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Tor

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Homological algebra

homological algebra

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nonabelian homological algebra

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Contents

Idea

In the context of homological algebra, the TorTor-functor is the
left derived functor of the tensor product of RR-modules, for RR a commutative ring.

Together with the Ext-functor it constitutes one of the central operations of interest in homological algebra.

Definition

Given a ring RR the bifunctor R:Mod R× RModAb\otimes_R : Mod_{R}\times _{R}Mod\to Ab from two copies of RR-Mod to Ab is a right exact functor. Its left derived functors are the Tor-functors

Tor(,B):Mod RAb Tor(-,B) : Mod_R \to Ab

and

Tor(A,):Mod RAb Tor(A,-) : Mod_R \to Ab

with respect to one argument with fixed another, if they exist, are parts of a bifunctor

Tor:Mod R× RModAb. Tor : Mod_{R}\times _{R}Mod\to Ab \,.

Properties

Existence and balancing

Given a right RR-module

AMod R A \in Mod_R

and a left RR-module

B RMod B \in {}_R Mod

there are in principle three different ways to compute their derived tensor product Tor (A,B)Tor_\bullet(A,B):

  1. keeping BB fixed and deriving the functor

    () RB:Mod RAb (-) \otimes_R B : Mod_R \to Ab
  2. keeping AA fixed and deriving the functor

    A R(): RModAb A \otimes_R (-) : {}_R Mod \to Ab
  3. deriving the functor

    () R():Mor R× RModAb (-) \otimes_R (-) : Mor_R \times {}_R Mod \to Ab

    in both arguments

Theorem

If both Mod RMod_{R} and RMod_{R}Mod have enough projectives, then all these three derived functors exist and all give the same result.

Proof

Existence is clear from the very definition of derived functor in homological algebra. So we show that deriving in the left argument gives the same result as deriving in the right argument.

Let Q A qiAQ^A_\bullet \stackrel{\simeq_{qi}}{\to} A and Q B qiBQ^B_\bullet \stackrel{\simeq_{qi}}{\to} B be projective resolutions of AA and BB, respectively. The corresponding tensor product of chain complexes]] Tot(Q AQ B)Tot (Q^A_\bullet\otimes Q^B_\bullet), hence by prop. \ref{AsTotalComplex} the total complex of the degreewise tensor product of modules double complex carries the filtration by horizontal degree as well as that by vertical degree.

Accordingly there are the corresponding two spectral sequences of a double complex, to be denoted here { AE p,q r} r,p,q\{{}^{A}E^r_{p,q}\}_{r,p,q} (for the filtering by AA-degree) and { BE p,q r} r,p,q\{{}^{B}E^r_{p,q}\}_{r,p,q} (for the filtering by BB-degree). By the discussion there, both converge to the chain homology of the total complex.

We find the value of both spectral sequences on low degree pages according to the general discussion at spectral sequence of a double complex - low degree pages.

The 0th page for both is

AE p,q 0= BE p,q 0Q p A RQ q B. {}^A E^0_{p,q} = {}^B E^0_{p,q} \coloneqq Q^A_p \otimes_R Q^B_q \,.

For the first page we have

AE p,q 1 H q(C p,) H q(Q p AQ B) \begin{aligned} {}^A E^1_{p,q} & \simeq H_q(C_{p,\bullet}) \\ & \simeq H_q( Q^A_p \otimes Q^B_\bullet ) \end{aligned}

and

BE p,q 1 H q(C ,p) H q(Q AQ p B). \begin{aligned} {}^B E^1_{p,q} & \simeq H_q(C_{\bullet,p}) \\ & \simeq H_q( Q^A_\bullet \otimes Q^B_p ) \end{aligned} \,.

Now using the universal coefficient theorem in homology and the fact that Q AQ^A_\bullet and Q BQ^B_\bullet is a resolution by projective objects, by construction, hence of tensor acyclic objects for which all Tor-modules vanish, this simplifies to

AE p,q 1 Q p AH q(Q B) {Q p A RB ifq=0 0 otherwise \begin{aligned} {}^A E^1_{p,q} & \simeq Q^A_p \otimes H_q(Q^B_\bullet) \\ & \simeq \left\{ \array{ Q^A_p \otimes_R B & if\; q = 0 \\ 0 & otherwise } \right. \end{aligned}

and similarly

BE p,q 1 H q(Q A) RQ p B {A RQ p B ifq=0 0 otherwise. \begin{aligned} {}^B E^1_{p,q} & \simeq H_q(Q^A_\bullet) \otimes_R Q^B_p \\ & \simeq \left\{ \array{ A \otimes_R Q^B_p & if\; q = 0 \\ 0 & otherwise } \right. \end{aligned} \,.

