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<!DOCTYPE html> <html lang="en"> <head> <title>AInfinity -- A-infinity algebra and module structures on free resolutions</title> <meta content="text/html; charset=utf-8" http-equiv="Content-Type"> <link type="text/css" rel="stylesheet" href="../../../../Macaulay2/Style/doc.css"> <link rel="stylesheet" href="../../../../Macaulay2/Style/katex/katex.min.css"> <script defer="defer" src="../../../../Macaulay2/Style/katex/katex.min.js"></script> <script defer="defer" src="../../../../Macaulay2/Style/katex/contrib/auto-render.min.js"></script> <script> var macros = { "\\break": "\\\\", "\\ZZ": "\\mathbb{Z}", "\\NN": "\\mathbb{N}", "\\QQ": "\\mathbb{Q}", "\\RR": "\\mathbb{R}", "\\CC": "\\mathbb{C}", "\\PP": "\\mathbb{P}" }, delimiters = [ { left: "$$", right: "$$", display: true}, { left: "\\[", right: "\\]", display: true}, { left: "$", right: "$", display: false}, { left: "\\(", right: "\\)", display: false} ], ignoredTags = [ "kbd", "var", "samp", "script", "noscript", "style", "textarea", "pre", "code", "option" ]; document.addEventListener("DOMContentLoaded", function() { renderMathInElement(document.body, { delimiters: delimiters, macros: macros, ignoredTags: ignoredTags, trust: true }); }); </script> <style>.katex { font-size: 1em; }</style> <script defer="defer" src="../../../../Macaulay2/Style/katex/contrib/copy-tex.min.js"></script> <script defer="defer" src="../../../../Macaulay2/Style/katex/contrib/render-a11y-string.min.js"></script> <script src="../../../../Macaulay2/Style/prism.js"></script> <script>var current_version = '1.25.06';</script> <script src="../../../../Macaulay2/Style/version-select.js"></script> <link type="image/x-icon" rel="icon" href="../../../../Macaulay2/Style/icon.gif"> </head> <body> <div id="buttons"> <div> <a href="https://macaulay2.com/">Macaulay2</a> <span id="version-select-container"></span> » <a title="Macaulay2 documentation" href="../../Macaulay2Doc/html/index.html">Documentation </a> <br><a href="../../Macaulay2Doc/html/_packages_spprovided_spwith_sp__Macaulay2.html">Packages</a> » <span><a title="A-infinity algebra and module structures on free resolutions" href="index.html">AInfinity</a> :: <a title="A-infinity algebra and module structures on free resolutions" href="index.html">AInfinity</a></span> </div> <div class="right"> <form method="get" action="https://www.google.com/search"> <input placeholder="Search" type="text" name="q" value=""> <input type="hidden" name="q" value="site:macaulay2.com/doc"> </form> <a href="_a__Infinity.html">next</a> | previous | <a href="_a__Infinity.html">forward</a> | backward | up | <a href="master.html">index</a> | <a href="toc.html">toc</a> </div> </div> <hr> <div> <h1>AInfinity -- A-infinity algebra and module structures on free resolutions</h1> <div> <h2>Description</h2> <div> <p>Following Jesse Burke's paper "Higher Homotopies and Golod Rings", given a polynomial ring S and a factor ring R = S/I and an R-module X, we compute (finite) A-infinity algebra structure mR on an S-free resolution of R and the A-infinity mR-module structure on an S-free resolution of X, and use them to give a finite computation of the maps in an R-free resolution of X that we call the Burke resolution. Here is an example with the simplest Golod non-hypersurface in 3 variables</p> </div> <table class="examples"> <tr> <td> <pre><code class="language-macaulay2">i1 : S = ZZ/101[a,b,c] o1 = S o1 : PolynomialRing</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i2 : R = S/(ideal(a)*ideal(a,b,c)) o2 = R o2 : QuotientRing</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i3 : mR = aInfinity R;</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i4 : res coker presentation R 1 3 3 1 o4 = S <-- S <-- S <-- S 0 1 2 3 o4 : Complex</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i5 : mR#{2,2} o5 = {3} | 0 -a 0 a 0 0 0 -c 0 | {3} | 0 0 -a 0 0 0 a b 0 | {3} | 0 0 0 0 0 -a 0 0 0 | 3 9 o5 : Matrix S <-- S</code></pre> </td> </tr> </table> <div> <p>Given a module X over R, Jesse Burke constructed a possibly non-minimal R-free resolution of any length from the finite data mR and mX:</p> </div> <table class="examples"> <tr> <td> <pre><code class="language-macaulay2">i6 : X = coker vars R o6 = cokernel | a b c | 1 o6 : R-module, quotient of R</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i7 : A = betti burkeResolution(X,8) 0 1 2 3 4 5 6 7 8 o7 = total: 1 3 6 13 28 60 129 277 595 0: 1 3 6 13 28 60 129 277 595 o7 : BettiTally</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i8 : B = betti