Bar complex explained

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 used a vertical bar | as a shortened form of the tensor product

⊗

in their notation for the complex.

Definition

Let

be an algebra over a field

, let

M₁

be a right

-module, and let

M₂

be a left

-module. Then, one can form the bar complex

\operatorname{Bar}_R(M_1,M₂₎

given by

… → M₁ ⊗ _kR ⊗ _kR ⊗ _kM₂ → M₁ ⊗ _kR ⊗ _kM₂ → M₁ ⊗ _kM₂ → 0,

with the differential

\begin{align} d(m₁ ⊗ r₁ ⊗ … ⊗ r_n ⊗ m₂₎&=m₁r₁ ⊗ … ⊗ r_n ⊗ m₂\\ &+

	n-1
\sum
	i=1

(-1)ⁱm₁ ⊗ r₁ ⊗ … ⊗ r_ir_i+1 ⊗ … ⊗ r_n ⊗ m₂+(-1)ⁿm₁ ⊗ r₁ ⊗ … ⊗ r_nm_{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

be a left

-module, with

a unital

-algebra. Then, the bar complex

\operatorname{Bar}_R(R,M)

gives a resolution of

by free left

-modules. Explicitly, the complex is

… → R ⊗ _kR ⊗ _kR ⊗ _kM → R ⊗ _kR ⊗ _kM → R ⊗ _kM → 0,

This complex is composed of free left

-modules, since each subsequent term is obtained by taking the free left

-module on the underlying vector space of the previous term.

To see that this gives a resolution of

, consider the modified complex

… → R ⊗ _kR ⊗ _kR ⊗ _kM → R ⊗ _kR ⊗ _kM → R ⊗ _kM → M → 0,

Then, the above bar complex being a resolution of

is equivalent to this extended complex having trivial homology. One can show this by constructing an explicit homotopy

h_n:

	⊗ _kn
R

⊗ _kM\to

	⊗ _k(n+1)
R

⊗ _kM

between the identity and 0. This homotopy is given by

\begin{align} h_n(r₁ ⊗ … ⊗ r_n ⊗ m)&=

	n-1
\sum
	i=1

(-1)ⁱ⁺¹r₁ ⊗ … ⊗ r_i-1 ⊗ 1 ⊗ r_i ⊗ … ⊗ r_n ⊗ m \end{align}

One can similarly construct a resolution of a right

-module

by free right modules with the complex

\operatorname{Bar}_R(N,R)

Notice that, in the case one wants to resolve

as a module over itself, the above two complexes are the same, and actually give a resolution of

-bimodules. This provides one with a slightly smaller resolution of

by free

-bimodules than the naive option

\operatorname{Bar}
	R^e

(R^e,M)

. Here we are using the equivalence between

-bimodules and

R^e

-modules, where

R^e=R ⊗ R^{\operatorname{op}}

, see bimodules for more details.

The Normalized Bar Complex

The normalized (or reduced) standard complex replaces

A ⊗ A ⊗ … ⊗ A ⊗ A

with

A ⊗ (A/K) ⊗ … ⊗ (A/K) ⊗ A

References

Ginzburg . Victor . Victor Ginzburg . Lectures on Noncommutative Geometry . math.AG/0506603 . 2005.

Notes and References

Eilenberg . Samuel . Lane . Saunders Mac . July 1953 . On the Groups H(Π, n), I . The Annals of Mathematics . 58 . 1 . 55 . 10.2307/1969820. subscription .
Book: Cartan, Henry . Homological Algebra (PMS-19) . Eilenberg . Samuel . 2016 . Princeton University Press . 978-0-691-04991-5 . Princeton Mathematical Series . Princeton, NJ.