Base change morphisms for higher pushforwards
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Consider a Cartesian diagram consisting of morphisms of schemes. StackExchange doesn't support TikzCD, but the following hopefully suffices.
$$begin{array}{ll}
X' & stackrel{psi'}{rightarrow} & X \
downarrow{f'} & & downarrow{f} \
S' & stackrel{psi}{rightarrow} & S \
end{array}$$
Let $mathcal{F}$ be a quasicoherent sheaf on $X$. I am looking for a construction of the morphism $psi^*(R^i f_* mathcal{F}) to R^i f'_* (psi')^* mathcal{F}$ of sheaves on $S'$. This is essentially the content of Exercise 18.8.B(a) of Vakil's FoAG. Despite being condescendingly called 'easy' I am entirely unable to find the construction.
What I know. One way to describe the map is by using the Grothendieck spectral sequence together with abstract pullback-pushforward adjunctions, and in fact, from this point of view one does not even need to assume that the diagram is Cartesian. But for the purposes of doing calculations, this is not of much help, and probably it's not the intended approach.
algebraic-geometry homology-cohomology sheaf-cohomology derived-functors
edited Nov 14 at 20:58
KReiser
9,02711233
asked Nov 14 at 10:38
guest
833
add a comment |
up vote
0
down vote
favorite
Consider a Cartesian diagram consisting of morphisms of schemes. StackExchange doesn't support TikzCD, but the following hopefully suffices.
$$begin{array}{ll}
X' & stackrel{psi'}{rightarrow} & X \
downarrow{f'} & & downarrow{f} \
S' & stackrel{psi}{rightarrow} & S \
end{array}$$
Let $mathcal{F}$ be a quasicoherent sheaf on $X$. I am looking for a construction of the morphism $psi^*(R^i f_* mathcal{F}) to R^i f'_* (psi')^* mathcal{F}$ of sheaves on $S'$. This is essentially the content of Exercise 18.8.B(a) of Vakil's FoAG. Despite being condescendingly called 'easy' I am entirely unable to find the construction.
What I know. One way to describe the map is by using the Grothendieck spectral sequence together with abstract pullback-pushforward adjunctions, and in fact, from this point of view one does not even need to assume that the diagram is Cartesian. But for the purposes of doing calculations, this is not of much help, and probably it's not the intended approach.
algebraic-geometry homology-cohomology sheaf-cohomology derived-functors
edited Nov 14 at 20:58
KReiser
9,02711233
asked Nov 14 at 10:38
guest
833
Maybe this could be of some help virtualmath1.stanford.edu/~conrad/248BPage/handouts/…
– random123
Nov 14 at 16:14
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
Consider a Cartesian diagram consisting of morphisms of schemes. StackExchange doesn't support TikzCD, but the following hopefully suffices.
$$begin{array}{ll}
X' & stackrel{psi'}{rightarrow} & X \
downarrow{f'} & & downarrow{f} \
S' & stackrel{psi}{rightarrow} & S \
end{array}$$
Let $mathcal{F}$ be a quasicoherent sheaf on $X$. I am looking for a construction of the morphism $psi^*(R^i f_* mathcal{F}) to R^i f'_* (psi')^* mathcal{F}$ of sheaves on $S'$. This is essentially the content of Exercise 18.8.B(a) of Vakil's FoAG. Despite being condescendingly called 'easy' I am entirely unable to find the construction.
What I know. One way to describe the map is by using the Grothendieck spectral sequence together with abstract pullback-pushforward adjunctions, and in fact, from this point of view one does not even need to assume that the diagram is Cartesian. But for the purposes of doing calculations, this is not of much help, and probably it's not the intended approach.
algebraic-geometry homology-cohomology sheaf-cohomology derived-functors
edited Nov 14 at 20:58
KReiser
9,02711233
asked Nov 14 at 10:38
guest
833
Consider a Cartesian diagram consisting of morphisms of schemes. StackExchange doesn't support TikzCD, but the following hopefully suffices.
$$begin{array}{ll}
X' & stackrel{psi'}{rightarrow} & X \
downarrow{f'} & & downarrow{f} \
S' & stackrel{psi}{rightarrow} & S \
end{array}$$
Let $mathcal{F}$ be a quasicoherent sheaf on $X$. I am looking for a construction of the morphism $psi^*(R^i f_* mathcal{F}) to R^i f'_* (psi')^* mathcal{F}$ of sheaves on $S'$. This is essentially the content of Exercise 18.8.B(a) of Vakil's FoAG. Despite being condescendingly called 'easy' I am entirely unable to find the construction.
What I know. One way to describe the map is by using the Grothendieck spectral sequence together with abstract pullback-pushforward adjunctions, and in fact, from this point of view one does not even need to assume that the diagram is Cartesian. But for the purposes of doing calculations, this is not of much help, and probably it's not the intended approach.
algebraic-geometry homology-cohomology sheaf-cohomology derived-functors
algebraic-geometry homology-cohomology sheaf-cohomology derived-functors
edited Nov 14 at 20:58
KReiser
9,02711233
asked Nov 14 at 10:38
guest
833
edited Nov 14 at 20:58
KReiser
9,02711233
asked Nov 14 at 10:38
guest
833
edited Nov 14 at 20:58
KReiser
9,02711233
edited Nov 14 at 20:58
KReiser
9,02711233
edited Nov 14 at 20:58
KReiser
9,02711233
9,02711233
asked Nov 14 at 10:38
guest
833
asked Nov 14 at 10:38
guest
833
asked Nov 14 at 10:38
guest
833
833
Maybe this could be of some help virtualmath1.stanford.edu/~conrad/248BPage/handouts/…
– random123
Nov 14 at 16:14
add a comment |
Maybe this could be of some help virtualmath1.stanford.edu/~conrad/248BPage/handouts/…
– random123
Nov 14 at 16:14
Maybe this could be of some help virtualmath1.stanford.edu/~conrad/248BPage/handouts/…
– random123
Nov 14 at 16:14
Maybe this could be of some help virtualmath1.stanford.edu/~conrad/248BPage/handouts/…
– random123
Nov 14 at 16:14
add a comment |
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Maybe this could be of some help virtualmath1.stanford.edu/~conrad/248BPage/handouts/…
– random123
Nov 14 at 16:14