Let \(X=\text{Spec}(R)\) be an affine scheme.

Let \(X'\) be an affine \(X\) scheme i.e., \(X'=\text{Spec}(R')\) for some ring \(R'\) with a morphism of schemes \(\pi: X'\rightarrow X\).  This \(\pi\) comes with morphism of global sections

\(R=\Gamma(X,\mathcal{O}_X)\rightarrow \Gamma(X',\mathcal{O}_X')=R'\)

giving \(R'\), structure of an \(R\) algebra. So, any affine scheme over \(X=\text{Spec}(R)\) is simply the specturm of an \(R\) algebra. Conversely, given an \(R\) algebra say \(R'\), we have an affine scheme \(X'=\text{Spec}(R')\) over \(X\) with morphism \(\pi:X'\rightarrow X\). Being a morphism of affine schemes, \(\pi: X\rightarrow X\) is an affine morphism. Let \(X\) be an arbitrary scheme. We want to associate an \(X\) scheme \(X'\) such that the structure morphism \(\pi:X'\rightarrow X\) is an affine morphism. To do this in case of \(X=\text{Spec}(R)\) we have fixed an \(R\) algebra and then associated an affine scheme for this. In case of an arbitrary scheme \(X\) unlike the case of affine scheme \(X=\text{Spec}(R)\) there is no single ring that has all information about  the scheme \(X\). It is only natural to consider the collection \(\{\mathcal{O}_X(U): U\subseteq X\}\) varying over all open subsets of \(X\) to get information about the scheme \(X\). Choosing an \(\mathcal{O}_X(U)\) algebra \(\mathcal{F}(U)\) for each open \(U\subseteq X\) we associate an \(X\) scheme \(X'\) for this collection \(\{\mathcal{F}(U)\}\) of \(\mathcal{O}_X(U)\) algebras. It is only natural to put a condition that this collection \(\{\mathcal{F}(U)\}\) to be compatible with structure sheaf \(\mathcal{O}_X\) i.e., we want \(U\mapsto \mathcal{F}(U)\) to give a structure of  sheaf of \(\mathcal{O}_X\) algebras on \(X\). So, given an arbitrary scheme \(X\) and a sheaf \(\mathcal{F}\) of \(\mathcal{O}_X\) algebras we associate an \(X\) scheme \(X'\) such that the structure map \(X'\rightarrow X\) is an affine morphism. It is not obvious at this point but we also want \(\mathcal{F}\) to be a quasicoherent sheaf of \(\mathcal{O}_X\) modules. We call this \(X'\),  Global spec or Relative spec of sheaf \(\mathcal{F}\) of \(\mathcal{O}_X\) algebras over \(X\) denoted by \(\textbf{Spec} (\mathcal{F})\). Here we make two important remarks :
  1. The \(\textbf{Spec}\) construction gives an important way to understand affine morphisms. Note that \(\textbf{Spec}(\mathcal{F})\rightarrow X\) is an affine morphism. The converse is also true. If \(f:X\rightarrow Y\) is an affine morphism then \(\mathcal{A}=f_*\mathcal{O}_X\) is a quasi coherent sheaf of \(\mathcal{O}_Y\) algebras and \(X\cong \text{Spec} (\mathcal{A})\).
  2. The \(\textbf{Spec}\) construction is used to assign a geometric vector bundle on a scheme \(Y\) to each locally free sheaf \(\mathcal{E}\) of rank \(n\) on a scheme \(Y\) which gives a bijection between isomorphism classes of locall free sheaves of rank \(n\) on \(Y\), and isomorphism classes of vector bundles of rank \(n\) on \(Y\).
  3. To define \(\textbf{Spec}(\mathcal{F})\) we do not need \(\mathcal{F}\) to be quasi coherent, but \(\mathcal{F}\) quasi coherent implies the structure map \(\textbf{Spec}(\mathcal{F})\rightarrow X\) is an affine morphism.
Now, we try to construct \(\textbf{Spec}(\mathcal{F})\) and structure map \(\textbf{Spec}(\mathcal{F})\rightarrow X\). One way to do this is gluing the schemes \(\text{Spec}(\mathcal{F}(U))\) over all open subsets \(U\subseteq X\). Another way is to use universal property of \(\text{Spec}\) of a ring. We have following result : Let \(A\) be a ring and let \((X,\mathcal{O}_X)\) be a scheme. Given a morphism \(f:X\rightarrow \text{Spec}(A)\) we have an associated map on sheaves \(f^{\#}:\mathcal{O}_{\text{Spec}(A)}\rightarrow f_*\mathcal{O}_X\). Taking global sections, we obtain a ring homomorphism \(A\rightarrow \mathcal{O}_X(X)\). Thus there is a natural map

