Tor functor Homological algebra

suppose r ring, , denoted r-mod category of left r-modules , mod-r category of right r-modules (if r commutative, 2 categories coincide). fix module b in r-mod. in mod-r, set t(a) = a⊗rb. t right exact functor mod-r category of abelian groups ab (in case when r commutative, right exact functor mod-r mod-r) , left derived functors lnt defined. set

t
o
r


n


r


(
a
,
b
)
=
(

l

n


t
)
(
a
)


{\displaystyle \mathrm {tor} _{n}^{r}(a,b)=(l_{n}t)(a)}

i.e., take projective resolution

⋯
→

p

2


→

p

1


→

p

0


→
a
→
0


{\displaystyle \cdots \rightarrow p_{2}\rightarrow p_{1}\rightarrow p_{0}\rightarrow a\rightarrow 0}

then remove term , tensor projective resolution b complex

⋯
→

p

2



⊗

r


b
→

p

1



⊗

r


b
→

p

0



⊗

r


b
→
0


{\displaystyle \cdots \rightarrow p_{2}\otimes _{r}b\rightarrow p_{1}\otimes _{r}b\rightarrow p_{0}\otimes _{r}b\rightarrow 0}

(note a⊗rb not appear , last arrow 0 map) , take homology of complex.