## What is “augmented algebra”?

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:

Let B be an augmented algebra over a semi-simple algebra T.

But I am stuck already with "augmented algebra"... -- can not find a definition on the web.

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 The notion of an augmented algebra is defined, for example, in Mac Lane's book Homology. There an augmented algebra $A$ over a field $k$ is a $k$-algebra that is also equipped with an algebra homomorphism $A \rightarrow k$ (the augmentation map). So in your context, I would assume that $B$ is a $T$-algebra that is also equipped with an algebra homomorphism $B \rightarrow T$. – Christopher Drupieski Feb 12 2012 at 0:16 It probably means that there is a map $\iota : T \to B$ of rings ($B$ is a $T$-algebra) which admits a splitting $\epsilon : B \to T$ (i.e. $\epsilon \circ \iota = 1_T$) – Konstantin Ardakov Feb 12 2012 at 0:18

An augmented ring 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.

Often, $M$ is itself a ring and the map $\epsilon$ is also a ring morphism. This is probably the situation you have.

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C-E also define *supplemented algebras*: a supplemented $k$-algebra is a $k$-algebra augmented over $k$ with $k$-linear augmentation. This is the concept that appears in the comments above. The name comes from the fact that in a supplemented algebra $A$ the base ring $k$ is a summand with a supplement, namely the kernel of the augmentation. – Mariano Suárez-Alvarez Feb 12 2012 at 0:23
Mariano, thanks a lot for a super quick answer! Just one more question. Why one would introduce the special name for such type of triples. Do they often pop up? – aglearner Feb 12 2012 at 0:24
I don't know who introduced the concept, but they surely come up often enough. In the context of Cartan-Eilenberg (it may well be they who came up with the notion) it serves as a unifying concept used to set up several different cohomology theories. (The supplemented algebra term is mostly unused nowadays, I'd say, but supplemented algebras are of course also all over the place) – Mariano Suárez-Alvarez Feb 12 2012 at 0:27
It appears they were indeed the inventors, at least according to Mac Lane classic review of the book (see projecteuclid.org/…) «Another danger of shiny new notions is that sometimes the shine proves illusory. For example, the authors define an augmented ring [...] Hence "augmented ring" is a new name for "left ideal in a ring."» The review is a must read. – Mariano Suárez-Alvarez Feb 12 2012 at 0:31