\(\alpha: \text{Hom}_{\text{Schemes}}(X,\text{Spec}(A))\rightarrow\text{Hom}_{\text{Rings}}(A,\mathcal{O}_X(X)).\)

This map is bijective. So, we have

\(\text{Hom}_{\text{Schemes}}(X,\text{Spec}(A))\cong\text{Hom}_{\text{Rings}}(A,\mathcal{O}_X(X)).\)

We have the similar result for relative schemes over an affine scheme Let \(A\) be a ring and \(B\) be an \(A\) algebra. let \((X,\mathcal{O}_X)\) be an \(S=\text{Spec}(A)\) scheme. Given a morphism \(f:X\rightarrow \text{Spec}(B)\) of \(S\) schemes we have an associated map on sheaves \(f^{\#}:\mathcal{O}_{\text{Spec}(B)}\rightarrow f_*\mathcal{O}_X\). Taking global sections, we obtain an \(A\) algebra homomorphism \(B\rightarrow \mathcal{O}_X(X)\). Thus there is a natural map

\(\alpha: \text{Hom}_{S -  \text{Schemes}}(X,\text{Spec}(B))\rightarrow\text{Hom}_{A \text{algebra}}(B,\mathcal{O}_X(X)).\)

This map is bijective. So, we have

\(\text{Hom}_{S-\text{Schemes}}(X,\text{Spec}(B))\cong\text{Hom}_{A \text{algebra}}(B,\mathcal{O}_X(X)).\)

In terms of commutative diagram we have the following : ql_63245f850a823da7f4d9a1430614dc47_l3 ql_442a9448f1314e08b5cdb2b157d49463_l3 Now, for an arbitrary scheme \(X\) (which is generalization of \(S\) above), and a quasi coherent sheaf of \(\mathcal{O}_X\) algebras \(\mathcal{F}\) (which is generalization of \(B\) above), we want to define \(\text{Spec}(\mathcal{F})\)  that comes with a structure map \(\text{Spec}(\mathcal{F})\rightarrow X\) such that given any \(X\) scheme \(f:Y\rightarrow X\) and a morphism of \(X\) schemes \(\alpha :  Y \rightarrow \text{Spec}(\mathcal{F})\) something similar to that of the above situation happens. In terms of commutative diagram, we have ql_5e62c2113229e226dbeb655ce5d4e9a3_l3As we are generalizing the case of \(S=\text{Spec}(R)\) we expect to have similar commutative diagram as in the case of \(S\) except that in this case commutative diagram not be of ring homomorphism but it would be of sheaves of algebras. We expect to have something like ql_97afde5a457fb279d3701b254276a3de_l3As \(\mathcal{F}\) is a sheaf of \(\mathcal{O}_X\) modules and \(\mathcal{O}_Y\) is a sheaf of \(\mathcal{O}_Y\) modules, it does not make sense to talk about morphism between these two, we make a slight change by considering morphsim \(\mathcal{F}\rightarrow f_*\mathcal{O}_Y\) where \(f^*\mathcal{O}_Y\) is the push forward of $\mathcal{O}_Y$ under \(f:Y\rightarrow X\), thus a sheaf of $\mathcal{O}_X$ modules.   So, we want the following commutative diagram ql_c79f021eb2a75a235bf36ff13a6c3ea5_l3 So, our definition of \(\textbf{Spec}(\mathcal{F})\) would be such that it satisfy the universal property ql_c8ebcac76ce03d3a0f7f8a55d7fa4127_l3 ql_d9935afd7b88fce281e750905117ad22_l3 We will discuss properties of this construction in next blog post. There are still some loose ends here, we will fix that soon.