<?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>manipulating chain complexes</title>
<link rel="stylesheet" type="text/css" href="../../../../Macaulay2/Style/doc.css"/>
</head>
<body>
<table class="buttons">
<tr>
<td><div><a href="_maps_spbetween_spchain_spcomplexes.html">next</a> | <a href="_making_spchain_spcomplexes_spby_sphand.html">previous</a> | <a href="_maps_spbetween_spchain_spcomplexes.html">forward</a> | <a href="_making_spchain_spcomplexes_spby_sphand.html">backward</a> | <a href="_chain_spcomplexes.html">up</a> | <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>
<div><a href="index.html" title="">Macaulay2Doc</a> > <a href="_chain_spcomplexes.html" title="">chain complexes</a> > <a href="_manipulating_spchain_spcomplexes.html" title="">manipulating chain complexes</a></div>
<hr/>
<div><h1>manipulating chain complexes</h1>
<div>There are several natural ways to handle chain complexes; for details, see <a href="___Chain__Complex.html" title="the class of all chain complexes">ChainComplex</a>. Let's illustrate by making two chain complexes.<table class="examples"><tr><td><pre>i1 : R = QQ[x,y];</pre>
</td></tr>
<tr><td><pre>i2 : M = coker vars R
o2 = cokernel | x y |
1
o2 : R-module, quotient of R</pre>
</td></tr>
<tr><td><pre>i3 : N = coker matrix {{x}}
o3 = cokernel | x |
1
o3 : R-module, quotient of R</pre>
</td></tr>
<tr><td><pre>i4 : C = res M
1 2 1
o4 = R <-- R <-- R <-- 0
0 1 2 3
o4 : ChainComplex</pre>
</td></tr>
<tr><td><pre>i5 : D = res N
1 1
o5 = R <-- R <-- 0
0 1 2
o5 : ChainComplex</pre>
</td></tr>
</table>
We can form the direct sum as follows.<table class="examples"><tr><td><pre>i6 : C ++ D
2 3 1
o6 = R <-- R <-- R <-- 0
0 1 2 3
o6 : ChainComplex</pre>
</td></tr>
</table>
We can shift the degree, using the traditional notation.<table class="examples"><tr><td><pre>i7 : E = C[5]
1 2 1
o7 = R <-- R <-- R <-- 0
-5 -4 -3 -2
o7 : ChainComplex</pre>
</td></tr>
<tr><td><pre>i8 : E_-4 == C_1
o8 = true</pre>
</td></tr>
</table>
The same syntax can be used to make a chain complex from a single module.<table class="examples"><tr><td><pre>i9 : R^4[1]
4
o9 = R
-1
o9 : ChainComplex</pre>
</td></tr>
</table>
We can form various tensor products with <a href="__st_st.html" title="a binary operator, usually used for tensor product or Cartesian product">**</a>, and compute <a href="___Tor.html" title="Tor module">Tor</a> using them.<table class="examples"><tr><td><pre>i10 : M ** D
o10 = cokernel | x y | <-- cokernel {1} | x y |
0 1
o10 : ChainComplex</pre>
</td></tr>
<tr><td><pre>i11 : C ** D
1 3 3 1
o11 = R <-- R <-- R <-- R <-- 0 <-- 0
0 1 2 3 4 5
o11 : ChainComplex</pre>
</td></tr>
<tr><td><pre>i12 : prune HH_1(C ** D)
o12 = cokernel {1} | y x |
1
o12 : R-module, quotient of R</pre>
</td></tr>
<tr><td><pre>i13 : prune HH_1(M ** D)
o13 = cokernel {1} | y x |
1
o13 : R-module, quotient of R</pre>
</td></tr>
<tr><td><pre>i14 : prune HH_1(C ** N)
o14 = cokernel {1} | y x |
1
o14 : R-module, quotient of R</pre>
</td></tr>
</table>
Of course, we can use <a href="___Tor.html" title="Tor module">Tor</a> to get the same result.<table class="examples"><tr><td><pre>i15 : prune Tor_1(M,N)
o15 = cokernel {1} | y x |
1
o15 : R-module, quotient of R</pre>
</td></tr>
</table>
</div>
</div>
</body>
</html>