It follows for the second pages that

AE p,q 2 H p(H q vert(Q AQ B)) {(L p(() RB))(A) ifq=0 0 otherwise \begin{aligned} {}^A E^2_{p,q} & \simeq H_p(H^{vert}_q(Q^A_\bullet \otimes Q^B_\bullet)) \\ & \simeq \left\{ \array{ (L_p( (-)\otimes_R B ))(A) & if \; q = 0 \\ 0 & otherwise } \right. \end{aligned}

and

BE p,q 2 H p(H q hor(Q AQ B)) {(L p(A R()))(B) ifq=0 0otherwise. \begin{aligned} {}^B E^2_{p,q} & \simeq H_p(H^{hor}_q(Q^A_\bullet \otimes Q^B_\bullet)) \\ & \simeq \left\{ \array{ (L_p ( A \otimes_R (-) ))(B) & if \; q = 0 \\ 0 \; otherwise } \right. \end{aligned} \,.

Now both of these second pages are concentrated in a single row and hence have converged on that page already. Therefore, since they both converge to the same value:

L p(() RB)(A) AE p,0 2 AE p,0 BE p,0 2L p(A R())(B). L_p((-)\otimes_R B)(A) \simeq {}^A E^2_{p,0} \simeq {}^A E^\infty_{p,0} \simeq {}^B E^2_{p,0} \simeq L_p(A \otimes_R (-))(B) \,.

Respect for direct sums and filtered colimits

Proposition

Each Tor n R(,N)Tor_n^R(-,N) respects direct sums.

Proof

Let SS \in Set and let {N s} sS\{N_s\}_{s \in S} be an SS-family of RR-modules. Observe that

  1. if {(F s) } sS\{(F_s)_\bullet\}_{s \in S} is an family of projective resolutions, then their degreewise direct sum ( sSF) (\oplus_{s \in S} F)_\bullet is a projective resolution of sSN s\oplus_{s \in S} N_s.

  2. the tensor product functor distributes over direct sums (this is discussed at tensor product of modules – monoidal category structure)

  3. the chain homology functor preserves direct sums (this is discussed at chain homology - respect for direct sums).

Using this we have

Tor n R( sSN s,N) H n(( sSF)N) H n( sS(F sN)) sSH n(F sN) sSTor n(N s,N). \begin{aligned} Tor_n^R(\oplus_{s \in S} N_s, N) & \simeq H_n\left( \left(\oplus_{s \in S} F\right) \otimes N \right) \\ & \simeq H_n\left( \oplus_{s \in S} \left(F_s \otimes N \right) \right) \\ & \simeq \oplus_{s \in S} H_n( F_s \otimes N ) \\ & \simeq \oplus_{s \in S} Tor_n(N_s, N) \end{aligned} \,.
Proposition

Each Tor n R(,N)Tor_n^R(-,N) respects filtered colimits.

Proof

Let hence A:IRModA \colon I \to R Mod be a filtered diagram of modules. For each A iA_i, iIi \in I we may find a projective resolution and in fact a free resolution (Y i) qiA(Y_i)_\bullet \stackrel{\simeq_{qi}}{\to} A. Since chain homology commutes with filtered colimits (this is discussed at chain homology - respect for filtered colimits), this means that

(lim iY i) A (\underset{\to_i}{\lim} Y_i)_\bullet \to A

is still a quasi-isomorphism. Moreover, by Lazard's criterion the degreewise filtered colimits of free modules lim i(Y i) n\underset{\to_i}{\lim} (Y_i)_n for each nn \in \mathbb{N} are flat modules. This means that lim i(Y i) A\underset{\to_i}{\lim} (Y_i)_\bullet \to A is flat resolution of AA. By the very definition or else by the basic properties of flat modules, this means that it is a ()N(-)\otimes N-acyclic resolution. By the discussion there it follows that

Tor n (A,N)H n((lim iY i)N). Tor_n^\mathbb{Z}(A,N) \simeq H_n( (\underset{\to_i}{\lim} Y_i) \otimes N ) \,.

