Editing Tor functor

{{Short description|Construction in homological algebra}}
In [[mathematics]], the '''Tor functors''' are the [[derived functor]]s of the [[tensor product of modules]] over a [[ring (mathematics)|ring]]. Along with the [[Ext functor]], Tor is one of the central concepts of [[homological algebra]], in which ideas from [[algebraic topology]] are used to construct invariants of algebraic structures. The [[group cohomology#Group homology|homology of groups]], [[Lie algebra homology|Lie algebra]]s, and [[Hochschild homology|associative algebras]] can all be defined in terms of Tor. The name comes from a relation between the first Tor group Tor<sub>1</sub> and the [[torsion subgroup]] of an [[abelian group]].

In the special case of abelian groups, Tor was introduced by [[Eduard Čech]] (1935) and named by [[Samuel Eilenberg]] around 1950.<ref>Weibel (1999).</ref> It was first applied to the [[Künneth theorem]] and [[universal coefficient theorem]] in topology. For modules over any ring, Tor was defined by [[Henri Cartan]] and Eilenberg in their 1956 book ''Homological Algebra''.<ref>Cartan & Eilenberg (1956), section VI.1.</ref>

==Definition==
Let ''R'' be a [[ring (mathematics)|ring]]. Write ''R''-Mod for the [[category theory|category]] of [[module (mathematics)|left ''R''-modules]] and Mod-''R'' for the category of right ''R''-modules. (If ''R'' is [[commutative ring|commutative]], the two categories can be identified.) For a fixed left ''R''-module ''B'', let <math>T(A) = A\otimes_R B</math> for ''A'' in Mod-''R''. This is a [[right exact functor]] from Mod-''R'' to the [[category of abelian groups]] Ab, and so it has left [[derived functor]]s <math>L_i T</math>. The Tor groups are the abelian groups defined by
<math display="block">\operatorname{Tor}_i^R(A,B) = (L_iT)(A),</math>
for an [[integer]] ''i''. By definition, this means: take any [[projective module#Projective resolutions|projective resolution]]
<math display="block">\cdots\to P_2 \to P_1 \to P_0 \to A\to 0,</math>
and remove ''A'', and form the [[chain complex]]:
<math display="block">\cdots \to P_2\otimes_R B \to P_1\otimes_R B \to P_0\otimes_R B \to 0</math>

For each integer ''i'', the group <math>\operatorname{Tor}_i^R(A,B)</math> is the [[chain complex|homology]] of this complex at position ''i''. It is zero for ''i'' negative. Moreover, <math>\operatorname{Tor}_0^R(A,B)</math> is the [[cokernel]] of the map <math>P_1\otimes_R B \to P_0\otimes_R B</math>, which is [[isomorphic]] to <math>A \otimes_R B</math>.

Alternatively, one can define Tor by fixing ''A'' and taking the left derived functors of the right exact functor <math>G(B)=A\otimes_RB</math>. That is, tensor ''A'' with a projective resolution of ''B'' and take homology. Cartan and Eilenberg showed that these constructions are independent of the choice of projective resolution, and that both constructions yield the same Tor groups.<ref>Weibel (1994), section 2.4 and Theorem 2.7.2.</ref> Moreover, for a fixed ring ''R'', Tor is a functor in each variable (from ''R''-modules to abelian groups).

For a commutative ring ''R'' and ''R''-modules ''A'' and ''B'', <math>\operatorname{Tor}^R_i(A,B)</math> is an ''R''-module (using that <math>A\otimes_RB</math> is an ''R''-module in this case). For a non-commutative ring ''R'', <math>\operatorname{Tor}^R_i(A,B)</math> is only an abelian group, in general. If ''R'' is an [[algebra over a ring]] ''S'' (which means in particular that ''S'' is commutative), then <math>\operatorname{Tor}^R_i(A,B)</math> is at least an ''S''-module.

