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Bar complex
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<h2 id="definition">Definition</h2> <p>Let R {\displaystyle R} be an <a href="/facts/Algebra_over_a_field/8T8ulWJb">algebra</a> over a field k {\displaystyle k} , let M 1 {\displaystyle M_{1}} be a right R {\displaystyle R} -<a href="/facts/Module_(mathematics)/jP0LIhFL">module</a>, and let M 2 {\displaystyle M_{2}} be a left R {\displaystyle R} -module. Then, one can form the bar <a href="/facts/Chain_complex/YAlchtCT">complex</a> Bar R ( M 1 , M 2 ) {\displaystyle \operatorname {Bar} _{R}(M_{1},M_{2})} given by </p> ⋯ → M 1 ⊗ k R ⊗ k R ⊗ k M 2 → M 1 ⊗ k R ⊗ k M 2 → M 1 ⊗ k M 2 → 0 , {\displaystyle \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\,,} <p>with the differential </p> d ( m 1 ⊗ r 1 ⊗ ⋯ ⊗ r n ⊗ m 2 ) = m 1 r 1 ⊗ ⋯ ⊗ r n ⊗ m 2 + ∑ i = 1 n − 1 ( − 1 ) i m 1 ⊗ r 1 ⊗ ⋯ ⊗ r i r i + 1 ⊗ ⋯ ⊗ r n ⊗ m 2 + ( − 1 ) n m 1 ⊗ r 1 ⊗ ⋯ ⊗ r n m 2 {\displaystyle {\begin{aligned}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{aligned}}} <h2 id="resolutions">Resolutions</h2> <p>The bar complex is useful because it provides a canonical way of producing (<a href="/facts/Free_module/o6rC6yoJ">free</a>) resolutions of modules over a ring. However, often these resolutions are very large, and can be prohibitively difficult to use for performing actual computations. </p> <h3>Free Resolution of a Module</h3> <p>Let M {\displaystyle M} be a left R {\displaystyle R} -module, with R {\displaystyle R} a unital k {\displaystyle k} -algebra. Then, the bar complex Bar R ( R , M ) {\displaystyle \operatorname {Bar} _{R}(R,M)} gives a resolution of M {\displaystyle M} by free left R {\displaystyle R} -modules. Explicitly, the complex is<a class="footnote-ref" id="fnref:1" href="#fn:1"><sup>1</sup></a> </p> ⋯ → R ⊗ k R ⊗ k R ⊗ k M → R ⊗ k R ⊗ k M → R ⊗ k M → 0 , {\displaystyle \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\,,} <p>This complex is composed of free left R {\displaystyle R} -modules, since each subsequent term is obtained by taking the free left R {\displaystyle R} -module on the underlying vector space of the previous term. </p><p>To see that this gives a resolution of M {\displaystyle M} , consider the modified complex </p> ⋯ → R ⊗ k R ⊗ k R ⊗ k M → R ⊗ k R ⊗ k M → R ⊗ k M → M → 0 , {\displaystyle \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\,,} <p>Then, the above bar complex being a resolution of M {\displaystyle M} is equivalent to this extended complex having trivial homology. One can show this by constructing an explicit homotopy h n : R ⊗ k n ⊗ k M → R ⊗ k ( n + 1 ) ⊗ k M {\displaystyle 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 </p> h n ( r 1 ⊗ ⋯ ⊗ r n ⊗ m ) = ∑ i = 1 n − 1 ( − 1 ) i + 1 r 1 ⊗ ⋯ ⊗ r i − 1 ⊗ 1 ⊗ r i ⊗ ⋯ ⊗ r n ⊗ m {\displaystyle {\begin{aligned}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{aligned}}} <p>One can similarly construct a resolution of a right R {\displaystyle R} -module N {\displaystyle N} by free right modules with the complex Bar R ( N , R ) {\displaystyle \operatorname {Bar} _{R}(N,R)} . </p><p>Notice that, in the case one wants to resolve R {\displaystyle R} as a module over itself, the above two complexes are