Namespaces
Variants
Actions

Difference between revisions of "Differential-algebra(2)"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
m (link)
 
(One intermediate revision by the same user not shown)
Line 1: Line 1:
An algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031820/d0318201.png" /> over a field (ring) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031820/d0318202.png" /> which is a [[Differential ring|differential ring]] and such that, moreover, any derivation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031820/d0318203.png" /> commutes with multiplications by elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031820/d0318204.png" />, i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031820/d0318205.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031820/d0318206.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031820/d0318207.png" />.
+
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 [[Derivation in a ring|Derivation in a ring]]. A differential graded algebra (or DGA) over a ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031820/d0318208.png" /> is a graded algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031820/d0318209.png" /> equipped with a graded <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031820/d03182010.png" />-module homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031820/d03182011.png" /> of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031820/d03182012.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031820/d03182013.png" /> and such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031820/d03182014.png" /> is a derivation in the graded sense, i.e. such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031820/d03182015.png" />. They are of importance in (co)homology theory.
+
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====
 
====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}}

Latest revision as of 17:32, 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=15881
This article was adapted from an original article by O.A. Ivanova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article