Koszul cohomology and support of local cohomology modules of complete intersections

Abstract

Let $R$ be a noetherian commutative ring and $f_1,\dots ,f_c$ be a regular sequence in $R$. We introduce a framework to study $Supp(H_I^j(R/(f_1,\dots ,f_c)))$ by linking the Koszul cohomology of $H_I^j(R)$ on the sequence $f_1,\dots ,f_c$ and local cohomology modules $H_I^j(R/(f_1,\dots ,f_c))$. As an application, we prove that if $R$ is a noetherian regular ring of prime characteristic $p$ and $f_1,f_2$ form a regular sequence in $R$ then $Supp(H_I^j(R/(f_1,f_2)))$ is Zariski-closed for each integer $j$ and each ideal $I$.

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