Now the tensor product of modules is a left adjoint functor (the right adjoint being the internal hom of modules) and so it commutes over the filtered colimit to yield, using again that chain homology commutes with filtered colimits,

H n(lim i(Y iN)) lim iH n(Y iN) lim iTor n(A i,N). \begin{aligned} \cdots & \simeq H_n( \underset{\to_i}{\lim} (Y_i \otimes N) ) \\ & \simeq \underset{\to_i}{\lim} H_n( Y_i \otimes N ) \\ & \simeq \underset{\to_i}{\lim} Tor_n( A_i, N) \end{aligned} \,.

Relation to torsion groups

An abelian group is called torsion if its elements are “nilpotent”, hence if all its elements have finite order.

Definition

For AA \in Ab and pp \in \mathbb{N}, write

pA{aApa=0} {}_p A \coloneqq \{ a \in A | p \cdot a = 0 \}

for the pp-torsion subgroup consisting of all those elements whose pp-fold sum with themselves gives 0.

For nn \in \mathbb{N} with n1n \geq 1, write n=/n\mathbb{Z}_n = \mathbb{Z}/n\mathbb{Z} for the cyclic group of order nn, as usual.

Proposition

For pp \in \mathbb{N}, p1p \geq 1, and AA \in Ab \simeq \mathbb{Z}Mod any abelian group, we have an isomorphism

Tor 1 ( p,A) pA Tor_1^\mathbb{Z}(\mathbb{Z}_p, A) \simeq {}_p A

of the Tor 1Tor_1-group with the pp-torsion subgroup of AA.

For p=0p = 0 we have

Tor 1 (,A)0. Tor_1^{\mathbb{Z}}(\mathbb{Z}, A) \simeq 0 \,.
Proof

For the first statement, the short exact sequence

0pmodp p0 0 \to \mathbb{Z} \stackrel{\cdot p}{\to} \mathbb{Z} \stackrel{mod\, p}{\to} \mathbb{Z}_p \to 0

constitutes a projective resolution (even a free resolution) of p\mathbb{Z}_p. Accordingly we have

Tor 1 ( p,A) H 1([0A(p)AA) ker((p)A) {aApa=0}. \begin{aligned} Tor_1^\mathbb{Z}(\mathbb{Z}_p, A) &\simeq H_1( [\cdots\to 0 \to \mathbb{Z}\otimes A \stackrel{(\cdot p) \otimes A}{\to} \mathbb{Z} \otimes A ) \\ & \simeq ker( (\cdot p) \otimes A ) \\ & \simeq \{ a\in A | p\cdot a = 0 \} \end{aligned} \,.

Here in the last step we use that (p)A(\cdot p)\otimes A acts as

(1,a) (p,a) =p(1,a) =(1,pa). \begin{aligned} (1, a) &\mapsto (p,a) \\ & = p \cdot (1,a) \\ & = (1, p \cdot a) \end{aligned} \,.

For the second statement, \mathbb{Z} is already free hence [00][\cdots \to 0 \to 0 \to \mathbb{Z}] is already a projective resolution and hence Tor 1(,A)H 1(0)0Tor_1(\mathbb{Z}, A) \simeq H_1(0) \simeq 0.

Proposition

Let AA be a finite abelian group and BB any abelian group. Then Tor 1(A,B)Tor_1(A,B) is a torsion group. Specifically, Tor 1(A,B)Tor_1(A,B) is a direct sum of torsion subgroups of AA.

Proof

By a fundamental fact about finite abelian groups (see this theorem), AA is a direct sum of cyclic group Aoplusk p kA \simeq \oplusk \mathbb{Z}_{p_k}. By prop. 1 Tor 1Tor_1 respects this direct sum, so that

Tor 1(A,B) kTor 1( p k,B). Tor_1(A,B) \simeq \oplus_k Tor_1(\mathbb{Z}_{p_k}, B) \,.

By prop. 3 every direct summand on the right is a torsion group and hence so is the whole direct sum.

More generally we have:

Proposition

Let AA and BB be abelian groups. Write Tor Tor^\mathbb{Z} for the left derived functor of tensoring over R=R = \mathbb{Z}. Then

  1. Tor 1 (A,B)Tor^\mathbb{Z}_1(A,B) is a torsion group. Specifically it is a filtered colimit of torsion subgroups of BB.

  2. Tor 1 (/,A)Tor^{\mathbb{Z}}_1(\mathbb{Q}/\mathbb{Z}, A) is the torsion subgroup of AA.

  3. AA is a torsion-free group precisely if Tor 1 (A,)=0Tor^\mathbb{Z}_1(A,-) = 0, equivalently if Tor 1 (,A)=0Tor^\mathbb{Z}_1(-,A) = 0.

