Editing Tensor product (section)

=== Computing the tensor product ===
For vector spaces, the tensor product <math>V \otimes W</math> is quickly computed since bases of {{math|''V''}} of {{math|''W''}} immediately determine a basis of {{tmath|1= V \otimes W }}, as was mentioned above. For modules over a general (commutative) ring, not every module is free. For example, {{math|'''Z'''/''n'''''Z'''}} is not a free abelian group ({{math|'''Z'''}}-module). The tensor product with {{math|'''Z'''/''n'''''Z'''}} is given by:
<math display="block">M \otimes_\mathbf{Z} \mathbf{Z}/n\mathbf{Z} = M/nM.</math>

More generally, given a [[presentation of a module|presentation]] of some {{math|''R''}}-module {{math|''M''}}, that is, a number of generators <math>m_i \in M, i \in I</math> together with relations:
<math display="block">\sum_{j \in J} a_{ji} m_i = 0,\qquad a_{ij} \in R,</math>
the tensor product can be computed as the following [[cokernel]]:
<math display="block">M \otimes_R N = \operatorname{coker} \left(N^J \to N^I\right)</math>

Here {{tmath|1= N^J = \oplus_{j \in J} N }}, and the map <math>N^J \to N^I</math> is determined by sending some <math>n \in N</math> in the {{math|''j''}}th copy of <math>N^J</math> to <math>a_{ij} n</math> (in {{tmath|1= N^I }}). Colloquially, this may be rephrased by saying that a presentation of {{math|''M''}} gives rise to a presentation of {{tmath|1= M \otimes_R N }}. This is referred to by saying that the tensor product is a [[right exact functor]]. It is not in general left exact, that is, given an injective map of {{math|''R''}}-modules {{tmath|1= M_1 \to M_2 }}, the tensor product:
<math display="block">M_1 \otimes_R N \to M_2 \otimes_R N</math>
is not usually injective. For example, tensoring the (injective) map given by multiplication with {{math|''n''}}, {{math|''n'' : '''Z''' → '''Z'''}} with {{math|'''Z'''/''n'''''Z'''}} yields the zero map {{math|0 : '''Z'''/''n'''''Z''' → '''Z'''/''n'''''Z'''}}, which is not injective. Higher [[Tor functor]]s measure the defect of the tensor product being not left exact. All higher Tor functors are assembled in the [[derived tensor product]].