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Consider a recollement situation like the following

enter image description here

by the very definition of the various functors it follows that $i^* j_*=0$, and $j^! i_* = 0 = j^* i_!$. Also, $j^! i_! = 0 = j^* i_*$ by inspection.

Are these "kernel properties" true in a general recollement situation? More precisely, let $$ \mathbf{D}^0 \underset{\underset{i_R}\leftarrow}{\overset{\overset{i_L}\leftarrow}\to} \mathbf{D} \underset{\underset{q_R}\leftarrow}{\overset{\overset{q_L}\leftarrow}\to} \mathbf{D}^1 $$ be a recollement where $i_L\dashv i\dashv i_R$ and $q_L\dashv q\dashv q_R$. From the axioms of recollement it follows that $qi=0$ implies $i_L q_L = 0 =i_R q_R$.

Is it true that also $i_R q_L = 0 = i_L q_R$?

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  • $\begingroup$ where is that image from? $\endgroup$ Mar 6, 2015 at 18:59
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    $\begingroup$ Banagl, Markus. Topological invariants of stratified spaces. Springer Science & Business Media, 2007. I should have mentioned $\endgroup$
    – fosco
    Mar 6, 2015 at 19:03
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    $\begingroup$ Maybe I am getting confused here, but it is not true that $j^\ast i_\ast = 0$ (where $i$ is the open embedding and $j$ is the closed complement - opposite to my usual convention!). For example if you take the direct image of the constant sheaf under the open embedding $i:\mathbb R - \{0\} \hookrightarrow \mathbb R$, then the stalk $j^\ast i_\ast \mathbb Z_{\mathbb R - \{0\}}$ is 2-dimensional. Similarly, $j^!i_!$ is not zero in general. $\endgroup$ Mar 6, 2015 at 22:52
  • $\begingroup$ Note that if you have a recollement with the property that $i_R q_L = 0 = i_L q_R$, then the category $\mathbb D$ splits as an orthogonal sum of $\mathbb D^0$ and $\mathbb D^1$; i.e. there are no Homs in either direction. This is certainly not the case for sheaves on a locally closed decomposition of a space... $\endgroup$ Mar 6, 2015 at 23:04
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    $\begingroup$ I agree with Sam. $\endgroup$ Mar 6, 2015 at 23:36

1 Answer 1

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I'll write my comment again here, so it appears as an answer.

If you have a recollement as in the question with the property that $i_Rq_L=0=i_Lq_R$, then the category $D$ splits as an orthogonal sum of $D^0$ and $D^1$; i.e. every object $d$ of $D$ can be written as a direct sum $d_0 \oplus d_1$ with $d_i \in D^i$, and there are no nonzero morphisms between $D^0$ and $D^1$ either direction. You can see this by looking at one of the distinguished triangles associated to the recollement and observing that the connecting morphism must be zero, and thus the triangle is split.

In particular, it is not true that $j^\ast i_\ast = 0$ in the derived category of sheaves on a space (where $i: U \hookrightarrow X$ is an open embedding and $j: F \hookrightarrow X$ the closed complement). In fact, the (hyper)cohomology of the complex $j^\ast i_\ast \mathbb Z_U$ computes the cohomology of the link of $F$ inside $X$ - an important invariant of the stratification. This is never zero, unless $U$ and $F$ are disjoint components.

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    $\begingroup$ I agree with your counterexample but I want to think about it calmly, since this has a series of implications that I want to meditate on. For the time being... Thanks a lot! $\endgroup$
    – fosco
    Mar 8, 2015 at 0:43
  • $\begingroup$ I love how in the comment above you actually took the time to write \mathbb{D}, while in the actual answer you just could not be bothered. $\endgroup$ Mar 8, 2015 at 3:16
  • $\begingroup$ @Sam Gunningham "In fact, the (hyper)cohomology of the complex $j^∗i_∗Z_U$ computes the cohomology of the link of $F$ inside $X$" I'm really curious what this means (not familiar with knots or cohomology of links). Could you elaborate a little bit if you have time? $\endgroup$
    – ಠ_ಠ
    Apr 25, 2017 at 9:22
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    $\begingroup$ @ಠ_ಠ Check out Dimca, "Sheaves in Topology", Example 2.3.18 $\endgroup$ Apr 25, 2017 at 17:00

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