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Is there a good definition of a 1-Cocycle of an algebra A that is relatively easy to understand? I am rather new to cohomology and representation theory, but it seems like this is a fundamental concept. Everywhere I look seems to refer to 1-cocycles, but I have yet to find a definition.

Given field $k$ and representations V,W of A, it would seem like if $f: A \rightarrow Hom_k(W,V)$ has the property $f(ab) = f(a) + \rho_V(a)f(b)$ it is a 1-cocycle. Is this definition close to being a start of the concept?

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Co-cycle are a general concept which arise in the context of homological algebra. It would be helpful if you could also add the reference to where are you studying about this co-cycles. – Giorgio Mossa Nov 17 '13 at 15:48
You need to tell us what cohomology theory you have in mind, for the answer very much depends on that! (the formula you gave is not the cocycle condition for Hochschild coohomology, which would have been tha natural cohomology theory to consider for algebras...) – Mariano Suárez-Alvarez Nov 17 '13 at 17:37
up vote 1 down vote accepted

Co-cycles are a much more general concept that belong to the area named homological algebra.

In general you have to deal with a co-chain complex, which is a sequence of modules index over the integer and a family of morphisms between then forming a diagram as the following

$$\require{AMScd} \begin{CD} \dots C_n @>\delta_{n+1}>> C_{n+1} @>\delta_{n+1}>> C_{n+2} @>\delta_{n+3}>> \dots \end{CD}$$ such that the $C_n$ are modules and the $\delta_n$ are module homomorphisms satisfying the condition $\delta_{n+1} \circ \delta_n = 0$. This condition is equivalent to require $\text{Im} \delta_n \subseteq \ker \delta_{n+1}$ for all $n \in \mathbb Z$, and so it makes sense to consider the modules $H^n(C)=\ker \delta_{n+1}/\text{Im}\delta_n$. These modules measure in some sense how the sequence of the $C_n$s and of the $\delta_n$s differ from being exact.

In this context the elements of $\ker \delta_{n+1}$, which represent equivalence classes in $H^n(C)$, are called $n$-cocycles, while the elements of $\text{Im} \delta_n$ are called $n$-coboundaries.

$1$-cocycles are simply elements of $\ker \delta_2$. What a cocycle is depend on the cochain complex considered.

In particular you have to tell us what are the modules you're considering in order to make us possible to understand what are the co-cycle you're looking for.

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