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For $\mathcal{F}$ a sheaf on $U,$ for $U$ an open subset of a space $X,$ is it true that $H^q(X, j_!\mathcal{F}) = H^q_c(U, \mathcal{F})$ for $j : U\rightarrow X$ the open inclusion?

  • How do you define $H^q_c$? The definition I know simply defines it as the cohomology of lower shriek along an inclusion as an open subspace of a compact space (or an open subscheme of a complete variety), in which case your statement follows trivially. – Sergey Guminov Jul 28 '22 at 18:28
  • @SergeyGuminov The definition I saw was as the right derived functors of $p_!$ for $p : U\rightarrow {\text{pt}}$. Do you have a reference for your source? I imagine it would contain exactly what I want! –  Jul 28 '22 at 21:36
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    You are right, what I wrote was a trick to define lower shriek for morphisms other than open immersions. I think page 58 here should be relevant: https://www.google.com/url?sa=t&source=web&rct=j&url=https://web.math.princeton.edu/~takumim/MATH731.pdf&ved=2ahUKEwjPyvzdq535AhWRAxAIHT4DCmsQFnoECBAQAQ&usg=AOvVaw0XIHSkEH46aYgdc4s6rQDq – Sergey Guminov Jul 29 '22 at 05:17

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