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In [[homological algebra]], the '''Tor functors''' are the [[derived functor]]s of the [[tensor product]] functor. They were first defined in generality to express the [[Künneth theorem]] and [[universal coefficient theorem]] in [[algebraic topology]].{{Citation needed|date=December 2008}} | |||
Specifically, suppose ''R'' is a [[ring (mathematics)|ring]], and denoted by ''R''-'''Mod''' the [[category theory|category]] of [[module (mathematics)|left ''R''-modules]] and by '''Mod'''-''R'' the category of right ''R''-modules (if ''R'' is [[commutative ring|commutative]], the two categories coincide). Pick a fixed module ''B'' in ''R''-'''Mod'''. For ''A'' in '''Mod'''-''R'', set ''T''(''A'') = ''A''⊗<sub>''R''</sub>''B''. Then ''T'' is a [[right exact functor]] from '''Mod'''-''R'' to the [[category of abelian groups]] '''Ab''' (in the case when ''R'' is commutative, it is a right exact functor from '''Mod'''-''R'' to '''Mod'''-''R'') and its [[derived functor|left derived functor]]s ''L<sub>n</sub>T'' are defined. We set | |||
: <math>\mathrm{Tor}_n^R(A,B)=(L_nT)(A)</math> | |||
i.e., we take a [[Projective module#Projective resolutions|projective resolution]] | |||
: <math>\cdots\rightarrow P_2 \rightarrow P_1 \rightarrow P_0 \rightarrow A\rightarrow 0</math> | |||
then remove the ''A'' term and tensor the projective resolution with ''B'' to get the complex | |||
: <math>\cdots \rightarrow P_2\otimes_R B \rightarrow P_1\otimes_R B \rightarrow P_0\otimes_R B \rightarrow 0</math> | |||
In the | (note that ''A''⊗<sub>''R''</sub>''B'' does not appear and the last arrow is just the zero map) and take the [[homology (mathematics)|homology]] of this complex. | ||
== Properties == | |||
* For every ''n'' ≥ 1, Tor{{su|b=''n''|p=''R''}} is an [[additive functor]] from '''Mod'''-''R'' × ''R''-'''Mod''' to '''Ab'''. In the case when ''R'' is commutative, we have additive functors from '''Mod'''-''R'' × '''Mod'''-''R'' to '''Mod'''-''R''. | |||
* As is true for every family of derived functors, every [[short exact sequence]] 0 → ''K'' → ''L'' → ''M'' → 0 induces a [[long exact sequence]] of the form | |||
::<math>\cdots\rightarrow\mathrm{Tor}_2^R(M,B)\rightarrow\mathrm{Tor}_1^R(K,B)\rightarrow\mathrm{Tor}_1^R(L,B)\rightarrow\mathrm{Tor}_1^R(M,B)\rightarrow K\otimes B\rightarrow L\otimes B\rightarrow M\otimes B\rightarrow 0</math>. | |||
* If ''R'' is commutative and ''r'' in ''R'' is not a [[zero divisor]] then | |||
::<math>\mathrm{Tor}_1^R(R/(r),B)=\{b\in B:rb=0\},</math> | |||
from which the terminology ''Tor'' (that is, ''Torsion'') comes: see [[torsion subgroup]]. | |||
* Tor{{su|b=''n''|p='''Z'''}}(''A'',''B'') = 0 for all ''n'' ≥ 2. The reason: every [[abelian group]] ''A'' has a [[free resolution]] of length 1, since subgroups of [[free abelian group]]s are free abelian. So in this important special case, the higher Tor functors are invisible. In addition, Tor{{su|b=1|p='''Z'''}}('''Z'''/''k'''''Z'''</sub>,''A'') = Ker(''f'') where ''f'' represents "multiplication by ''k''". | |||
* Furthermore, every free module has a free resolution of length zero, so by the argument above, if ''F'' is a free ''R''-module, then Tor{{su|b=''n''|p=''R''}}(''F,B'') = 0 for all ''n'' ≥ 1. | |||
* The Tor functors preserve [[filtered colimit]]s and arbitrary [[direct sum of modules|direct sums]]: there is a [[natural isomorphism]] | |||
::<math>\mathrm{Tor}_n^R \left (\bigoplus_i A_i, \bigoplus_j B_j \right) \simeq \bigoplus_i \bigoplus_j \mathrm{Tor}_n^R(A_i,B_j)</math> | |||
: | |||
* From the [[Finitely generated abelian group|classification of finitely generated abelian groups]], we know that every finitely generated abelian group is the direct sum of copies of '''Z''' and '''Z'''<sub>''k''</sub>. This together with the previous three points allows us to compute Tor{{su|b=1|p='''Z'''}}(''A'', ''B'') whenever ''A'' is finitely generated. | |||
