Return to Answer

The Zariski spectrum is essentially the classifying topos for prime ideal of $A$, or to be more precise, the classifying topos for subsets of $A$ that are "complement of prime ideals of $A$". The precise geometric theory is given by:

$$x+y \in U \Rightarrow x \in U \text{ or } y \in U$$ $$xy \in U \Leftrightarrow x \in U \text{ and } y \in U$$ $$ 1 \in U$$ $$ 0 \notin U$$

where the $x \in U$ are the basic proposition of the theory (and $U$ represents this "subset of $A$ which is the complement of a prime ideals").

This can be immediately translated into the fact that that (the frame corresponding to) the Zariski spectrum is the frame freely generated by the symbols $D(a)$ for $a \in A$ subject to the relations:

$$D(x+y) \leqslant D(x) \cup D(y)$$ $$D(xy)= D(x) \cap D(y)$$ $$D(1) = \top$$ $$D(0) = \bot$$

You can rewrite this as a site if you want - but generally, the presentation as a free locale is more convenient. Of course, the relation $D(f^n)=D(f)$ immediately follow from the second condition.

The structure sheaf of the Zariski spectrum can easily be obtained form this as well: internally in the topos of sheaf over the Zariski spectrum, you have the generic such subsheaf $U \subset A$ (where $A$ is the constant sheaf). The Structure sheaf is $A[U^{-1}]$.

Note that this definitely appears in a few places in the literature. But I'm not sure what is the appropriate reference to attribute it to. My Guess is Johnstone's Stone Space, but I can't check it right now.

The Zariski spectrum is essentially the classifying topos for prime ideal of $A$, or to be more precise, the classifying topos for subsets of $A$ that are "complement of prime ideals of $A$". The precise geometric theory is given by:

$$x+y \in U \Rightarrow x \in U \text{ or } y \in U$$ $$xy \in U \Leftrightarrow x \in U \text{ and } y \in U$$ $$ 1 \in U$$ $$ 0 \notin U$$

where the $x \in U$ are the basic proposition of the theory (and $U$ represents this "subset of $A$ which is the complement of a prime ideals").

This can be immediately translated into the fact that that (the frame corresponding to) the Zariski spectrum is the frame freely generated by the symbols $D(a)$ for $a \in A$ subject to the relations:

$$D(x+y) \leqslant D(x) \cup D(y)$$ $$D(xy)= D(x) \cap D(y)$$ $$D(1) = \top$$ $$D(0) = \bot$$

You can rewrite this as a site if you want - but generally, the presentation as a free locale is more convenient. Of course, the relation $D(f^n)=D(f)$ immediately follow from the second condition.

The structure sheaf of the Zariski spectrum can easily be obtained form this as well: internally in the topos of sheaf over the Zariski spectrum, you have the generic such subsheaf $U \subset A$ (where $A$ is the constant sheaf). The Structure sheaf is $A[U^{-1}]$.

Note that this definitely appears in a few places in the literature. But I'm not sure what is the appropriate reference to attribute it to. My Guess is Johnstone's Stone Space, but I can't check it right now.Edit: as comfirmed by Zhen Lin in the comment this appears in Johnstone's Stone Space, where it is attributed to Joyal without reference.

The Zariski spectrum is essentially the classifying topos for prime ideal of $A$, or to be more precise, the classifying topos for subsets of $A$ that are "complement of prime ideals of $A$". The precise geometric theory is given by:

$$x+y \in U \Rightarrow x \in U \text{ or } y \in U$$ $$xy \in U \Leftrightarrow x \in U \text{ and } y \in U$$ $$ 1 \in U$$ $$ 0 \notin U$$

where the $x \in U$ are the basic proposition of the theory (and $U$ represents this "subset of $A$ which is the complement of a prime ideals").

This can be immediately translated into the fact that that (the frame corresponding to) the Zariski spectrum is the frame freely generated by the symbols $D(a)$ for $s \in A$ subject to the relations:

$$D(x+y) \leqslant D(x) \cup D(y)$$ $$D(xy)= D(x) \cap D(y)$$ $$D(1) = \top$$ $$D(0) = \bot$$

You can rewrite this as a site if you want - but generally, the presentation as a free locale is more convenient. Of course, the relation $D(f^n)=D(f)$ immediately follow from the second condition.

The structure sheaf of the Zariski spectrum can easily be obtained form this as well: internally in the topos of sheaf over the Zariski spectrum, you have the generic such subsheaf $U \subset A$ (where $A$ is the constant sheaf). The Structure sheaf is $A[U^{-1}]$.

