Bar complex

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 $\backslash operatorname\{Bar\}\_R(M\_1,M\_2)$ given by

:$\backslash cdots\backslash rightarrow\; M\_1\; \backslash otimes\_k\; R\; \backslash otimes\_k\; R\; \backslash otimes\_k\; M\_2\; \backslash rightarrow\; M\_1\; \backslash otimes\_k\; R\; \backslash otimes\_k\; M\_2\; \backslash rightarrow\; M\_1\; \backslash otimes\_k\; M\_2\; \backslash rightarrow\; 0\backslash ,,$

with the differential

:$\backslash begin\{align\}$

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}

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.

Let $M$ be a left $R$-module, with $R$ a unital $k$-algebra. Then, the bar complex $\backslash operatorname\{Bar\}\_R(R,M)$ gives a resolution of $M$ by free left $R$-modules. Explicitly, the complex is{{sfn|Weibel|1994|p=283}}

:$\backslash cdots\backslash rightarrow\; R\; \backslash otimes\_k\; R\; \backslash otimes\_k\; R\; \backslash otimes\_k\; M\; \backslash rightarrow\; R\; \backslash otimes\_k\; R\; \backslash otimes\_k\; M\; \backslash rightarrow\; R\; \backslash otimes\_k\; M\; \backslash rightarrow\; 0\backslash ,,$

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

:$\backslash cdots\backslash rightarrow\; R\; \backslash otimes\_k\; R\; \backslash otimes\_k\; R\; \backslash otimes\_k\; M\; \backslash rightarrow\; R\; \backslash otimes\_k\; R\; \backslash otimes\_k\; M\; \backslash rightarrow\; R\; \backslash otimes\_k\; M\; \backslash rightarrow\; M\; \backslash rightarrow\; 0\backslash ,,$

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 $h\_n\; :\; R^\{\backslash otimes\_k\; n\}\; \backslash otimes\_k\; M\; \backslash to\; R^\{\backslash otimes\_k\; (n+1)\}\; \backslash otimes\_k\; M$ between the identity and 0. This homotopy is given by

:$\backslash begin\{align\}$

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 $\backslash operatorname\{Bar\}\_R(N,R)$.

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

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

• Koszul complex

{{reflist}}

• {{Citation | last1=Cartan | first1=Henri | author1-link=Henri Cartan|last2=Eilenberg | first2=Samuel | author2-link=Samuel Eilenberg | title=Homological algebra | url=https://books.google.com/books?id=0268b52ghcsC | publisher=Princeton University Press | series=Princeton Mathematical Series | isbn=((978-0-691-04991-5)) | mr=0077480 | year=1956 | volume=19}}
• {{Citation | last1=Eilenberg | first1=Samuel | author1-link=Samuel Eilenberg | last2=Mac Lane | first2=Saunders | author2-link=Saunders Mac Lane | title=On the groups of $H(\backslash Pi,n)$. I | jstor=1969820 | mr=0056295 | year=1953 | journal=Annals of Mathematics |series=Second Series | issn=0003-486X | volume=58 | pages=55–106 | doi=10.2307/1969820}}
• {{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

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