Let $R$ be a normal noetherian domain. Write it as intersection of discrete valued domains $\cap_p R_p$. Assume I have schemes $X_p\rightarrow Spec(R_p)$. Via the inclusion $B_{p,p^{\prime}}:=A_p\cap A_{p^{\prime}}\rightarrow A_p$, we can consider the structure sheaf $\mathcal{O}_{X_p}$ as $B_{p,p^{\prime}}$-algebra. Assume that for any pairs of such primes $p,p^{\prime}$ we have an isomorphism $\varphi_{p,p^{\prime}}:X_p\rightarrow X_{p^{\prime}}$ as $Spec(B_{p,p^{\prime}})$-schemes, satisfying the usual cocycle condition. Under which conditions is it true that we can find a scheme $X\rightarrow Spec(R)$ such that $X_p$ is the base change of $X$ via the morphism $Spec(R_p)\rightarrow Spec(R)$??

Let $R$ be a normal noetherian domain. Write it as intersection of discrete valued domains $\bigcap_p R_p$. Assume I have schemes $X_p\rightarrow \operatorname{Spec}(R_p)$. Via the inclusion $B_{p,p^{\prime}}:=A_p\cap A_{p^{\prime}}\rightarrow A_p$, we can consider the structure sheaf $\mathcal{O}_{X_p}$ as $B_{p,p^{\prime}}$-algebra. Assume that for any pairs of such primes $p,p^{\prime}$ we have an isomorphism $\varphi_{p,p^{\prime}}:X_p\rightarrow X_{p^{\prime}}$ as $\operatorname{Spec}(B_{p,p^{\prime}})$-schemes, satisfying the usual cocycle condition. Under which conditions is it true that we can find a scheme $X\rightarrow \operatorname{Spec}(R)$ such that $X_p$ is the base change of $X$ via the morphism $\operatorname{Spec}(R_p)\rightarrow \operatorname{Spec}(R)$??