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augmented algebra

Contents

Definition

For RR a ring, an associative algebra over RR is a ring AA equipped with a ring inclusion RAR \hookrightarrow A.

Definition

If the RR-algebra AA is equipped with a ring homomorphism the other way round,

ϵ:AR \epsilon \colon A \to R

then it is called an augmented algebra.

Remark

In Cartan-Eilenberg this is called a supplemented algebra.

Definition

The kernel of ϵ\epsilon is called the corresponding augmentation ideal in AA.

Examples

Example

An augmentation of a bare ring itself, being an associative algebra over the ring of integers \mathbb{Z}, is a ring homomorphism to the integers

ϵ:R \epsilon \colon R \to \mathbb{Z}
Example

Every group algebra R[G]R[G] is canonically augmented, the augmentation map being the operation that forms the sum of coefficients of the canonical basis elements.

Example

If XX is a variety over an algebraically closed field kk and xX(k)x\in X(k) is a closed point, then the local ring 𝒪 X,x\mathcal{O}_{X,x} naturally has the structure of an augmented kk-algebra. The augmentation map 𝒪 X,xk\mathcal{O}_{X,x}\rightarrow k is the evaluation map, and the augmentation ideal is the maximal ideal of 𝒪 X,x\mathcal{O}_{X,x}.

References

Revised on August 13, 2015 11:46:47 by Alexander Betts? (129.67.187.128)