|
|
|
Viewing Version
6
of
'Tor'
|
[ view 'Tor'
|
back to history
]
| Title of object: |
Tor |
| Canonical Name: |
Tor |
| Type: |
Definition |
| Created on: |
2004-08-09 09:12:08 |
| Modified on: |
2005-03-04 15:02:49 |
| Classification: |
msc:16E30, msc:18G15 |
| Keywords: |
homology, homological algebra |
| Defines: |
Tor, Ext |
Revision comment (for changes between this and next version):
| Changes for correction #10386 ('formatting'). |
Preamble:
% this is the default PlanetMath preamble. as your knowledge
% of TeX increases, you will probably want to edit this, but
% it should be fine as is for beginners.
% almost certainly you want these
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}
% used for TeXing text within eps files
%\usepackage{psfrag}
% need this for including graphics (\includegraphics)
%\usepackage{graphicx}
% for neatly defining theorems and propositions
%\usepackage{amsthm}
% making logically defined graphics
%\usepackage{xypic}
% there are many more packages, add them here as you need them
% define commands here |
Content:
Let $R$ be a ring with multiplicative identity. Let $M$ be a (right) module over R. We may assume there exists an exact sequence $C_*$:
$$
\dots\dots\rightarrow P_2\rightarrow P_1\rightarrow P_0
$$
with the $P_n$ projective and the cokernel of the last map $M$. Given $M$, this sequence is unique up to chain homotopy. Hence we may make the following definitions.
For a (right) $R$- module $A$ we may define
$$
Ext_R^n(M,A)=H^n(C_*; A)
$$
For a (left) $R$- module $A$ we may define
$$
Tor_R^n(M,A)=H_n(C_*; A)
$$ |
|
|
|
|
|
