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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 \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{Bar}_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{align} 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{align}

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{Bar}_R(R,M) gives a resolution of M by free left R-modules. Explicitly, the complex is[3]

\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 n} \otimes_k M \to R^{\otimes_k (n+1)} \otimes_k M between the identity and 0. This homotopy is given by

\begin{align} 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{align}

One can similarly construct a resolution of a right R-module N by free right modules with the complex \operatorname{Bar}_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{Bar}_{R^e}(R^e,M). Here we are using the equivalence between R-R-bimodules and R^e-modules, where R^e = R \otimes R^\operatorname{op}, 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

Notes

  1. ^ Eilenberg, Samuel & Lane, Saunders Mac (July 1953). "On the Groups H(Π, n), I". The Annals of Mathematics. 58 (1): 55. doi:10.2307/1969820
  2. ^ Cartan, Henry & Eilenberg, Samuel (2016). Homological Algebra (PMS-19). Princeton Mathematical Series. Princeton, NJ: Princeton University Press. ISBN 978-0-691-04991-5.
  3. ^ Weibel 1994, p. 283.

References

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