Bar complex

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In mathematics, the bar complex, also called the bar resolution, bar construction, standard resolution, or standard complex, is a way of constructing resolutions in homological algebra. It was first introduced for the special case of algebras over a commutative ring by Samuel Eilenberg and Saunders Mac Lane,^[1] and Henri Cartan and Eilenberg^[2] and has since been generalized in many ways. The name "bar complex" comes from the fact that Eilenberg and Mac Lane^[1] used a vertical bar | as a shortened form of the tensor product ${\displaystyle \otimes }$ in their notation for the complex.

Let ${\displaystyle R}$ be an algebra over a field ${\displaystyle k}$, let ${\displaystyle M_{1}}$ be a right ${\displaystyle R}$-module, and let ${\displaystyle M_{2}}$ be a left ${\displaystyle R}$-module. Then, one can form the bar complex ${\displaystyle \operatorname {Bar} _{R}(M_{1},M_{2})}$ given by

${\displaystyle \cdots \rightarrow M_{1}\otimes _{k}R\otimes _{k}R\otimes _{k}M_{2}\rightarrow M_{1}\otimes _{k}R\otimes _{k}M_{2}\rightarrow M_{1}\otimes _{k}M_{2}\rightarrow 0\,,}$

with the differential

${\displaystyle {\begin{aligned}d(m_{1}\otimes r_{1}\otimes \cdots \otimes r_{n}\otimes m_{2})&=m_{1}r_{1}\otimes \cdots \otimes r_{n}\otimes m_{2}\\&+\sum _{i=1}^{n-1}(-1)^{i}m_{1}\otimes r_{1}\otimes \cdots \otimes r_{i}r_{i+1}\otimes \cdots \otimes r_{n}\otimes m_{2}+(-1)^{n}m_{1}\otimes r_{1}\otimes \cdots \otimes r_{n}m_{2}\end{aligned}}}$

The bar complex is useful because it provides a canonical way of producing (free) resolutions of modules over a ring. However, often these resolutions are very large, and can be prohibitively difficult to use for performing actual computations.

Let ${\displaystyle M}$ be a left ${\displaystyle R}$-module, with ${\displaystyle R}$ a unital ${\displaystyle k}$-algebra. Then, the bar complex ${\displaystyle \operatorname {Bar} _{R}(R,M)}$ gives a resolution of ${\displaystyle M}$ by free left ${\displaystyle R}$-modules. Explicitly, the complex is^[3]

${\displaystyle \cdots \rightarrow R\otimes _{k}R\otimes _{k}R\otimes _{k}M\rightarrow R\otimes _{k}R\otimes _{k}M\rightarrow R\otimes _{k}M\rightarrow 0\,,}$

This complex is composed of free left ${\displaystyle R}$-modules, since each subsequent term is obtained by taking the free left ${\displaystyle R}$-module on the underlying vector space of the previous term.

To see that this gives a resolution of ${\displaystyle M}$, consider the modified complex

${\displaystyle \cdots \rightarrow R\otimes _{k}R\otimes _{k}R\otimes _{k}M\rightarrow R\otimes _{k}R\otimes _{k}M\rightarrow R\otimes _{k}M\rightarrow M\rightarrow 0\,,}$

Then, the above bar complex being a resolution of ${\displaystyle M}$ is equivalent to this extended complex having trivial homology. One can show this by constructing an explicit homotopy ${\displaystyle h_{n}:R^{\otimes _{k}n}\otimes _{k}M\to R^{\otimes _{k}(n+1)}\otimes _{k}M}$ between the identity and 0. This homotopy is given by

${\displaystyle {\begin{aligned}h_{n}(r_{1}\otimes \cdots \otimes r_{n}\otimes m)&=\sum _{i=1}^{n-1}(-1)^{i+1}r_{1}\otimes \cdots \otimes r_{i-1}\otimes 1\otimes r_{i}\otimes \cdots \otimes r_{n}\otimes m\end{aligned}}}$

One can similarly construct a resolution of a right ${\displaystyle R}$-module ${\displaystyle N}$ by free right modules with the complex ${\displaystyle \operatorname {Bar} _{R}(N,R)}$.

Notice that, in the case one wants to resolve ${\displaystyle R}$ as a module over itself, the above two complexes are the same, and actually give a resolution of ${\displaystyle R}$ by ${\displaystyle R}$-${\displaystyle R}$-bimodules. This provides one with a slightly smaller resolution of ${\displaystyle R}$ by free ${\displaystyle R}$-${\displaystyle R}$-bimodules than the naive option ${\displaystyle \operatorname {Bar} _{R^{e}}(R^{e},M)}$. Here we are using the equivalence between ${\displaystyle R}$-${\displaystyle R}$-bimodules and ${\displaystyle R^{e}}$-modules, where ${\displaystyle R^{e}=R\otimes R^{\operatorname {op} }}$, see bimodules for more details.

The normalized (or reduced) standard complex replaces ${\displaystyle A\otimes A\otimes \cdots \otimes A\otimes A}$ with ${\displaystyle A\otimes (A/K)\otimes \cdots \otimes (A/K)\otimes A}$.

• Koszul complex

• Ginzburg, Victor (2005). "Lectures on Noncommutative Geometry". arXiv:math.AG/0506603.
• Weibel, Charles (1994), An Introduction to Homological Algebra, Cambridge Studies in Advanced Mathematics, vol. 38, Cambridge: Cambridge University Press, ISBN 0-521-43500-5