Standard Resolution

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 and and has since been generalized in many ways. The name "bar complex" comes from the fact that used a vertical bar , as a shortened form of the tensor product $\otimes$ in their notation for the complex.

Definition

Let $R$ be an algebra over a field $k$, let $M_1$ be a right $R$- module, and let $M_2$ be a left $R$-module. Then, one can form the bar complex $\operatorname_R(M_1,M_2)$ given by
:$\cdots\rightarrow M_1 \otimes_k R \otimes_k R \otimes_k M_2 \rightarrow M_1 \otimes_k R \otimes_k M_2 \rightarrow M_1 \otimes_k M_2 \rightarrow 0\,,$
with the differential
:$\begin d(m_1 \otimes r_1 \otimes \cdots \otimes r_n \otimes m_2) &= m_1 r_1 \otimes \cdots \otimes r_n \otimes m_2 \\ &+ \sum_^ (-1)^i m_1 \otimes r_1 \otimes \cdots \otimes r_i r_ \otimes \cdots \otimes r_n \otimes m_2 + (-1)^n m_1 \otimes r_1 \otimes \cdots \otimes r_n m_2 \end$

Resolutions

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.

Free Resolution of a Module

Let $M$ be a left $R$-module, with $R$ a unital $k$-algebra. Then, the bar complex $\operatorname_R(R,M)$ gives a resolution of $M$ by free left $R$-modules. Explicitly, the complex is
:$\cdots\rightarrow R \otimes_k R \otimes_k R \otimes_k M \rightarrow R \otimes_k R \otimes_k M \rightarrow R \otimes_k M \rightarrow 0\,,$
This complex is composed of free left $R$-modules, since each subsequent term is obtained by taking the free left $R$-module on the underlying vector space of the previous term.

To see that this gives a resolution of $M$, consider the modified complex
:$\cdots\rightarrow R \otimes_k R \otimes_k R \otimes_k M \rightarrow R \otimes_k R \otimes_k M \rightarrow R \otimes_k M \rightarrow M \rightarrow 0\,,$
Then, the above bar complex being a resolution of $M$ is equivalent to this extended complex having trivial homology. One can show this by constructing an explicit homotopy $h_n : R^ \otimes_k M \to R^ \otimes_k M$ between the identity and 0. This homotopy is given by
:$\begin h_n(r_1 \otimes \cdots \otimes r_n \otimes m) &= \sum_^ (-1)^ r_1 \otimes \cdots \otimes r_ \otimes 1 \otimes r_i \otimes \cdots \otimes r_n \otimes m \end$

One can similarly construct a resolution of a right $R$-module $N$ by free right modules with the complex $\operatorname_R(N,R)$.

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

The Normalized Bar Complex

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

See also

*Koszul complex

Notes

References

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Homological algebra

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