==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>

==Important special cases==

*[[Group homology]] is defined by <math>H_*(G,M)=\operatorname{Tor}^{\Z[G]}_*(\Z, M),</math> where ''G'' is a group, ''M'' is a [[group representation|representation]] of ''G'' over the integers, and <math>\Z[G]</math> is the [[group ring]] of ''G''.
*For an [[algebra over a field|algebra]] ''A'' over a field ''k'' and an ''A''-[[bimodule]] ''M'', [[Hochschild homology]] is defined by <math display="block">HH_*(A,M)=\operatorname{Tor}_*^{A\otimes_k A^{\text{op}}}(A, M).</math>
*[[Lie algebra homology]] is defined by <math>H_*(\mathfrak g,M)=\operatorname{Tor}_*^{U\mathfrak g}(R,M)</math>, where <math>\mathfrak g</math> is a [[Lie algebra]] over a commutative ring ''R'', ''M'' is a <math>\mathfrak g</math>-module, and <math>U\mathfrak g</math> is the [[universal enveloping algebra]].
*For a commutative ring ''R'' with a homomorphism onto a field ''k'', <math>\operatorname{Tor}_*^R(k,k)</math> is a graded-commutative [[Hopf algebra]] over ''k''.<ref>Avramov & Halperin (1986), section 4.7.</ref> (If ''R'' is a [[Noetherian local ring]] with residue field ''k'', then the dual Hopf algebra to <math>\operatorname{Tor}_*^R(k,k)</math> is [[Ext functor#Important special cases|Ext]]{{supsub|*|''R''}}(''k'',''k'').) As an algebra, <math>\operatorname{Tor}_*^R(k,k)</math> is the free graded-commutative divided power algebra on a graded vector space π<sub>*</sub>(''R'').<ref>Gulliksen & Levin (1969), Theorem 2.3.5; Sjödin (1980), Theorem 1.</ref> When ''k'' has [[characteristic of a field|characteristic]] zero, π<sub>*</sub>(''R'') can be identified with the [[André-Quillen homology]] ''D''<sub>*</sub>(''k''/''R'',''k'').<ref>Quillen (1970), section 7.</ref>

==See also==
*[[Flat morphism]]
*[[Serre's intersection formula]]
*[[Derived tensor product]]
*[[Eilenberg–Moore spectral sequence]]

==Notes==
{{reflist|30em}}

==References==
*{{Citation | author1-last=Avramov | author1-first=Luchezar | author1-link=Luchezar L. Avramov | author2-last=Halperin | author2-first=Stephen | author2-link=Stephen Halperin | chapter=Through the looking glass: a dictionary between rational homotopy theory and local algebra | title=Algebra, algebraic topology, and their interactions (Stockholm, 1983) | editor=J.-E. Roos | series=Lecture Notes in Mathematics | volume=1183 | publisher=[[Springer Nature]] | year=1986 | pages=1–27 | isbn=978-3-540-16453-1 | doi=10.1007/BFb0075446 | mr=0846435}}
*{{Citation | author1-last=Cartan | author1-first=Henri | author1-link=Henri Cartan | author2-last=Eilenberg | author2-first=Samuel | author2-link=Samuel Eilenberg | title=Homological algebra | orig-year=1956 | year=1999 | publisher=[[Princeton University Press]] | location=Princeton | mr=0077480 | isbn=0-691-04991-2}}
*{{Citation | author1-last=Čech | author1-first=Eduard | author1-link=Eduard Čech | title=Les groupes de Betti d'un complexe infini | journal=[[Fundamenta Mathematicae]] | volume=25 | year=1935 | pages=33–44 | doi=10.4064/fm-25-1-33-44 | jfm=61.0609.02| url=http://dml.cz/bitstream/handle/10338.dmlcz/501039/Cech_01-0000-67_1.pdf | doi-access=free }}
*{{Citation | author1-last=Gulliksen | author1-first=Tor | author2-last=Levin | author2-first=Gerson | title=Homology of local rings | series=Queen's Papers in Pure and Applied Mathematics | publisher=Queen's University | volume=20 | year=1969 | mr=0262227}}
*{{Citation | last1=Quillen | first1=Daniel | author1-link=Daniel Quillen | chapter=On the (co-)homology of commutative rings | title=Applications of categorical algebra | pages=65–87 | series=Proc. Symp. Pure Mat. | volume=17 | publisher=[[American Mathematical Society]] | year=1970 | mr=0257068}}
*{{Citation | author1-last=Sjödin | author1-first=Gunnar | title=Hopf algebras and derivations | journal=[[Journal of Algebra]] | volume=64 |year=1980 | pages=218–229 | doi=10.1016/0021-8693(80)90143-X | mr=0575792| doi-access=free }} 
* {{Weibel IHA}}
*{{Citation | author1-last=Weibel | author1-first=Charles | author1-link=Charles Weibel | chapter=History of homological algebra | title=History of topology | pages=797–836 | publisher=North-Holland | location=Amsterdam | year=1999 | mr=1721123 | url=http://sites.math.rutgers.edu/~weibel/HA-history.pdf}}

==External links==
*{{Citation | author1=The Stacks Project Authors | title=The Stacks Project  | url=http://stacks.math.columbia.edu/}}

[[Category:Homological algebra]]
[[Category:Binary operations]]
[[Category:Functors]]