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In Broadmann and Sharp's book, Local Cohomology: An Algebraic Introduction with Geometric Applications, the exercise $3.2.4$ is about an exact sequence of the form $\DeclareMathOperator{\Hom}{Hom}$

$$0\rightarrow \Hom_R(a+b,M)\rightarrow \Hom_R(a,M)\oplus \Hom_R(b,M)\rightarrow \Hom_R(a\cap b,M)$$

where $R$ is a commutative unitary ring, $a$ and $b$ ideals of $R$ and $M$ an $R$-module (all $R$-homomorphisms are restrictions, up to signs).

I would like to know if one may extends this result for more than two ideals. Precisely, if $a_1,...,a_n$ are ideals of $R$ and $M$ an $R$-module then there exists exact sequence

$$0\rightarrow \Hom_R(a_1+...+a_n,M)\rightarrow\bigoplus_i \Hom_R(a_i,M)\rightarrow\bigoplus_{i,j}\Hom_R\left(a_i\cap a_j,M\right).$$

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  • $\begingroup$ Try with all $a_i=R$ and see what goes wrong if $n>2$. $\endgroup$
    – Mohan
    Commented Mar 15, 2020 at 22:40
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    $\begingroup$ Exactness in the case of 2 ideals means that a pair of homomorphisms $a\to M$ and $b\to M$ can be combined into $a+b\to M$ iff they agree on $a\cap b$. Your attempted analog for bigger $n$ would try to combine $n$ homomorphisms if they "agree" on the intersection of all $n$ of the $a_i$'s. But for the combination to make sense, you need agreement of each pair on the intersection of the common domain $a_i\cap a_j$, which is a much stronger requirement than agreement on $\bigcap_{i=1}^na_i$. $\endgroup$
    – Andreas Blass
    Commented Mar 15, 2020 at 22:41
  • $\begingroup$ In case you find it confusing that @Mohan suggests a scenario where the pairwise intersections and the intersection of all the ideals are the same, so the problem in my comment doesn't arise, the point here is that there's another problem, namely how to express "agreement" by just changing a sign of one of the restriction maps. That works for 2, but not for larger $n$. $\endgroup$
    – Andreas Blass
    Commented Mar 15, 2020 at 22:45
  • $\begingroup$ Related question: mathoverflow.net/questions/21782/… $\endgroup$
    – Filippo Alberto Edoardo
    Commented Mar 16, 2020 at 10:10
  • $\begingroup$ @AndreasBlass you're right. The question should have $\oplus_{i,j} Hom_R(a_i\cap a_j,M) $ instead of $Hom_R\left(\bigcap_i a_i,M\right)$ and last map being restrictions with suitable signs. In this case do we have an affirmative answer? $\endgroup$
    – Rafael
    Commented Mar 16, 2020 at 18:24

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