally_rp.html" title="is a Betti diagram pure?">isPure(BettiTally)</a> -- is a Betti diagram pure?</span></span></li>
<li><span>lowestDegrees, see <span><a href="_lowest__Degrees_lp__Betti__Tally_rp.html" title="list of lowest degree shifts">lowestDegrees(BettiTally)</a> -- list of lowest degree shifts</span></span></li>
<li><span>mat2betti, see <span><a href="_mat2betti_lp__Matrix_cm__Z__Z_rp.html" title="matrix to Betti diagram">mat2betti(Matrix,ZZ)</a> -- matrix to Betti diagram</span></span></li>
<li><span><tt>mat2cohom</tt> (missing documentation<!-- tag: mat2cohom -->)</span></li>
<li><span><a href="_pure__All.html" title="Vector of first betti number of our three specific exact complexes">pureAll</a> -- Vector of first betti number of our three specific exact complexes</span></li>
<li><span>pureBetti, see <span><a href="_pure__Betti_lp__List_rp.html" title="list of smallest integral Betti numbers corresponding to a degree sequence">pureBetti(List)</a> -- list of smallest integral Betti numbers corresponding to a degree sequence</span></span></li>
<li><span>pureBettiDiagram, see <span><a href="_pure__Betti__Diagram_lp__List_rp.html" title="pure Betti diagram given a list of degrees">pureBettiDiagram(List)</a> -- pure Betti diagram given a list of degrees</span></span></li>
<li><span><a href="_pure__Char__Free.html" title="first betti number of specific exact complex">pureCharFree</a> -- first betti number of specific exact complex</span></li>
<li><span>pureCohomologyTable, see <span><a href="_pure__Cohomology__Table_lp__List_cm__Z__Z_cm__Z__Z_rp.html" title="pure cohomology table given zeros of Hilbert polynomial">pureCohomologyTable(List,ZZ,ZZ)</a> -- pure cohomology table given zeros of Hilbert polynomial</span></span></li>
<li><span><a href="_pure__Two__Invariant.html" title="first betti number of specific exact complex">pureTwoInvariant</a> -- first betti number of specific exact complex</span></li>
<li><span><a href="_pure__Weyman.html" title="first betti number of specific exact complex">pureWeyman</a> -- first betti number of specific exact complex</span></li>
<li><span>randomModule, see <span><a href="_random__Module_lp__List_cm__Z__Z_rp.html" title="module with random relations in prescribed degrees">randomModule(List,ZZ)</a> -- module with random relations in prescribed degrees</span></span></li>
<li><span>randomSocleModule, see <span><a href="_random__Socle__Module_lp__List_cm__Z__Z_rp.html" title="random finite length module with prescribed number of socle elements in single degree">randomSocleModule(List,ZZ)</a> -- random finite length module with prescribed number of socle elements in single degree</span></span></li>
<li><span><tt>supportFunctional</tt> (missing documentation<!-- tag: supportFunctional -->)</span></li>
</ul>
</div>
</li>
<li><div class="single">Methods<ul><li><span><tt>BettiTally * CohomologyTally</tt> (missing documentation<!-- tag: (*,BettiTally,CohomologyTally) -->)</span></li>
<li><span><tt>CohomologyTally * BettiTally</tt> (missing documentation<!-- tag: (*,CohomologyTally,BettiTally) -->)</span></li>
<li><span><tt>CohomologyTally ++ CohomologyTally</tt> (missing documentation<!-- tag: (++,CohomologyTally,CohomologyTally) -->)</span></li>
<li><span><tt>CohomologyTally == CohomologyTally</tt> (missing documentation<!-- tag: (==,CohomologyTally,CohomologyTally) -->)</span></li>
<li><span><tt>CohomologyTally ZZ</tt> (missing documentation<!-- tag: (SPACE,CohomologyTally,ZZ) -->)</span></li>
<li><span><tt>ZZ * CohomologyTally</tt> (missing documentation<!-- tag: (*,ZZ,CohomologyTally) -->)</span></li>
</ul>
</div>
</li>
</ul>
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