What is "augmented algebra"? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T19:07:16Z http://mathoverflow.net/feeds/question/88247 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/88247/what-is-augmented-algebra What is "augmented algebra"? aglearner 2012-02-12T00:10:23Z 2012-02-12T00:15:22Z <p>Really sorry for this question, but googling for some time did not help me. I was trying to understand the meaning of the following phrase:</p> <p><em>Let B be an augmented algebra over a semi-simple algebra T.</em></p> <p>But I am stuck already with "augmented algebra"... -- can not find a definition on the web.</p> http://mathoverflow.net/questions/88247/what-is-augmented-algebra/88248#88248 Answer by Mariano Suárez-Alvarez for What is "augmented algebra"? Mariano Suárez-Alvarez 2012-02-12T00:15:22Z 2012-02-12T00:15:22Z <p>An <em>augmented ring</em> is simply a triple $(A,M,\epsilon)$ with $A$ ring, $M$ a left $A$-module and $\epsilon:A\to M$ a surjection of $A$ modules. One says then that $A$ is augmented over $M$. You can find this defined in Cartan-Eilenberg, for example.</p> <p>Often, $M$ is itself a ring and the map $\epsilon$ is also a ring morphism. This is probably the situation you have. </p>