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Categories, abelian categories and cohomologies.
Yoshifumi Tsuchimoto
Tensor products and Tor.
DEFINITION 09.1
Let
be an associative unital (but not necessarily commutative) ring.
Let
be a right
module. Let
be a left
module.
For any (
)module
, an map
is called an
balanced biadditive map if

.

.

.
DEFINITION 09.3
We employ the assumption of the proposition above.
By a standard argument on universal objects, we see that such object is
unique up to a unique isomorphism. We call it
the
tensor product of
and
and denote it by
DEFINITION 09.5
For any left
module
,
the left derived functor
of
is called the Tor functor
and denoted by
.
By definition,
may be computed by using projective resolutions of
.
DEFINITION 09.6
For any group
, the derived functor of a functor
defined by
span
is called the homology of
with coefficients in
. We denote the
homology group
by
.
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