{{Short description|Technique for constructing resolutions in homological algebra}} {{redirect|Standard resolution|the television monitor size|Standard-definition television}} 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,<ref name=":0">{{Cite journal |last=Eilenberg |first=Samuel |last2=Lane |first2=Saunders Mac |date=July 1953 |title=On the Groups H(Π, n), I |url=https://www.jstor.org/stable/1969820?origin=crossref |journal=The Annals of Mathematics |volume=58 |issue=1 |pages=55 |doi=10.2307/1969820|url-access=subscription }}</ref> and Henri Cartan and Eilenberg<ref>{{Cite book |last=Cartan |first=Henry |title=Homological Algebra (PMS-19) |last2=Eilenberg |first2=Samuel |date=2016 |publisher=Princeton University Press |isbn=978-0-691-04991-5 |series=Princeton Mathematical Series |location=Princeton, NJ}}</ref> and has since been generalized in many ways. The name "bar complex" comes from the fact that Eilenberg and Mac Lane<ref name=":0" /> used a vertical bar | as a shortened form of the tensor product <math>\otimes</math> in their notation for the complex.
==Definition==
Let <math>R</math> be an algebra over a field <math>k</math>, let <math>M_1</math> be a right <math>R</math>-module, and let <math>M_2</math> be a left <math>R</math>-module. Then, one can form the bar complex <math>\operatorname{Bar}_R(M_1,M_2)</math> given by :<math>\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\,,</math> with the differential :<math>\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}</math>
==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 <math>M</math> be a left <math>R</math>-module, with <math>R</math> a unital <math>k</math>-algebra. Then, the bar complex <math>\operatorname{Bar}_R(R,M)</math> gives a resolution of <math>M</math> by free left <math>R</math>-modules. Explicitly, the complex is{{sfn|Weibel|1994|p=283}} :<math>\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\,,</math> This complex is composed of free left <math>R</math>-modules, since each subsequent term is obtained by taking the free left <math>R</math>-module on the underlying vector space of the previous term.
To see that this gives a resolution of <math>M</math>, consider the modified complex :<math>\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\,,</math> Then, the above bar complex being a resolution of <math>M</math> is equivalent to this extended complex having trivial homology. One can show this by constructing an explicit homotopy <math>h_n : R^{\otimes_k n} \otimes_k M \to R^{\otimes_k (n+1)} \otimes_k M</math> between the identity and 0. This homotopy is given by :<math>\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}</math>
One can similarly construct a resolution of a right <math>R</math>-module <math>N</math> by free right modules with the complex <math>\operatorname{Bar}_R(N,R)</math>.
Notice that, in the case one wants to resolve <math>R</math> as a module over itself, the above two complexes are the same, and actually give a resolution of <math>R</math> by <math>R</math>-<math>R</math>-bimodules. This provides one with a slightly smaller resolution of <math>R</math> by free <math>R</math>-<math>R</math>-bimodules than the naive option <math>\operatorname{Bar}_{R^e}(R^e,M)</math>. Here we are using the equivalence between <math>R</math>-<math>R</math>-bimodules and <math>R^e</math>-modules, where <math>R^e = R \otimes R^\operatorname{op}</math>, see bimodules for more details.
==The Normalized Bar Complex==
The normalized (or reduced) standard complex replaces <math>A\otimes A\otimes \cdots \otimes A\otimes A</math> with <math>A\otimes(A/K) \otimes \cdots \otimes (A/K)\otimes A</math>.
==See also==
==Notes== {{reflist}}
==References==
* {{cite arXiv |last=Ginzburg |first=Victor |authorlink=Victor Ginzburg |title=Lectures on Noncommutative Geometry |eprint=math.AG/0506603 |year=2005}} * {{Citation | last1=Weibel | first1=Charles | author1-link=Charles Weibel | title=An Introduction to Homological Algebra | series=Cambridge Studies in Advanced Mathematics | volume=38 | publisher=Cambridge University Press | location=Cambridge | isbn=0-521-43500-5 | year=1994}}
Category:Homological algebra