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added a description of the Grothendieck spectral sequence
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algori
algori
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The purpose of this question is to find out whether one can view the Atiyah-Hirzebruch spectral sequence as a particular case of the "composition of derived functors" spectral sequence.

The Leray spectral sequence of a continuous map $f:X\to Y$ of topological spaces can be constructed as follows. Let $a_X:X\to pt$ and $a_Y:Y\to pt$ be the maps from $X$ and $Y$ respectively to the one point space $pt$; we obviously have $a_X=a_Y\circ f$. So for any sheaf $L$ on $X$ we have $(a_X)_\ast L=(a_Y)_*f_\ast L$. But $(a_X)_\ast$ is just the functor of the global sections and so is $(a_Y)_*$. Deriving thisRecall that if $A,B$ and taking the Grothendieck$C$ are abelian categories and $F:A\to B,G:B\to C$ are left exact functors, then (under mild hypotheses) $R_\ast(G\circ F)$ is isomorphic to $R_\ast\circ G_\ast$ and for any object $X$ of $A$ there exists a spectral sequence abutting $R^\ast (G\circ F) X$ with the $E_2$ sheet given by $E_2^{pq}=R^pG(R^q F(X))$. See e.g Gelfand-Manin, Methods of homological algebra, 3.7.

Applying this to the case when $A$, $B$ and $C$ are the categories of sheaves of $X$, $Y$ and the point respectively we get a spectral sequence $(E^{pq}_r,d_d)$ abutting to $H^*(X,F)$ with the $E_2$ term given by $$E_2^{pq}=H^p(Y,R^q f_\ast L).$$

If $X$ and $Y$ are sufficiently nice (say finite CW complexes), $F$ is constant and $f$ is a locally trivial fibration with fiber $F$, then we get (assuming for simplicity that $Y$ is simply-connected) $$E_2^{pq}=H^q(Y,H^q(F)).$$

Now, if we have an extraordinary cohomology theory $h^\ast$, we can construct (under the hypotheses of the previous paragraph) the Atiyah-Hirzebruch spectral sequence: the $E_2$ sheet is given by $$E_2^{pq}=H^q(Y,h^q(F))$$ and the spectral sequence abuts to $h^*(X)$. This looks pretty similar to the Leray spectral sequence, so it seems natural to ask whether it can be obtained in a way similar to the one described above.

Namely, given an extraordinary cohomology theory $h^\ast$ and a continuous map $f:X\to Y$ of topological spaces, is there an "extraordinary direct image" functor $f^{ex}_*$ from sheaves on $X$ to sheaves on $Y$ which would be "functorial in $f$" and which would give $h^{\ast}(X)$ after deriving when $Y$ is a point?

If not, is there still a way to view the Atiyah-Hirzebruch spectral sequence as (a version of) the spectral sequence of the composition of two derived functors? (It may happen that one has to consider something other than the categories of sheaves, but I have no idea what this could be.)

The purpose of this question is to find out whether one can view the Atiyah-Hirzebruch spectral sequence as a particular case of the "composition of derived functors" spectral sequence.

The Leray spectral sequence of a continuous map $f:X\to Y$ of topological spaces can be constructed as follows. Let $a_X:X\to pt$ and $a_Y:Y\to pt$ be the maps from $X$ and $Y$ respectively to the one point space $pt$; we obviously have $a_X=a_Y\circ f$. So for any sheaf $L$ on $X$ we have $(a_X)_\ast L=(a_Y)_*f_\ast L$. But $(a_X)_\ast$ is just the functor of the global sections and so is $(a_Y)_*$. Deriving this and taking the Grothendieck spectral sequence we get a spectral sequence $(E^{pq}_r,d_d)$ abutting to $H^*(X,F)$ with the $E_2$ term given by $$E_2^{pq}=H^p(Y,R^q f_\ast L).$$

If $X$ and $Y$ are sufficiently nice (say finite CW complexes), $F$ is constant and $f$ is a locally trivial fibration with fiber $F$, then we get (assuming for simplicity that $Y$ is simply-connected) $$E_2^{pq}=H^q(Y,H^q(F)).$$

Now, if we have an extraordinary cohomology theory $h^\ast$, we can construct (under the hypotheses of the previous paragraph) the Atiyah-Hirzebruch spectral sequence: the $E_2$ sheet is given by $$E_2^{pq}=H^q(Y,h^q(F))$$ and the spectral sequence abuts to $h^*(X)$. This looks pretty similar to the Leray spectral sequence, so it seems natural to ask whether it can be obtained in a way similar to the one described above.

Namely, given an extraordinary cohomology theory $h^\ast$ and a continuous map $f:X\to Y$ of topological spaces, is there an "extraordinary direct image" functor $f^{ex}_*$ from sheaves on $X$ to sheaves on $Y$ which would be "functorial in $f$" and which would give $h^{\ast}(X)$ after deriving when $Y$ is a point?

If not, is there still a way to view the Atiyah-Hirzebruch spectral sequence as (a version of) the spectral sequence of the composition of two derived functors? (It may happen that one has to consider something other than the categories of sheaves, but I have no idea what this could be.)