the same, and actually give a resolution of R {\displaystyle R} by R {\displaystyle R} - R {\displaystyle R} -bimodules. This provides one with a slightly smaller resolution of R {\displaystyle R} by free R {\displaystyle R} - R {\displaystyle R} -bimodules than the naive option Bar R e ( R e , M ) {\displaystyle \operatorname {Bar} _{R^{e}}(R^{e},M)} . Here we are using the equivalence between R {\displaystyle R} - R {\displaystyle R} -bimodules and R e {\displaystyle R^{e}} -modules, where R e = R ⊗ R op {\displaystyle R^{e}=R\otimes R^{\operatorname {op} }} , see <a href="/facts/Bimodules/NskmNGQ1">bimodules</a> for more details. </p> <h2 id="the-normalized-bar-complex">The Normalized Bar Complex</h2> <p>The normalized (or reduced) standard complex replaces A ⊗ A ⊗ ⋯ ⊗ A ⊗ A {\displaystyle A\otimes A\otimes \cdots \otimes A\otimes A} with A ⊗ ( A / K ) ⊗ ⋯ ⊗ ( A / K ) ⊗ A {\displaystyle A\otimes (A/K)\otimes \cdots \otimes (A/K)\otimes A} . </p> <h2 id="see-also">See also</h2> <ul><li><a href="/facts/Koszul_complex/G8N1ugQK">Koszul complex</a></li></ul> <h2 id="notes">Notes</h2> <ul><li><a href="/facts/Henri_Cartan/IGrXDS4a">Cartan, Henri</a>; <a href="/facts/Samuel_Eilenberg/vs6aksKX">Eilenberg, Samuel</a> (1956), <a href="https://books.google.com/books?id=0268b52ghcsC"><i>Homological algebra</i></a>, Princeton Mathematical Series, vol. 19, <a href="/facts/Princeton_University_Press/kwvFlKEj">Princeton University Press</a>, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-691-04991-5, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0077480">0077480</a>{{citation}}: CS1 maint: ignored ISBN errors (link)</li> <li><a href="/facts/Samuel_Eilenberg/vs6aksKX">Eilenberg, Samuel</a>; <a href="/facts/Saunders_Mac_Lane/rjgUqfkA">Mac Lane, Saunders</a> (1953), "On the groups of H ( Π , n ) {\displaystyle H(\Pi ,n)} . I", <i><a href="/facts/Annals_of_Mathematics/CUYktUqD">Annals of Mathematics</a></i>, Second Series, 58: 55–106, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.2307%2F1969820">10.2307/1969820</a>, <a href="/facts/ISSN_(identifier)/DPAflDvU">ISSN</a> <a href="https://search.worldcat.org/issn/0003-486X">0003-486X</a>, <a href="/facts/JSTOR_(identifier)/YTeVmaJ7">JSTOR</a> <a href="https://www.jstor.org/stable/1969820">1969820</a>, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0056295">0056295</a></li> <li><a href="/facts/Victor_Ginzburg/z3t61ccf">Ginzburg, Victor</a> (2005). "Lectures on Noncommutative Geometry". <a href="/facts/ArXiv_(identifier)/H6EtgnBe">arXiv</a>:<a href="https://arxiv.org/abs/math.AG/0506603">math.AG/0506603</a>.</li> <li><a href="/facts/Charles_Weibel/ERYbVCB9">Weibel, Charles</a> (1994), <i>An Introduction to Homological Algebra</i>, Cambridge Studies in Advanced Mathematics, vol. 38, Cambridge: <a href="/facts/Cambridge_University_Press/fgEBSSRq">Cambridge University Press</a>, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-521-43500-5</li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Weibel 1994, p. 283. - Weibel, Charles (1994), An Introduction to Homological Algebra, Cambridge Studies in Advanced Mathematics, vol. 38, Cambridge: Cambridge University Press, ISBN 0-521-43500-5 <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> </ol>