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Tony Scholl
Tony Scholl
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I believe the question is: "when are the complexes $A^\bullet$ and $\bigoplus H^i(A^\bullet)[-i]$ (with trivial differential) isomorphic in the derived category"? - which as you say is not very profound for an acyclic complex...

A complex with only two nonzero cohomology groups, $H^0$ and $H^n$ say, can be rewritten after truncation as a Yoneda $(n+1)$-extension of $H^n$ by $H^0$ $$ 0 \to H^0 \to A^0 \to \dots A^n \to H^n \to 0 $$ so the obstruction in this case is just an element of $Ext^{n+1}(H^n,H^0)$. For a more general complex one can proceed inductively: first split the 2-extensions $H^{i-1} \to * \to * \to H^i$, then the 3-extensions... but that's not very pleasant. Cohomological dimension may simplify the considerations.

A useful starting point is Deligne's thesis: "Theoreme de Lefschetz et criteres de degenerescence..." (Publ Math IHES 35 (1968) 107-126) where he proves some important splitting results of this kind in a geometric setting, and a later sequel article by Deligne "Decomposition dans la categorie derivee" (in volume 1 of the Motives proceedings, Proc Symp AMS 55). There is also a study of some splittings in de Rham cohomology in the Deligne-Illusie paper on the degerenerationdegeneration of the Hodge-de Rham spectral sequence and Frobenius (Inventiones vol.89). I am afraid these may be too geometric for what you have in mind, though.

I believe the question is: "when are the complexes $A^\bullet$ and $\bigoplus H^i(A^\bullet)[-i]$ (with trivial differential) isomorphic in the derived category"? - which as you say is not very profound for an acyclic complex...

A complex with only two nonzero cohomology groups, $H^0$ and $H^n$ say, can be rewritten after truncation as a Yoneda $(n+1)$-extension of $H^n$ by $H^0$ $$ 0 \to H^0 \to A^0 \to \dots A^n \to H^n \to 0 $$ so the obstruction in this case is just an element of $Ext^{n+1}(H^n,H^0)$. For a more general complex one can proceed inductively: first split the 2-extensions $H^{i-1} \to * \to * \to H^i$, then the 3-extensions... but that's not very pleasant. Cohomological dimension may simplify the considerations.

A useful starting point is Deligne's thesis: "Theoreme de Lefschetz et criteres de degenerescence..." (Publ Math IHES 35 (1968) 107-126) where he proves some important splitting results of this kind in a geometric setting, and a later sequel article by Deligne "Decomposition dans la categorie derivee" (in volume 1 of the Motives proceedings, Proc Symp AMS 55). There is also a study of some splittings in de Rham cohomology in the Deligne-Illusie paper on the degereneration of the Hodge-de Rham spectral sequence and Frobenius (Inventiones vol.89). I am afraid these may be too geometric for what you have in mind, though.

I believe the question is: "when are the complexes $A^\bullet$ and $\bigoplus H^i(A^\bullet)[-i]$ (with trivial differential) isomorphic in the derived category"? - which as you say is not very profound for an acyclic complex...

A complex with only two nonzero cohomology groups, $H^0$ and $H^n$ say, can be rewritten after truncation as a Yoneda $(n+1)$-extension of $H^n$ by $H^0$ $$ 0 \to H^0 \to A^0 \to \dots A^n \to H^n \to 0 $$ so the obstruction in this case is just an element of $Ext^{n+1}(H^n,H^0)$. For a more general complex one can proceed inductively: first split the 2-extensions $H^{i-1} \to * \to * \to H^i$, then the 3-extensions... but that's not very pleasant. Cohomological dimension may simplify the considerations.

A useful starting point is Deligne's thesis: "Theoreme de Lefschetz et criteres de degenerescence..." (Publ Math IHES 35 (1968) 107-126) where he proves some important splitting results of this kind in a geometric setting, and a later sequel article by Deligne "Decomposition dans la categorie derivee" (in volume 1 of the Motives proceedings, Proc Symp AMS 55). There is also a study of some splittings in de Rham cohomology in the Deligne-Illusie paper on the degeneration of the Hodge-de Rham spectral sequence and Frobenius (Inventiones vol.89). I am afraid these may be too geometric for what you have in mind, though.

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Tony Scholl
Tony Scholl
  • 2.4k
  • 18
  • 15

I believe the question is: "when are the complexes $A^\bullet$ and $\bigoplus H^i(A^\bullet)[-i]$ (with trivial differential) isomorphic in the derived category"? - which as you say is not very profound for an acyclic complex...

A complex with only two nonzero cohomology groups, $H^0$ and $H^n$ say, can be rewritten after truncation as a Yoneda $(n+1)$-extension of $H^n$ by $H^0$ $$ 0 \to H^0 \to A^0 \to \dots A^n \to H^n \to 0 $$ so the obstruction in this case is just an element of $Ext^{n+1}(H^n,H^0)$. For a more general complex one can proceed inductively: first split the 2-extensions $H^{i-1} \to * \to * \to H^i$, then the 3-extensions... but that's not very pleasant. Cohomological dimension may simplify the considerations.

A useful starting point is Deligne's thesis: "Theoreme de Lefschetz et criteres de degenerescence..." (Publ Math IHES 35 (1968) 107-126) where he proves some important splitting results of this kind in a geometric setting, and a later sequel article by Deligne "Decomposition dans la categorie derivee" (in volume 1 of the Motives proceedings, Proc Symp AMS 55). There is also a study of some splittings in de Rham cohomology in the Deligne-Illusie paper on the degereneration of the Hodge-de Rham spectral sequence and Frobenius (Inventiones vol.89). I am afraid these may be too geometric for what you have in mind, though.