The purpose of this question is to find out whether one can view the Atiyah-Hirzebruch spectral sequence as a particular case of the "composition of derived functors" spectral sequence.

The Leray spectral sequence of a continuous map $f:X\to Y$ of topological spaces can be constructed as follows. Let $a_X:X\to pt$ and $a_Y:Y\to pt$ be the maps from $X$ and $Y$ respectively to the one point space $pt$; we obviously have $a_X=a_Y\circ f$. So for any sheaf $L$ on $X$ we have $(a_X)_\ast L=(a_Y)_*f_\ast L$. But $(a_X)_\ast$ is just the functor of the global sections and so is $(a_Y)_*$. Recall that if $A,B$ and $C$ are abelian categories and $F:A\to B,G:B\to C$ are left exact functors, then (under mild hypotheses) $R_\ast(G\circ F)$ is isomorphic to $R_\ast\circ G_\ast$ and for any object $X$ of $A$ there exists a spectral sequence abutting $R^\ast (G\circ F) X$ with the $E_2$ sheet given by $E_2^{pq}=R^pG(R^q F(X))$. See e.g Gelfand-Manin, Methods of homological algebra, 3.7.

Applying this to the case when $A$, $B$ and $C$ are the categories of sheaves of $X$, $Y$ and the point respectively we get a spectral sequence $(E^{pq}_r,d_d)$ abutting to $H^*(X,F)$ with the $E_2$ term given by $$E_2^{pq}=H^p(Y,R^q f_\ast L).$$

If $X$ and $Y$ are sufficiently nice (say finite CW complexes), $F$ is constant and $f$ is a locally trivial fibration with fiber $F$, then we get (assuming for simplicity that $Y$ is simply-connected) $$E_2^{pq}=H^q(Y,H^q(F)).$$

Now, if we have an extraordinary cohomology theory $h^\ast$, we can construct (under the hypotheses of the previous paragraph) the Atiyah-Hirzebruch spectral sequence: the $E_2$ sheet is given by $$E_2^{pq}=H^q(Y,h^q(F))$$ and the spectral sequence abuts to $h^*(X)$. This looks pretty similar to the Leray spectral sequence, so it seems natural to ask whether it can be obtained in a way similar to the one described above.

Namely, given an extraordinary cohomology theory $h^\ast$ and a continuous map $f:X\to Y$ of topological spaces, is there an "extraordinary direct image" functor $f^{ex}_*$ from sheaves on $X$ to sheaves on $Y$ which would be "functorial in $f$" and which would give $h^{\ast}(X)$ after deriving when $Y$ is a point?

If not, is there still a way to view the Atiyah-Hirzebruch spectral sequence as (a version of) the spectral sequence of the composition of two derived functors? (It may happen that one has to consider something other than the categories of sheaves, but I have no idea what this could be.)

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algori
algori
  • 23.5k
  • 3
  • 100
  • 152

Extraordinary cohomology as a derived functor?

The purpose of this question is to find out whether one can view the Atiyah-Hirzebruch spectral sequence as a particular case of the "composition of derived functors" spectral sequence.

The Leray spectral sequence of a continuous map $f:X\to Y$ of topological spaces can be constructed as follows. Let $a_X:X\to pt$ and $a_Y:Y\to pt$ be the maps from $X$ and $Y$ respectively to the one point space $pt$; we obviously have $a_X=a_Y\circ f$. So for any sheaf $L$ on $X$ we have $(a_X)_\ast L=(a_Y)_*f_\ast L$. But $(a_X)_\ast$ is just the functor of the global sections and so is $(a_Y)_*$. Deriving this and taking the Grothendieck spectral sequence we get a spectral sequence $(E^{pq}_r,d_d)$ abutting to $H^*(X,F)$ with the $E_2$ term given by $$E_2^{pq}=H^p(Y,R^q f_\ast L).$$

If $X$ and $Y$ are sufficiently nice (say finite CW complexes), $F$ is constant and $f$ is a locally trivial fibration with fiber $F$, then we get (assuming for simplicity that $Y$ is simply-connected) $$E_2^{pq}=H^q(Y,H^q(F)).$$

Now, if we have an extraordinary cohomology theory $h^\ast$, we can construct (under the hypotheses of the previous paragraph) the Atiyah-Hirzebruch spectral sequence: the $E_2$ sheet is given by $$E_2^{pq}=H^q(Y,h^q(F))$$ and the spectral sequence abuts to $h^*(X)$. This looks pretty similar to the Leray spectral sequence, so it seems natural to ask whether it can be obtained in a way similar to the one described above.

Namely, given an extraordinary cohomology theory $h^\ast$ and a continuous map $f:X\to Y$ of topological spaces, is there an "extraordinary direct image" functor $f^{ex}_*$ from sheaves on $X$ to sheaves on $Y$ which would be "functorial in $f$" and which would give $h^{\ast}(X)$ after deriving when $Y$ is a point?

If not, is there still a way to view the Atiyah-Hirzebruch spectral sequence as (a version of) the spectral sequence of the composition of two derived functors? (It may happen that one has to consider something other than the categories of sheaves, but I have no idea what this could be.)