Note that this definitely appears in a few places in the literature. But I'm not sure what is the appropriate reference to attribute it to. My Guess is Johnstone's Stone Space, but I can't check it right now.

The Zariski spectrum is essentially the classifying topos for prime ideal of $A$, or to be more precise, the classifying topos for subsets of $A$ that are "complement of prime ideals of $A$". The precise geometric theory is given by:

$$x+y \in U \Rightarrow x \in U \text{ or } y \in U$$ $$xy \in U \Leftrightarrow x \in U \text{ and } y \in U$$ $$ 1 \in U$$ $$ 0 \notin U$$

where the $x \in U$ are the basic proposition of the theory (and $U$ represents this "subset of $A$ which is the complement of a prime ideals").

This can be immediately translated into the fact that that (the frame corresponding to) the Zariski spectrum is the frame freely generated by the symbols $D(a)$ for $a \in A$ subject to the relations:

$$D(x+y) \leqslant D(x) \cup D(y)$$ $$D(xy)= D(x) \cap D(y)$$ $$D(1) = \top$$ $$D(0) = \bot$$

You can rewrite this as a site if you want - but generally, the presentation as a free locale is more convenient. Of course, the relation $D(f^n)=D(f)$ immediately follow from the second condition.

The structure sheaf of the Zariski spectrum can easily be obtained form this as well: internally in the topos of sheaf over the Zariski spectrum, you have the generic such subsheaf $U \subset A$ (where $A$ is the constant sheaf). The Structure sheaf is $A[U^{-1}]$.

Note that this definitely appears in a few places in the literature. But I'm not sure what is the appropriate reference to attribute it to. My Guess is Johnstone's Stone Space, but I can't check it right now.

The Zariski spectrum is essentially the classifying topos for prime ideal of $A$, or to be more precise, the classifying topos for subsets of A that are "complement of prime ideals of $A$". The precise geometric theory is given by:

$$x+y \in U \Rightarrow x \in U \text{ or } y \in U$$ $$xy \in U \Leftrightarrow x \in U \text{ and } y \in U$$ $$ 1 \in U$$ $$ 0 \notin U$$

This can be immediately translated into the fact that that (the frame corresponding to) the Zariski spectrum is the frame freely generated by the symbols $D(a)$ for $s \in A$ subject to the relations:

$$D(x+y) \leqslant D(x) \cup D(y)$$ $$D(xy)= D(x) \cap D(y)$$ $$D(1) = \top$$ $$D(0) = \bot$$

You can rewrite this as a site if you want - but generally, the presentation as a free locale is more convenient. Of course, the relation $D(f^n)=D(f)$ immediately follow from the second condition.

The structure sheaf of the Zariski spectrum can easily be obtained form this as well: internally in the topos of sheaf over the Zariski spectrum, you have the generic such subsheaf $U \subset A$ (where $A$ is the constant sheaf). The Structure sheaf is $A[U^{-1}]$.

Note that this definitely appears in a few places in the literature. But I'm not sure what is the appropriate reference to attribute it to. My Guess is Johnstone's Stone Space, but I can't check it right now.

The Zariski spectrum is essentially the classifying topos for prime ideal of $A$, or to be more precise, the classifying topos for subsets of $A$ that are "complement of prime ideals of $A$". The precise geometric theory is given by:

$$x+y \in U \Rightarrow x \in U \text{ or } y \in U$$ $$xy \in U \Leftrightarrow x \in U \text{ and } y \in U$$ $$ 1 \in U$$ $$ 0 \notin U$$

where the $x \in U$ are the basic proposition of the theory (and $U$ represents this "subset of $A$ which is the complement of a prime ideals").

This can be immediately translated into the fact that that (the frame corresponding to) the Zariski spectrum is the frame freely generated by the symbols $D(a)$ for $s \in A$ subject to the relations:

$$D(x+y) \leqslant D(x) \cup D(y)$$ $$D(xy)= D(x) \cap D(y)$$ $$D(1) = \top$$ $$D(0) = \bot$$

You can rewrite this as a site if you want - but generally, the presentation as a free locale is more convenient. Of course, the relation $D(f^n)=D(f)$ immediately follow from the second condition.

The structure sheaf of the Zariski spectrum can easily be obtained form this as well: internally in the topos of sheaf over the Zariski spectrum, you have the generic such subsheaf $U \subset A$ (where $A$ is the constant sheaf). The Structure sheaf is $A[U^{-1}]$.

Note that this definitely appears in a few places in the literature. But I'm not sure what is the appropriate reference to attribute it to. My Guess is Johnstone's Stone Space, but I can't check it right now.