coefficient (changes) in nLab
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Definition
There are two different meansing meanings of the termcoefficient in mathematics, one
and one
The two notion do however conincide coincide forordinary homology/ordinary cohomology expressed as singular homology/singular cohomology. See remark 1 below.
In algebra and analysis
In algebra and analysis, a coefficient is an element of a ring (or rig) RR that appears in scalar multiplication; more generally, coefficients are elements of RR that appear in a linear combinations. Thus, we multiply a coefficient by an element of an RR-module MM (which may even be an algebra, associative or not) to get another element of MM.
For example, given a polynomial P=∑ na nx nP = \sum_n a_n x^n over RR in the variable xx, each a n∈Ra_n \in R is the coefficient on x nx^n in PP. The module MM here is the symmetric algebra over RR.
In cohomology and homology
In cohomology coefficients are what the cohomology takes values in. For ordinary cohomology H •(−,A)H^\bullet(-,A) the abelian group AA is the coefficient group. For generalized (Eilenberg-Steenrod) cohomology H •(−,E)H^\bullet(-,E) the given spectrum EE that represents it is the coefficient spectrum. Dually for homology.
It is this notion of “coefficient” that appears in terms like
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