Spectral sequence for ExtIf I have a morphism of schemes f : X → Y and sheaves F,G on X, then is there a spectral sequence which relates the Ext-groupsExt(f∗F,f∗G) on Y and Ext(F,G) on X?I should add, that if this is not possible in general, that my morphism f is actually an affine morphism and that I want to compare the two Ext-groups in any way. The first thing I thought of was a spectral sequence, but perhaps there are other ways you know.

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Spectral sequence for ExtIf I have a morphism of schemes   f  :  X  →  Y and sheaves F,G on X, then is there a spectral sequence which relates the Ext-groupsExt(f∗F,f∗G) on Y and Ext(F,G) on X?I should add, that if this is not possible in general, that my morphism f is actually an affine morphism and that I want to compare the two Ext-groups in any way. The first thing I thought of was a spectral sequence, but perhaps there are other ways you know.

Savanah Morton · Accepted Answer

Step 1Lemma Suppose that   f  :  X  →  Y is flat, then the push forward of an injective sheaf I is injective.Indeed, we need to show that   H  o      m    Y    (  −  ,      f    ∗    I  ) is exact. Since f is flat, the pull-back functor f∗ is exact, so the exactness of   H  o      m    Y    (  −  ,      f    ∗    I  ) follows from the adjointness of f∗ and f∗.Corollay Suppose that   f  :  X  →  Y is flat, then there is an spectral sequence  E  x      t    Y    p    (      f    ∗        F    ,      R    q        f    ∗        G    )  ⇒  E  x      t    X          p      +      q        (      f    ∗        f    ∗        F    ,      G    )  .Here we consider the two functors f∗ and   H  o      m    Y    (      f    ∗        F    ,  −  ). Their composition is the functor      G    ↦  H  o      m    Y    (      f    ∗        F    ,      f    ∗        G    )  =  H  o      m    X    (      f    ∗        f    ∗        F    ,      G    )  .So the Corollary is a direct consequence of Grothendieck spectral sequence.The edge morphism gives rise to a map   E  x      t    Y    p    (      f    ∗        F    ,      f    ∗        G    )  →  E  x      t    X    p    (      f    ∗        f    ∗        F    ,      G    ). On the other hand, we have the adjunction map       f    ∗        f    ∗        F    →      F  . So there are maps  E  x      t    Y    p    (      f    ∗        F    ,      f    ∗        G    )  →  E  x      t    X    p    (      f    ∗        f    ∗        F    ,      G    )  ←  E  x      t    X    p    (      F    ,      G    )  .I wonder if one can relate the two &quot;external&quot; terms in the above formula in general.However, if f is further assumed to be affine, then f∗ is exact, so the spectral sequence degenerates, and we get   E  x      t    Y    p    (      f    ∗        F    ,      f    ∗        G    )  =  E  x      t    X    p    (      f    ∗        f    ∗        F    ,      G    ), and therefore a map   E  x      t    X    p    (      F    ,      G    )  →  E  x      t    Y    p    (      f    ∗        F    ,      f    ∗        G    ). Again, since f∗ is exact, the existence of this map is obvious from Yoneda&#039;s description of elements of   E  x      t    n   as extensions of length n (It seems flatness is not needed here?).Anyway, say X is an affine algebra over a field k, and let   Y  =  S  p  e  c  (  k  ). Then   E  x      t    Y    n    (      f    ∗        F    ,      f    ∗        G    )  =  0 for all   n  &amp;gt;  1 since all k-vector spaces are injective k-modules. There are certainly &quot;lose of information&quot; by going to the Ext of the push forward.

If I have a morphism of schemes f:X rightarrow Y and sheaves F, G on X, then is there a spectral sequence which relates the Ext-groups

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