res(X, LengthLimit => 8) 0 1 2 3 4 5 6 7 8 o8 = total: 1 3 6 13 28 60 129 277 595 0: 1 3 6 13 28 60 129 277 595 o8 : BettiTally</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i9 : A == B o9 = true</code></pre> </td> </tr> </table> </div> <div> <h2>See also</h2> <ul> <li><span><a title="aInfinity algebra and module structures on free resolutions" href="_a__Infinity.html">aInfinity</a> -- aInfinity algebra and module structures on free resolutions</span></li> </ul> </div> <div> <div> <div> <h2>Authors</h2> <ul> <li><a href="http://www.msri.org/~de">David Eisenbud</a><span> <<a href="mailto:de%40msri.org">de@msri.org</a>></span></li> <li><a href="http://pi.math.cornell.edu/~mike">Mike Stillman</a><span> <<a href="mailto:mike%40math.cornell.edu">mike@math.cornell.edu</a>></span></li> </ul> </div> <div> <h2>Version</h2> <p>This documentation describes version <b>0.1</b> of AInfinity, released <b>October 4, 2020, rev Feb 2021, rev May 2021</b>.</p> </div> <div> <h2>Citation</h2> <p>If you have used this package in your research, please cite it as follows:</p> <table class="examples"> <tr> <td> <pre><code class="language-bib">@misc{AInfinitySource, title = {{AInfinity: AInfinity structures on free resolutions. Version~0.1}}, author = {David Eisenbud and Mike Stillman}, howpublished = {A \emph{Macaulay2} package available at \url{https://github.com/Macaulay2/M2/tree/stable/M2/Macaulay2/packages}} } </code></pre> </td> </tr> </table> </div> <div> <h2>Exports</h2> <div class="exports"> <ul> <li>Functions and commands <ul> <li><span><a title="aInfinity algebra and module structures on free resolutions" href="_a__Infinity.html">aInfinity</a> -- aInfinity algebra and module structures on free resolutions</span></li> <li><span><kbd>burkeDifferential</kbd> -- see <span><a title="compute a resolution from A-infinity structures" href="_burke__Resolution.html">burkeResolution</a> -- compute a resolution from A-infinity structures</span></span></li> <li><span><a title="compute a resolution from A-infinity structures" href="_burke__Resolution.html">burkeResolution</a> -- compute a resolution from A-infinity structures</span></li> <li><span><a title="prints a matrix showing the source and target decomposition" href="_display__Blocks.html">displayBlocks</a> -- prints a matrix showing the source and target decomposition</span></li> <li><span><a title="displays components of a map in a labeled complex" href="_extract__Blocks.html">extractBlocks</a> -- displays components of a map in a labeled complex</span></li> <li><span><a title="list the ranks of the free modules in the resolution of a Golod module" href="_golod__Betti.html">golodBetti</a> -- list the ranks of the free modules in the resolution of a Golod module</span></li> <li><span><a title="Determines if the A-infinity multiplication is minimal" href="_has__Minimal__Mult.html">hasMinimalMult</a> -- Determines if the A-infinity multiplication is minimal</span></li> <li><span><a title="Determines if the ring is Golod or not" href="_is__Golod__A__Inf.html">isGolodAInf</a> -- Determines if the ring is Golod or not</span></li> <li><span><a title="displays information about the blocks of a map or maps between direct sum modules" href="_picture.html">picture</a> -- displays information about the blocks of a map or maps between direct sum modules</span></li> </ul> </li> <li>Methods <ul> <li><span><kbd>aInfinity(HashTable,Module)</kbd> -- see <span><a title="aInfinity algebra and module structures on free resolutions" href="_a__Infinity.html">aInfinity</a> -- aInfinity algebra and module structures on free resolutions</span></span></li> <li><span><kbd>aInfinity(Module)</kbd> -- see <span><a title="aInfinity algebra and module structures on free resolutions" href="_a__Infinity.html">aInfinity</a> -- aInfinity algebra and module structures on free resolutions</span></span></li> <li><span><kbd>aInfinity(Ring)</kbd> -- see <span><a title="aInfinity algebra and module structures on free resolutions" href="_a__Infinity.html">aInfinity</a> -- aInfinity algebra and module structures on free resolutions</span></span></li> <li><span><kbd>burkeDifferential(HashTable,HashTable,ZZ)</kbd> -- see <span><a title="compute a resolution from A-infinity structures" href="_burke__Resolution.html">burkeResolution</a> -- compute a resolution from A-infinity structures</span></span></li> <li><span><kbd>burkeResolution(Module,ZZ)</kbd> -- see <span><a title="compute a resolution from A-infinity structures" href="_burke__Resolution.html">burkeResolution</a> -- compute a resolution from A-infinity