For instance (Weibel, prop. 3.1.2, prop. 3.1.3, cor. 3.1.5).

Proof

The group AA may be expressed as a filtered colimit

Alim iA i A \simeq \underset{\to_i}{\lim} A_i

of finitely generated subgroups (this is discussed at Mod - Limits and colimits). Each of these is a direct sum of cyclic groups.

By prop. 2 Tor 1 (,B)Tor_1^\mathbb{Z}(-,B) preserves these colimits. By prop. 3 every cyclic group is sent to a torsion group (of either AA or B).Thereforebyprop.refTorOutOfCyclicGroupB). Therefore by prop. \ref{TorOutOfCyclicGroup} Tor_1(A,B)$ is a filtered colimit of direct sums of torsion groups. This is itself a torsion group.

Remark

Analogous results fail, in general, for \mathbb{Z} replaced by another ring RR.

Corollary

An abelian group is torsion free precisely if regarded as a \mathbb{Z}-module it is a flat module.

See at flat module - Examples for more.

Symmetry in the two arguments

Proposition

For N 1,N 2RModN_1, N_2 \in R Mod and nn \in \mathbb{N} there is a natural isomorphism

Tor n(A,B)Tor n(B,A). Tor_n(A,B) \simeq Tor_n(B,A) \,.

We first give a proof for RR a principal ideal domain such as \mathbb{Z}.

Proof

Let RR be a principal ideal domain such as \mathbb{Z} (in the latter case RRMod\simeq Ab). Then by the discussion at projective resolution – length-1 resolutions there is always a short exact sequence

0F 1F 0N0 0 \to F_1 \to F_0 \to N \to 0

exhibiting a projective resolution of any module NN. It follows that Tor n2(,)=0Tor_{n \geq 2}(-,-) = 0.

Let then 0F 1F 2N 200 \to F_1 \to F_2 \to N_2 \to 0 be such a short resolution for N 2N_2. Then by the long exact sequence of a derived functor this induces an exact sequence of the form

0Tor 1(N 1,F 1)Tor 1(N 1,F 0)Tor 1(N 1,N 2)N 1F 1N 1F 0N 1N 20. 0 \to Tor_1(N_1, F_1) \to Tor_1(N_1, F_0) \to Tor_1(N_1, N_2) \to N_1 \otimes F_1 \to N_1 \otimes F_0 \to N_1 \otimes N_2 \to 0 \,.

Since by construction F 0F_0 and F 1F_1 are already projective modules themselves this collapses to an exact sequence

0Tor 1(N 1,N 2)N 1F 1N 1F 0N 1N 20. 0 \to Tor_1(N_1, N_2) \hookrightarrow N_1 \otimes F_1 \to N_1 \otimes F_0 \to N_1 \otimes N_2 \to 0 \,.

To the last three terms we apply the natural symmetric braiding isomorphism in (RMod, R)(R Mod, \otimes_R) to get

0 Tor 1(N 1,N 2) N 1F 1 N 1F 0 N 1N 2 0 0 Tor 1(N 2,N 1) F 1N 1 F 0N 1 N 2N 1 0. \array{ 0 &\to& Tor_1(N_1, N_2) &\hookrightarrow& N_1 \otimes F_1 &\to& N_1 \otimes F_0 &\to& N_1 \otimes N_2 &\to& 0 \\ && \downarrow && \downarrow^{\mathrlap{\simeq}} && \downarrow^{\mathrlap{\simeq}} && \downarrow^{\mathrlap{\simeq}} && \\ 0 &\to& Tor_1(N_2, N_1) &\hookrightarrow& F_1 \otimes N_1 &\to& F_0 \otimes N_1 &\to& N_2 \otimes N_1 &\to& 0 } \,.

This exhibits a morphism Tor 1(N 1,N 2)Tor 1(N 2,N 1)Tor_1(N_1,N_2) \to Tor_1(N_2, N_1) as the morphism induced on kernels from an isomorphism between two morphisms. Hence this is itself an isomorphism. (This is just by the universal property of the kernel, but one may also think of it as a simple application of the the four lemma/five lemma.)

Localization

(…)

For instance (Weibel, cor. 3.2.13).

References

Standard textbook accounts include the following:

Lecture notes include

  • Daniel Murfet, Tor (pdf)

section 3 of

Original articles include

  • Patrick Keef, On the Tor functor and some classes of abelian groups, Pacific J. Math. Volume 132, Number 1 (1988), 63-84. (Euclid)

Revised on October 29, 2012 17:37:59 by Urs Schreiber (131.174.188.167)