Difference between revisions of "Differential-algebra(2)"
From Encyclopedia of Mathematics
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| − | An algebra | + | An algebra $A$ over a field (ring) $K$ which is a [[differential ring]] and such that, moreover, any derivation $\partial$ commutes with multiplications by elements of $K$, i.e. $\partial(\alpha x) = \alpha \partial(x)$, where $\alpha \in K$, $x \in A$. |
====Comments==== | ====Comments==== | ||
| − | Cf. also [[ | + | Cf. also [[Derivation in a ring]]. |
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| + | A '''differential graded algebra''' (or DGA) over a ring $K$ is a graded algebra $A$ equipped with a graded $K$-module homomorphism $\partial : A \rightarrow A$ of degree $-1$ such that $\partial^2 = 0$ and such that $\partial$ is a derivation in the graded sense, i.e. such that $\partial(uv) = \partial(u) v + (-1)^{\deg u} u \partial(v)$. They are of importance in (co)homology theory. | ||
====References==== | ====References==== | ||
| − | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> S. MacLane, "Homology" , Springer (1963)</TD></TR></table> | + | <table> |
| + | <TR><TD valign="top">[a1]</TD> <TD valign="top"> S. MacLane, "Homology" , Springer (1963)</TD></TR> | ||
| + | </table> | ||
| + | |||
| + | {{TEX|done}} | ||
Revision as of 17:28, 1 November 2016
An algebra $A$ over a field (ring) $K$ which is a differential ring and such that, moreover, any derivation $\partial$ commutes with multiplications by elements of $K$, i.e. $\partial(\alpha x) = \alpha \partial(x)$, where $\alpha \in K$, $x \in A$.
Comments
Cf. also Derivation in a ring.
A differential graded algebra (or DGA) over a ring $K$ is a graded algebra $A$ equipped with a graded $K$-module homomorphism $\partial : A \rightarrow A$ of degree $-1$ such that $\partial^2 = 0$ and such that $\partial$ is a derivation in the graded sense, i.e. such that $\partial(uv) = \partial(u) v + (-1)^{\deg u} u \partial(v)$. They are of importance in (co)homology theory.
References
| [a1] | S. MacLane, "Homology" , Springer (1963) |
How to Cite This Entry:
Differential-algebra(2). Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Differential-algebra(2)&oldid=39579
Differential-algebra(2). Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Differential-algebra(2)&oldid=39579
This article was adapted from an original article by O.A. Ivanova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article