Editing Tor functor (section)

==Properties==
Here are some of the basic properties and computations of Tor groups.<ref>Weibel (1994), Chapters 2 and 3.</ref>

*Tor{{supsub|''R''|0}}(''A'', ''B'') ≅ ''A'' ⊗<sub>''R''</sub> ''B'' for any right ''R''-module ''A'' and left ''R''-module ''B''.
*Tor{{su|b=''i''|p=''R''}}(''A'', ''B'') = 0 for all ''i'' > 0 if either ''A'' or ''B'' is [[flat module|flat]] (for example, [[free module|free]]) as an ''R''-module. In fact, one can compute Tor using a flat resolution of either ''A'' or ''B''; this is more general than a projective (or free) resolution.<ref>Weibel (1994), Lemma 3.2.8.</ref>
*There are converses to the previous statement:
**If Tor{{su|b=1|p=''R''}}(''A'', ''B'') = 0 for all ''B'', then ''A'' is flat (and hence Tor{{su|b=''i''|p=''R''}}(''A'', ''B'') = 0 for all ''i'' > 0).
**If Tor{{su|b=1|p=''R''}}(''A'', ''B'') = 0 for all ''A'', then ''B'' is flat (and hence Tor{{su|b=''i''|p=''R''}}(''A'', ''B'') = 0 for all ''i'' > 0).
*By the general properties of [[derived functor]]s, every [[short exact sequence]] 0 → ''K'' → ''L'' → ''M'' → 0 of right ''R''-modules induces a [[long exact sequence]] of the form<ref>Weibel (1994), Definition 2.1.1.</ref> <math display="block">\cdots \to \operatorname{Tor}_2^R(M,B) \to \operatorname{Tor}_1^R(K,B) \to \operatorname{Tor}_1^R(L,B) \to \operatorname{Tor}_1^R (M,B) \to K\otimes_R B\to L\otimes_R B\to M\otimes_R B\to 0,</math> for any left ''R''-module ''B''. The analogous exact sequence also holds for Tor with respect to the second variable.
*Symmetry: for a commutative ring ''R'', there is a [[natural isomorphism]] Tor{{su|b=''i''|p=''R''}}(''A'', ''B'') ≅ Tor{{su|b=''i''|p=''R''}}(''B'', ''A'').<ref>Weibel (1994), Remark in section 3.1.</ref> (For ''R'' commutative, there is no need to distinguish between left and right ''R''-modules.)
*If ''R'' is a commutative ring and ''u'' in ''R'' is not a [[zero divisor]], then for any ''R''-module ''B'', <math display="block">\operatorname{Tor}^R_i(R/(u),B)\cong\begin{cases} B/uB & i=0\\ B[u] & i=1\\ 0 &\text{otherwise}\end{cases}</math> where <math display="block">B[u] = \{x \in B : ux =0 \}</math> is the ''u''-torsion subgroup of ''B''. This is the explanation for the name Tor. Taking ''R'' to be the ring <math>\Z</math> of integers, this calculation can be used to compute <math>\operatorname{Tor}^{\Z}_1(A,B)</math> for any [[finitely generated abelian group]] ''A''.
*Generalizing the previous example, one can compute Tor groups that involve the quotient of a commutative ring by any [[regular sequence]], using the [[Koszul complex]].<ref>Weibel (1994), section 4.5.</ref> For example, if ''R'' is the [[polynomial ring]] ''k''[''x''<sub>1</sub>, ..., ''x''<sub>''n''</sub>] over a field ''k'', then <math>\operatorname{Tor}_*^R(k,k)</math> is the [[exterior algebra]] over ''k'' on ''n'' generators in Tor<sub>1</sub>.
* <math>\operatorname{Tor}^{\Z}_i(A,B)=0</math> for all ''i'' ≥ 2. The reason: every [[abelian group]] ''A'' has a free resolution of length 1, since every subgroup of a [[free abelian group]] is free abelian. 
* Generalizing the previous example, <math>\operatorname{Tor}^{R}_i(A,B)=0</math> for all ''i'' ≥ 2 if <math>R</math> is a [[principal ideal domain]] (PID). The reason: every module ''A'' over a PID has a free resolution of length 1, since every submodule of a [[free module]] over a PID is free. 
*For any ring ''R'', Tor preserves [[direct sum of modules|direct sums]] (possibly infinite) and [[filtered colimit]]s in each variable.<ref>Weibel (1994), Corollary 2.6.17.</ref> For example, in the first variable, this says that <math display="block">\begin{align}
\operatorname{Tor}_i^R \left (\bigoplus_{\alpha} M_{\alpha}, N \right ) &\cong \bigoplus_{\alpha} \operatorname{Tor}_i^R(M_{\alpha},N) \\
\operatorname{Tor}_i^R \left (\varinjlim_{\alpha} M_{\alpha}, N \right ) &\cong \varinjlim_{\alpha} \operatorname{Tor}_i^R(M_{\alpha},N)
\end{align}</math>
*Flat base change: for a commutative flat ''R''-algebra ''T'', ''R''-modules ''A'' and ''B'', and an integer ''i'',<ref>Weibel (1994), Corollary 3.2.10.</ref> <math display="block">\mathrm{Tor}_i^R(A,B)\otimes_R T \cong \mathrm{Tor}_i^T(A\otimes_R T,B\otimes_R T).</math> It follows that Tor commutes with [[localization of a ring|localization]]. That is, for a [[multiplicatively closed set]] ''S'' in ''R'', <math display="block">S^{-1} \operatorname{Tor}_i^R(A, B) \cong \operatorname{Tor}_i^{S^{-1} R} \left (S^{-1} A, S^{-1} B \right ).</math>
*For a commutative ring ''R'' and commutative ''R''-algebras ''A'' and ''B'', Tor{{supsub|''R''|*}}(''A'',''B'') has the structure of a [[graded-commutative]] algebra over ''R''. Moreover, elements of odd degree in the Tor algebra have square zero, and there are [[divided power]] operations on the elements of positive even degree.<ref>Avramov & Halperin (1986), section 2.16; {{Citation | title=Stacks Project, Tag 09PQ | url=http://stacks.math.columbia.edu/tag/09PQ}}.</ref>