* A module ''M'' in '''Mod'''-''R'' is [[flat module|flat]] if and only if Tor{{su|b=1|p=''R''}}(''M'', -) = 0. In this case, we even have Tor{{su|b=''n''|p=''R''}}(''M'', -) = 0 for all ''n'' ≥ 1 . In fact, to compute Tor{{su|b=''n''|p=''R''}}(''A,B''), one may use a ''[[flat resolution]]'' of ''A'' or ''B'', instead of a projective resolution (note that a projective resolution is automatically a flat resolution, but the converse isn't true, so allowing flat resolutions is more flexible). | |||
==See also== | |||
*[[Ext functor]] | |||
==References== | |||
* {{Weibel IHA}} | |||
{{DEFAULTSORT:Tor Functor}} | |||
[[Category:Homological algebra]] | |||
[[Category:Binary operations]] |
Revision as of 07:28, 14 January 2014
In homological algebra, the Tor functors are the derived functors of the tensor product functor. They were first defined in generality to express the Künneth theorem and universal coefficient theorem in algebraic topology.Potter or Ceramic Artist Truman Bedell from Rexton, has interests which include ceramics, best property developers in singapore developers in singapore and scrabble. Was especially enthused after visiting Alejandro de Humboldt National Park.
Specifically, suppose R is a ring, and denoted by R-Mod the category of left R-modules and by Mod-R the category of right R-modules (if R is commutative, the two categories coincide). Pick a fixed module B in R-Mod. For A in Mod-R, set T(A) = A⊗RB. Then T is a right exact functor from Mod-R to the category of abelian groups Ab (in the case when R is commutative, it is a right exact functor from Mod-R to Mod-R) and its left derived functors LnT are defined. We set
i.e., we take a projective resolution
then remove the A term and tensor the projective resolution with B to get the complex
(note that A⊗RB does not appear and the last arrow is just the zero map) and take the homology of this complex.
Properties
- For every n ≥ 1, TorTemplate:Su is an additive functor from Mod-R × R-Mod to Ab. In the case when R is commutative, we have additive functors from Mod-R × Mod-R to Mod-R.
- As is true for every family of derived functors, every short exact sequence 0 → K → L → M → 0 induces a long exact sequence of the form
- If R is commutative and r in R is not a zero divisor then
from which the terminology Tor (that is, Torsion) comes: see torsion subgroup.
- TorTemplate:Su(A,B) = 0 for all n ≥ 2. The reason: every abelian group A has a free resolution of length 1, since subgroups of free abelian groups are free abelian. So in this important special case, the higher Tor functors are invisible. In addition, TorTemplate:Su(Z/kZ,A) = Ker(f) where f represents "multiplication by k".
- Furthermore, every free module has a free resolution of length zero, so by the argument above, if F is a free R-module, then TorTemplate:Su(F,B) = 0 for all n ≥ 1.
- The Tor functors preserve filtered colimits and arbitrary direct sums: there is a natural isomorphism
- From the classification of finitely generated abelian groups, we know that every finitely generated abelian group is the direct sum of copies of Z and Zk. This together with the previous three points allows us to compute TorTemplate:Su(A, B) whenever A is finitely generated.
- A module M in Mod-R is flat if and only if TorTemplate:Su(M, -) = 0. In this case, we even have TorTemplate:Su(M, -) = 0 for all n ≥ 1 . In fact, to compute TorTemplate:Su(A,B), one may use a flat resolution of A or B, instead of a projective resolution (note that a projective resolution is automatically a flat resolution, but the converse isn't true, so allowing flat resolutions is more flexible).