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(i) Suppose we have a contravariant functor
We would like to extend this to a contravariant functor
Let
be a chain complex in
, applying
we get the double complex
:
This can be converted into a chain complex in the usual way:
This defines what of an object is, so now we specify its behaviour on morphisms:
where denotes the component of from . The claim is that this defines a contravariant functor.
Since
is a chain map we have
so by the contravariance of the functor
we observe that
Also, because
sends morphisms to morphisms,
is a chain map, thus
These formulae prove precisely that
is a chain map by the commutativity of the following diagram:
Thus sends objects to objects and morphisms to morphisms. Functoriality of follows from that of :
(ii) Now suppose that the chain complex
is concentrated between dimensions
and
and that
. Then looking at
we observe that the only non-zero contributions occur when
and when
is between
and
, in other words
is only non-zero for
between
and
as required.
$ and $n$ and that $T_\mathbb{A}: \mathbb{A} \to \mathbb{A}$. Then looking at $$ T_{\mathbb{B}}(C)_k := \sum_{-i+j=k}{T_\mathbb{A}(C_i)_j} $$ we observe that the only non-zero contributions occur when $j=0$ and when $i$ is between be a chain complex in
, applying
we get the double complex
:
This can be converted into a chain complex in the usual way:
This defines what of an object is, so now we specify its behaviour on morphisms:
where denotes the component of from . The claim is that this defines a contravariant functor.
Since
is a chain map we have
so by the contravariance of the functor
we observe that
Also, because
sends morphisms to morphisms,
is a chain map, thus
These formulae prove precisely that
is a chain map by the commutativity of the following diagram:
Thus sends objects to objects and morphisms to morphisms. Functoriality of follows from that of :
(ii) Now suppose that the chain complex
is concentrated between dimensions
and
and that
. Then looking at
we observe that the only non-zero contributions occur when
and when
is between
and
, in other words
is only non-zero for
between
and
as required.
$ and $n$, in other words $T_{\mathbb{B}}(C)_k$ is only non-zero for $k$ between $-n$ and be a chain complex in
, applying
we get the double complex
:
This can be converted into a chain complex in the usual way:
This defines what of an object is, so now we specify its behaviour on morphisms:
where denotes the component of from . The claim is that this defines a contravariant functor.
Since
is a chain map we have
so by the contravariance of the functor
we observe that
Also, because
sends morphisms to morphisms,
is a chain map, thus
These formulae prove precisely that
is a chain map by the commutativity of the following diagram:
Thus sends objects to objects and morphisms to morphisms. Functoriality of follows from that of :
(ii) Now suppose that the chain complex
is concentrated between dimensions
and
and that
. Then looking at
we observe that the only non-zero contributions occur when
and when
is between
and
, in other words
is only non-zero for
between
and
as required.
$ as required.
C be a chain complex in
, applying
we get the double complex
:
This can be converted into a chain complex in the usual way:
This defines what of an object is, so now we specify its behaviour on morphisms:
where denotes the component of from . The claim is that this defines a contravariant functor.
Since
is a chain map we have
so by the contravariance of the functor
we observe that
Also, because
sends morphisms to morphisms,
is a chain map, thus
These formulae prove precisely that
is a chain map by the commutativity of the following diagram:
Thus sends objects to objects and morphisms to morphisms. Functoriality of follows from that of :
(ii) Now suppose that the chain complex
is concentrated between dimensions
and
and that
. Then looking at
we observe that the only non-zero contributions occur when
and when
is between
and
, in other words
is only non-zero for
between
and
as required.