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Given a quasi-projective $X$ variety that is the union of two affines $\text{Spec(A)}$, $\text{Spec(B)}$ with intersection $\text{Spec}(C)$. Let $f\in C$. Then $\text{Spec}(C_f)$ is an open in both of $\text{Spec(A)}$ and $\text{Spec(B)}$. What is the gluing of $\text{Spec(A)}$, $\text{Spec(B)}$ along $\text{Spec}(C_f)$ and its relation to $X$? I can only imagine this object very intuitively and I'm not sure if it is correct or not. It seems to me this is not a separated scheme and it resembles $X$ with double the portion of zeros of $f$ that are in $\text{Spec}(C)$. Specially it seems to me there are two maps from $X$ to this scheme. Is it possibile to make these more clear in case they are true?

Edit: Here are some related questions. Let $\text{Spec}(A)^f$ be the complement of zeros of $f$ that are in $\text{Spec}(C)$. Is it possible to make sense of $\text{Spec}(A)^f$ as a scheme? (explicit description would be ideal). In a way that it admits $\text{Spec}(C_f)$ as an open and gluing $\text{Spec}(A)^f$ and $\text{Spec}(B)$ along $\text{Spec}(C_f)$ is same as gluing $\text{Spec}(A)$ and $\text{Spec}(B)$ along $\text{Spec}(C)$.

Edit2: Let's assume for simplicity that $\text{Spec}(C)=\text{Spec}(A_g)$ for some $g$. Let $f'$ be minimal number of times required to multiply $f$ with $g$ so it is in $A$. Then we can look at $\text{Spec}(A)^f$ as gluing $\text{Spec}(A_{f'})$ and $\text{Spec}(A/gA)$ along $\text{Spec}(A_{f'}/(g))$. This makes sense according to Karl Schwede's article. It is an affine scheme. But with this candidate for $\text{Spec}(A)^f$ I'm not sure whether gluing it to $\text{Spec}(B)$ along $\text{Spec}(C_f)$ gives back $X$. Even more simplistically by gluing $\text{Spec}(C)$ to $\text{Spec}(A)^f$ along $\text{Spec}(C_f)$, does it give back $\text{Spec}(A)$?

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    $\begingroup$ Karl Schwede's article (math.stanford.edu/~vakil/files/schwede03.pdf) has probably been for sometimes a canonical reference on matters related to gluing schemes - what you want is a special case of the construction in Section 2 $\endgroup$
    – pinaki
    Commented Dec 7, 2020 at 18:54
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    $\begingroup$ Title of Schwede's article referenced by @auniket: Gluing schemes and a scheme without closed points. $\endgroup$
    – LSpice
    Commented Dec 7, 2020 at 19:37
  • $\begingroup$ Isn't Karl Schwede's article good when gluing along closed subschemes? It might be helpful to give meaning to $\text{Spec}(A)^f$. Let the zeros of $f$ extend in $\text{Spec}(A)$, it intersects the complement of $\text{Spec}(C)$ in a closed subscheme. Then we can glue the complement of the zero set in $\text{Spec}(A)$ and the complement of $\text{Spec}(C)$ to get an affine scheme. This is a candidate for $\text{Spec}(A)^f$. But I'm not sure whether gluing it to $\text{Spec}(B)$ along $\text{Spec}(C_f)$ gives back $X$. $\endgroup$
    – user127776
    Commented Dec 7, 2020 at 20:32
  • $\begingroup$ The general construction given in Section 2 of Schwede's article is very general - I think it works with any collection of "ringed subspaces" of ringed spaces. $\endgroup$
    – pinaki
    Commented Dec 7, 2020 at 23:16
  • $\begingroup$ I see. But here our gluings are along open affines that seems pretty easy by itself, I mean I can't see how Karl's article can make it more clear. I'm just not sure about the existence of the two maps from $X$ to the new gluing. Or the construction of $\text{Spec}(A)^f$ with aforementioned properties is also very helpful. $\endgroup$
    – user127776
    Commented Dec 7, 2020 at 23:20

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