Bar complex

In mathematics, the bar complex, also called the bar resolution, bar construction, standard resolution, or standard complex,^[1] 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,^[2] and has since been generalized in many ways. The name "bar complex" comes from the fact that Eilenberg and Mac Lane 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 ${Bar}_{R} (M_{1}, M_{2})$ given by

\dots \to M_{1} \otimes_{k} R \otimes_{k} R \otimes_{k} M_{2} \to M_{1} \otimes_{k} R \otimes_{k} M_{2} \to M_{1} \otimes_{k} M_{2} \to 0,

with the differential

\begin{matrix} d (m_{1} \otimes r_{1} \otimes \dots \otimes r_{n} \otimes m_{2}) & = m_{1} r_{1} \otimes \dots \otimes r_{n} \otimes m_{2} \\ + \sum_{i = 1}^{n - 1} (- 1)^{i} m_{1} \otimes r_{1} \otimes \dots \otimes r_{i} r_{i + 1} \otimes \dots \otimes r_{n} \otimes m_{2} + (- 1)^{n} m_{1} \otimes r_{1} \otimes \dots \otimes r_{n} m_{2} \end{matrix}

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

\dots \to R \otimes_{k} R \otimes_{k} R \otimes_{k} M \to R \otimes_{k} R \otimes_{k} M \to R \otimes_{k} M \to 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

\dots \to R \otimes_{k} R \otimes_{k} R \otimes_{k} M \to R \otimes_{k} R \otimes_{k} M \to R \otimes_{k} M \to M \to 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{matrix} h_{n} (r_{1} \otimes \dots \otimes r_{n} \otimes m) & = \sum_{i = 1}^{n - 1} (- 1)^{i + 1} r_{1} \otimes \dots \otimes r_{i - 1} \otimes 1 \otimes r_{i} \otimes \dots \otimes r_{n} \otimes m \end{matrix}

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

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 ${Bar}_{R^{e}} (R^{e}, M)$ . Here we are using the equivalence between $R$ - $R$ -bimodules and $R^{e}$ -modules, where $R^{e} = R \otimes R^{op}$ , see bimodules for more details.

Normalized bar complex

The normalized (or reduced) standard complex replaces $A \otimes A \otimes \dots \otimes A \otimes A$ with $A \otimes (A / K) \otimes \dots \otimes (A / K) \otimes A$ .^[4]

References

Sources

Cartan, Henri; Eilenberg, Samuel (1956). Homological Algebra. Princeton Mathematical Series. Vol. 19. Princeton University Press. Zbl 75.24305.

Eilenberg, Samuel; Mac Lane, Saunders (1953). "On the groups H(Π, n), I". Annals of Mathematics. 58 (1): 55–106. JSTOR 1969820.

Ginzburg, Victor (2005). "Lectures on noncommutative geometry". arXiv:math.AG/0506603.

Weibel, Charles A. (1994). An Introduction to Homological Algebra. Cambridge Studies in Advanced Mathematics. Vol. 38. Cambridge University Press. ISBN 0-521-43500-5.

[FOOTNOTECartanEilenberg1956[httpswwwmathstonybrookedu~mmovshevBOOKShomologicalalgeb033541mbppdfpage196_174]-1] Cartan & Eilenberg (1956), p. 174.

[FOOTNOTEEilenbergMac_Lane1953-2] Eilenberg & Mac Lane (1953).

[FOOTNOTEWeibel1994283-3] Weibel (1994), p. 283.

[FOOTNOTECartanEilenberg1956[httpswwwmathstonybrookedu~mmovshevBOOKShomologicalalgeb033541mbppdfpage198_176]-4] Cartan & Eilenberg (1956), p. 176.

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