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Difference between revisions of "Étale algebra"

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(Start article: Étale algebra)
 
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====References====
 
====References====
* Tsit-Yuen Lam, "Lectures on Modules and Rings" Graduate Texts in Mathematics '''189''' Springer (2012) ISBN 1461205255 {{ZBL|0911.16001}}
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* Tsit-Yuen Lam, "Lectures on Modules and Rings", Graduate Texts in Mathematics '''189''', Springer (2012) {{ISBN|1461205255}} {{ZBL|0911.16001}}

Latest revision as of 07:29, 24 November 2023

2020 Mathematics Subject Classification: Primary: 13B Secondary: 12F [MSN][ZBL]

A commutative algebra $A$ finite-dimensional over a field $K$ for which the bilinear form induced by the trace $$ \langle x,y \rangle = \mathrm{tr}_{A/K} (x\cdot y) $$ is non-singular. Equivalently, an algebra which is isomorphic to a product of fields $A \sim K_1 \times \cdots \times K_r$ with each $K_i$ a field extension of $K$.

Since $\langle xy,z \rangle = \mathrm{tr}(xyz) = \langle x,yz \rangle$, an étale algebra is a Frobenius algebra over $K$.

References

  • Tsit-Yuen Lam, "Lectures on Modules and Rings", Graduate Texts in Mathematics 189, Springer (2012) ISBN 1461205255 Zbl 0911.16001
How to Cite This Entry:
Étale algebra. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=%C3%89tale_algebra&oldid=54649