<?xml version="1.0" encoding="utf-8" ?> <!-- for emacs: -*- coding: utf-8 -*- -->
<!-- Apache may like this line in the file .htaccess: AddCharset utf-8 .html -->
<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.1 plus MathML 2.0 plus SVG 1.1//EN" "http://www.w3.org/2002/04/xhtml-math-svg/xhtml-math-svg-flat.dtd" >
<html xmlns="http://www.w3.org/1999/xhtml" xml:lang="en">
<head><title>bott(List,ZZ) -- cohomology of Schur functor of tautological bundle on P^n</title>
<link rel="stylesheet" type="text/css" href="../../../../Macaulay2/Style/doc.css"/>
</head>
<body>
<table class="buttons">
<tr>
<td><div><a href="_bott_lp__List_cm__Z__Z_cm__Z__Z_rp.html">next</a> | <a href="_bott.html">previous</a> | <a href="_bott_lp__List_cm__Z__Z_cm__Z__Z_rp.html">forward</a> | <a href="_bott.html">backward</a> | up | <a href="index.html">top</a> | <a href="master.html">index</a> | <a href="toc.html">toc</a> | <a href="http://www.math.uiuc.edu/Macaulay2/">Macaulay2 web site</a></div>
</td>
</tr>
</table>
<hr/>
<div><h1>bott(List,ZZ) -- cohomology of Schur functor of tautological bundle on P^n</h1>
<div class="single"><h2>Synopsis</h2>
<ul><li><div class="list"><dl class="element"><dt class="heading">Usage: </dt><dd class="value"><div><tt>bott(L,u)</tt></div>
</dd></dl>
</div>
</li>
<li><span>Function: <a href="_bott.html" title="cohomology of Schur functors of tautological bundle on P^n">bott</a></span></li>
<li><div class="single">Inputs:<ul><li><span><tt>L</tt>, <span>a <a href="../../Macaulay2Doc/html/___List.html">list</a></span>, a non-increasing sequence of integers</span></li>
<li><span><tt>u</tt>, <span>an <a href="../../Macaulay2Doc/html/___Z__Z.html">integer</a></span></span></li>
</ul>
</div>
</li>
<li><div class="single">Outputs:<ul><li><span><span>a <a href="../../Macaulay2Doc/html/___List.html">list</a></span>, (List, ZZ, ZZ) or the integer 0. See below for details.</span></li>
</ul>
</div>
</li>
</ul>
</div>
<div class="single"><h2>Description</h2>
<div>Given a weakly decreasing list of integers L of length n and an integer u, uses Bott's algorithm to compute the cohomology of the vector bundle E=O(-u) \tensor S_L(Q), on P^n = PP(V) where Q is the tautological rank n quotient bundle in the sequence 0--> O(-1) --> O^(n+1) --> Q -->0 and S_L(Q) is the Schur functor with the convention S_(d,0..0) = Sym_d, S_(1,1,1) = \wedge^3 etc. Returns either 0, if all cohomology is zero, or a list of three elements: A weakly decreasing list of n+1 integers M; a number i such that H^i(E)=S_M(V); and the rank of this module. For more information on how the partition M is constructed, see math.AC/0709.1529v3, "The Existence of Pure Free Resolutions", section 3.<p/>
For example, on P^3, E = S_3(Q) has H^0(S_3(Q)) = S_3(kk^4) = kk^20.<table class="examples"><tr><td><pre>i1 : bott({3,0,0},0)
o1 = {{3, 0, 0, 0}, 0, 20}
o1 : List</pre>
</td></tr>
</table>
H^*(E(-1)) = H^*(E(-2)) = 0, and H^2(E(-3)) == S_2(kk^4) == kk^10.<table class="examples"><tr><td><pre>i2 : bott({3,0,0},1)
o2 = 0</pre>
</td></tr>
<tr><td><pre>i3 : bott({3,0,0},2)
o3 = 0</pre>
</td></tr>
<tr><td><pre>i4 : bott({3,0,0},3)
o4 = {{2, 0, 0, 0}, 2, 10}
o4 : List</pre>
</td></tr>
<tr><td><pre>i5 : bott({2,1,0},0)
o5 = {{2, 1, 0, 0}, 0, 20}
o5 : List</pre>
</td></tr>
</table>
</div>
</div>
<div class="single"><h2>See also</h2>
<ul><li><span><a href="_bott_lp__List_cm__Z__Z_cm__Z__Z_rp.html" title="cohomology table of Schur functor of tautolgical bundle on P^n">bott(List,ZZ,ZZ)</a> -- cohomology table of Schur functor of tautolgical bundle on P^n</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>
</ul>
</div>
</div>
</body>
</html>