structures</span></span></li> <li><span><kbd>displayBlocks(Matrix)</kbd> -- see <span><a title="prints a matrix showing the source and target decomposition" href="_display__Blocks.html">displayBlocks</a> -- prints a matrix showing the source and target decomposition</span></span></li> <li><span><kbd>extractBlocks(Matrix,List)</kbd> -- see <span><a title="displays components of a map in a labeled complex" href="_extract__Blocks.html">extractBlocks</a> -- displays components of a map in a labeled complex</span></span></li> <li><span><kbd>extractBlocks(Matrix,List,List)</kbd> -- see <span><a title="displays components of a map in a labeled complex" href="_extract__Blocks.html">extractBlocks</a> -- displays components of a map in a labeled complex</span></span></li> <li><span><kbd>golodBetti(Module,ZZ)</kbd> -- see <span><a title="list the ranks of the free modules in the resolution of a Golod module" href="_golod__Betti.html">golodBetti</a> -- list the ranks of the free modules in the resolution of a Golod module</span></span></li> <li><span><kbd>hasMinimalMult(Ideal)</kbd> -- see <span><a title="Determines if the A-infinity multiplication is minimal" href="_has__Minimal__Mult.html">hasMinimalMult</a> -- Determines if the A-infinity multiplication is minimal</span></span></li> <li><span><kbd>hasMinimalMult(Ideal,ZZ)</kbd> -- see <span><a title="Determines if the A-infinity multiplication is minimal" href="_has__Minimal__Mult.html">hasMinimalMult</a> -- Determines if the A-infinity multiplication is minimal</span></span></li> <li><span><kbd>hasMinimalMult(Ring)</kbd> -- see <span><a title="Determines if the A-infinity multiplication is minimal" href="_has__Minimal__Mult.html">hasMinimalMult</a> -- Determines if the A-infinity multiplication is minimal</span></span></li> <li><span><kbd>hasMinimalMult(Ring,InfiniteNumber)</kbd> -- see <span><a title="Determines if the A-infinity multiplication is minimal" href="_has__Minimal__Mult.html">hasMinimalMult</a> -- Determines if the A-infinity multiplication is minimal</span></span></li> <li><span><kbd>hasMinimalMult(Ring,ZZ)</kbd> -- see <span><a title="Determines if the A-infinity multiplication is minimal" href="_has__Minimal__Mult.html">hasMinimalMult</a> -- Determines if the A-infinity multiplication is minimal</span></span></li> <li><span><kbd>isGolodAInf(Ring)</kbd> -- see <span><a title="Determines if the ring is Golod or not" href="_is__Golod__A__Inf.html">isGolodAInf</a> -- Determines if the ring is Golod or not</span></span></li> <li><span><kbd>picture(ChainComplex)</kbd> -- see <span><a title="displays information about the blocks of a map or maps between direct sum modules" href="_picture.html">picture</a> -- displays information about the blocks of a map or maps between direct sum modules</span></span></li> <li><span><kbd>picture(Complex)</kbd> -- see <span><a title="displays information about the blocks of a map or maps between direct sum modules" href="_picture.html">picture</a> -- displays information about the blocks of a map or maps between direct sum modules</span></span></li> <li><span><kbd>picture(Matrix)</kbd> -- see <span><a title="displays information about the blocks of a map or maps between direct sum modules" href="_picture.html">picture</a> -- displays information about the blocks of a map or maps between direct sum modules</span></span></li> <li><span><kbd>picture(Module)</kbd> -- see <span><a title="displays information about the blocks of a map or maps between direct sum modules" href="_picture.html">picture</a> -- displays information about the blocks of a map or maps between direct sum modules</span></span></li> </ul> </li> <li>Symbols <ul> <li><span><a title="Option for burkeResolution" href="___Check.html">Check</a> -- Option for burkeResolution</span></li> <li><span><kbd>ShowRanks</kbd> -- see <span><a title="displays information about the blocks of a map or maps between direct sum modules" href="_picture.html">picture</a> -- displays information about the blocks of a map or maps between direct sum modules</span></span></li> </ul> </li> </ul> </div> </div> </div> <div class="waystouse"> <h2>For the programmer</h2> <p>The object <a title="A-infinity algebra and module structures on free resolutions" href="index.html">AInfinity</a> is <span>a <a title="the class of all packages" href="../../Macaulay2Doc/html/___Package.html">package</a></span>, defined in <span class="tt">AInfinity.m2</span>.</p> </div> <hr> <div class="waystouse"> <p>The source of this document is in <span class="tt">AInfinity.m2:1093:0</span>.</p> </div> </div> </div> </body> </html>