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Specifically, suppose ''R'' is a Ring , and denote 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''⊗''R''''B''. Then ''T'' is a Right Exact Functor from Mod-''R'' to the Category Of Abelian Groups '''Ab''' (in case ''R'' is commutative, it is a right exact functor from Mod-''R'' to Mod-''R'') and its Left Derived Functor s L''n''''T'' are defined. We set

: \mathrm{Tor}_n^R(A,B)=(L_nT)(A)

i.e., we take a Projective Resolution

: \cdots ightarrow P_3 ightarrow P_2 ightarrow P_1 ightarrow A ightarrow 0

then chop off the last term ''A'' and tensor it with ''B'' to get the complex

: \cdots ightarrow P_3\otimes B ightarrow P_2\otimes B ightarrow P_1\otimes B ightarrow 0

and take the Homology of this complex.


PROPERTIES


  • For every ''n'' ≥ 1, Tor''n''''R'' is an Additive Functor from Mod-''R'' × ''R''-Mod to '''Ab'''. In case ''R'' is commutative, we have additive functors from Mod-''R'' × Mod-''R'' to Mod-''R''.



:\cdots ightarrow\mathrm{Tor}_2^R(M,B) ightarrow\mathrm{Tor}_1^R(K,B) ightarrow\mathrm{Tor}_1^R(L,B) ightarrow\mathrm{Tor}_1^R(M,B) ightarrow K\otimes B ightarrow L\otimes B ightarrow M\otimes B ightarrow 0.

  • If ''R'' is commutative and ''r'' in ''R'' is not a Zero Divisor then

    • :\mathrm{Tor}_1^R(R/(r),B)=\{b\in B:rb=0\},

    from which the terminology ''Tor'' (that is, ''Torsion'') comes: see Torsion Subgroup .



  • In the case of s are free abelian. So in this important special case, the higher Tor functors are invisible.


  • The Tor functors commute with arbitrary

    • :\mathrm{Tor}_n^R(\oplus_i A_i, \oplus_j B_j) \simeq \oplus_i \oplus_j \mathrm{Tor}_n^R(A_i,B_j).


  • A module ''M'' in Mod-''R'' is Flat if and only if Tor1''R''(''M'', -) = 0. In this case, we even have Tor''n''''R''(''M'', -) = 0 for all ''n''. In fact, to compute Tor''n''''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).