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3 corrected direction of some arrows

In the category of sets there is no such thing as the initial local ring into which R maps, i.e. a local ring L and a map f:R-->L such that any map from R into a local ring factors through f.

But a ring R is a ring object in the topos of Sets. Now if you are willing to let the topos vary in which it should live, such a "free local ring on R" does exist: It is the ring object in the topos of sheaves on Spec(R) which is given by the structure sheaf of Spec(R). So the space you were wondering about is part of the solution of forming a free local ring over a given ring (you can reconstruct the space from the sheaf topos, so you could really say that it "is" the space).

Edit: I rephrase that less sloppily, in response to Lars' comment. So the universal property is about pairs (Topos, ring object in it). A map (T,R)-->(T',R') between such is a pair

(adjunction $f_:T \leftrightarrow T':f^$ , morphism of ring objects $f^*R'\rightarrow R$).

Note that by convention the direction of the map is the geometric direction, the one corresponding to the direction of a map topological spaces - in my "universal local ring" picture I was stressing the algebraic direction, which is given by $f^*$.

Now for a ring R there is a map $(Set,R)\rightarrow(Sh(Spec(R)), O_{Spec(R)})$(Sh(Spec(R)), O_{Spec(R)})\rightarrow(Set,R)$:$f^* R$is the constant sheaf with value R on Spec(R), the map$f^* R \rightarrow O_{Spec(R)}$is given by the inclusion of R into its localisations which occur in the definition of$O_{Spec(R)}$. This is the initial terminal map (Set,R)-->(T,L) into T,L)-->(Set,R) from pairs with L a local ring. For a simple example you might want to work out how such a map factors, if the target domain pair happens to be of the form (Set,L). This universal property of course determines the pair up to equivalence. It thus also determines the topos half of the pair up to equivalence, and thus also the space Spec(R) up to homeomorphism.(end of edit) An even nicer reformulation of this is the following (even more high brow, but to me it gives the true and most illuminating justification for the Zariski topology, since it singles out just the space Spec(R)): A ring R, i.e. a ring in the topos of sets, is the same as a topos morphism from the topos of sets into the classifying topos T of rings (by definition of classifying topos). There also is a classifying topos of local rings with a map to T (which is given by forgetting that the universal local ring is local). If you form the pullback (in an appropriate topos sense) of these two maps you get the topos of sheaves on Spec(R) (i.e. morally the space Spec(R)). The map from this into the classifying topos of local rings is what corresponds to the structure sheaf. Isn't that nice? See Monique Hakim's "Schemas relatifs et Topos anelles" for all this (the original reference, free of logic), or alternatively Moerdijk/MacLane's "Sheaves in Geometry and Logic" (with logic and formal languages). 2 included a more detailed explanation Edit: I rephrase that less sloppily, in response to Lars' comment. So the universal property is about pairs (Topos, ring object in it). A map (T,R)-->(T',R') between such is a pair (adjunction$f_:T \leftrightarrow T':f^ $, morphism of ring objects$f^*R'\rightarrow R$). Now for a ring R there is a map$(Set,R)\rightarrow(Sh(Spec(R)), O_{Spec(R)})$:$f^* R$is the constant sheaf with value R on Spec(R), the map$f^* R \rightarrow O_{Spec(R)}$is given by the inclusion of R into its localisations which occur in the definition of$O_{Spec(R)}\$.This is the initial map (Set,R)-->(T,L) into pairs with L a local ring. For a simple example you might want to work out how such a map factors, if the target pair happens to be of the form (Set,L).

This universal property of course determines the pair up to equivalence. It thus also determines the topos half of the pair up to equivalence, and thus also the space Spec(R) up to homeomorphism.(end of edit)

An even nicer reformulation of this is the following (even more high brow, but to me it gives the true and most illuminating justification for the Zariski topology)topology, since it singles out just the space Spec(R)):

1

In the category of sets there is no such thing as the initial local ring into which R maps, i.e. a local ring L and a map f:R-->L such that any map from R into a local ring factors through f.

But a ring R is a ring object in the topos of Sets. Now if you are willing to let the topos vary in which it should live, such a "free local ring on R" does exist: It is the ring object in the topos of sheaves on Spec(R) which is given by the structure sheaf of Spec(R). So the space you were wondering about is part of the solution of forming a free local ring over a given ring (you can reconstruct the space from the sheaf topos, so you could really say that it "is" the space).

An even nicer reformulation of this is the following (even more high brow, but to me it gives the true and most illuminating justification for the Zariski topology):

A ring R, i.e. a ring in the topos of sets, is the same as a topos morphism from the topos of sets into the classifying topos T of rings (by definition of classifying topos). There also is a classifying topos of local rings with a map to T (which is given by forgetting that the universal local ring is local). If you form the pullback (in an appropriate topos sense) of these two maps you get the topos of sheaves on Spec(R) (i.e. morally the space Spec(R)). The map from this into the classifying topos of local rings is what corresponds to the structure sheaf.

Isn't that nice? See Monique Hakim's "Schemas relatifs et Topos anelles" for all this (the original reference, free of logic), or alternatively Moerdijk/MacLane's "Sheaves in Geometry and Logic" (with logic and formal languages).