Let $X$ be the zero locus of $e_1, \dots, e_n$ sections of a vector bundle $\mathcal{E}$ of rank $r$ on $Y$. Assume that the codimension of $X$ is strictly less than $n$, then the Koszul complex associated to the sections is not exact. What can be said about the cohomologies of this complex in general? Are there nice cases in which, even though the complex is not exact, the cohomologies can be known?
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asked Feb 26, 2019 at 17:42
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Assume $X$ is a locally complete intersection of codimension $m < n$. Then the natural morphism $$ E^\vee|_X \to I_X \otimes O_X = I_X/I_X^2 = N^\vee_{X/Y} $$ is surjective, let $F$ be its kernel (it is a vector bundle on $X$ of rank $n - m$). Then $$ H_i(Kosz(E,e)) \cong \wedge^iF. $$
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$\begingroup$ I might be misunderstanding something, but shouldn't the map you wrote be zero? It is defined in terms of the sections $e_1, \dots, e_n$ and by the definition they vanish on $X$ and you're restricting the map to $X$. $\endgroup$ Commented Feb 27, 2019 at 18:33
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$\begingroup$ No, it is not zero. The resulting map to the conormal bundle is given by the differentials of the $e_i$. $\endgroup$– SashaCommented Feb 27, 2019 at 18:40
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$\begingroup$ I see, thank you. Another question: are you considering $\Lambda^i F$ as a sheaf on $Y$ via the pushforward? The point is that the cohomology sheaves will be annihilated by the action of $I_X$ and therefore it's enough to recover them as modules on $Y$, is it? $\endgroup$ Commented Feb 27, 2019 at 18:57
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$\begingroup$ Yes, you are right, $\wedge^iF$ is considered as a sheaf on $Y$ via pushforward. $\endgroup$– SashaCommented Feb 27, 2019 at 19:04