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I am in the following situation, I have two schemes $X$, $Y$ and two closed immersions $Z \rightarrow Y$, $Z \rightarrow X$. Everything is smooth. I am interested in calculating morphisms in the derived category of $X \cup_{Z} Y$ between two complexes, one of which is in the image of the pushforward via one of the two closed immersions $X \rightarrow X \cup_Z Y$, $Y \rightarrow X \cup_Z Y$. I know that there is a short exact sequence $$ 0 \rightarrow \mathcal{O}_{X \cup_Z Y} \rightarrow \mathcal{O}_{X} \oplus \mathcal{O}_{Y} \rightarrow \mathcal{O}_{Z} \rightarrow 0 $$ but this doesn't help very much. It would be very helpful if $X$ and $Y$ were cut in $X \cup_Z Y$ by some vector bundles, but I don't think that's possible as $X$ and $Y$ are smooth and $X \cup_Z Y$ might be not. Is there any general technique one tries to apply in such situation? Any help would be very appriaceted.

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    $\begingroup$ Lurie's SAG, section 16.2 will help you a lot. $\endgroup$
    – crystalline
    Commented Aug 15, 2019 at 13:50
  • $\begingroup$ @Federico : it would somehow be easier to answer to your question if you tell us what are the two objects in $D^b(X \cup_{Z} Y)$ you are considering. $\endgroup$
    – Libli
    Commented Aug 15, 2019 at 14:29
  • $\begingroup$ @Libli for example, I'm interested in the endomorphism algebra in the derived category of $\mathcal{O}_{X}$ and the same for $Y$. Also, I'm interested in the derived dual of $\mathcal{O}_{X}$ and $Y$. $\endgroup$
    – Federico Barbacovi
    Commented Aug 15, 2019 at 16:13
  • $\begingroup$ @crystalline Thank you for the advice. However, I had a look at SAG and I don't see how it helps with respect to the calculations I am trying to do (the ones in the previous comment). $\endgroup$
    – Federico Barbacovi
    Commented Aug 15, 2019 at 16:13
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    $\begingroup$ @Federico : There is a notion of Diagram of schemes (which is basically the data of the two arrows from $X$ and $Y$ to $X \cup_{Z} Y$) and Lipman-Hashimoto wrote a book on Grothendieck duality for such diagrams. $\endgroup$
    – Libli
    Commented Aug 16, 2019 at 9:47

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