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→References: Resolving Category:Harv and Sfn no-target errors. Created cite |
Resolving Category:Harv and Sfn no-target errors. Author and year are required |
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where Γ means the [[space of sections]] and the superscript <math>\otimes p</math> means tensoring ''p'' times over ''R''. By definition, an element of <math>\mathfrak{T}^p_q</math> is a [[tensor field]] of type (''p'', ''q'').
As ''R''-modules, <math>\mathfrak{T}^q_p</math> is the dual module of <math>\mathfrak{T}^p_q.</math><ref>{{harvnb|Helgason|1978|loc=Lemma 2.3'}}</ref>
To lighten the notation, put <math>E = \Gamma(M, T M)</math> and so <math>E^* = \Gamma(M, T^* M)</math>.<ref>This is actually the ''definition'' of differential one-forms, global sections of <math>T^*M</math>, in Helgason, but is equivalent to the usual definition that does not use module theory.</ref> When ''p'', ''q'' ≥ 1, for each (''k'', ''l'') with 1 ≤ ''k'' ≤ ''p'', 1 ≤ ''l'' ≤ ''q'', there is an ''R''-multilinear map:
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<math display="block">(X \otimes_R Y)_n = \sum_{i + j = n} X_i \otimes_R Y_j,</math>
with the differential given by: for ''x'' in ''X''<sub>''i''</sub> and ''y'' in ''Y''<sub>''j''</sub>,
<math display="block">d_{X \otimes Y} (x \otimes y) = d_X(x) \otimes y + (-1)^i x \otimes d_Y(y).</math><ref>{{harvnb|May|1999|loc=ch. 12 §3}}</ref>
For example, if ''C'' is a chain complex of flat abelian groups and if ''G'' is an abelian group, then the homology group of <math>C \otimes_{\Z } G</math> is the homology group of ''C'' with coefficients in ''G'' (see also: [[universal coefficient theorem]].)
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