Let $ R $ be a commutative unital ring, $ I $ an ideal of $ R $, and $ M $ an $ R $-module. The cohomological dimension of $ M $ with respect to $ I $ is defined as $$ \operatorname{cd}(I,M) \stackrel{\text{def}}{=} \sup(\{ i \in \mathbb{N} \mid {H_{I}^{i}}(M) \neq 0 \}). $$ […]

I read that Grothendieck developed Local Cohomology to answer a question of Pierre Samuel about when certain type of rings are UFDs. I know the basics of local cohomology but I have not seen a theorem which shows the connection between UFDs and Local Cohomology. My Question: Could someone tell me about a result which […]

I was trying to prove this theorem (problem): Suppose that $R$ is a commutative ring with identity, $I\unlhd R$, and $M$ an $R$-module. We define: $$\Gamma_I(M)=\bigcup_{n\geq0}\operatorname{Ann}_M(I^n)$$ in which for each natural $n\geq 0$: $$\operatorname{Ann}_M(I^n)=\{x\in M\;;\;I^nx=0\}.$$ Prove $$\Gamma_I\left(\frac{M}{\Gamma_I(M)}\right)=0.$$ Note that $\Gamma_I(\cdot)$ will be defined for any $R$-module naturally. I can show $\Gamma_I(M)\leq M$, but couldn’t prove […]

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