Want to take part in these discussions? Sign in if you have an account, or apply for one below
Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.
I edited group scheme and scheme a bit.
Forget, I wrote some nonsense
It looks correct. I fixed some funny typesetting of primes. The second axiom in the sheaf-style definition can probably be replaced by “there is an epimorphism $\coprod_i U_i \to X$ in the category of sheaves”, but I don’t immediately see how to show that the two conditions are equivalent without knowing that both definitions lead to schemes in the locally ringed space sense.
Not very helpful comment, but scheme doesn’t strike me as particularly clear. Couldn’t the functor-of-points approach be given greater prominence? This could be related to ’Examples and similar dualities’ of Isbell duality, no?
As a more concrete point, this
the category $Aff:=Psh(CRing^{op})$ of presheaves on the opposite category of commutative rings
should presumably be
the category $Psh(Aff):=Psh(CRing^{op})$ of presheaves on the opposite category of commutative rings.
Or the defn $Aff :=CRing^{op}$ happens earlier.
The page generalized schemes seems friendlier. But while on that page
To recall the equivalence between the two points of view, every scheme $X$ gives rise to a representable presheaf on the formal dual of commutative rings
$X(-) : CRing \to Set$ $A \mapsto Hom_{CRing}(Spec A, X)$
is taking Hom space in the wrong category, no?
affine scheme isn’t so clear either. E.g., in the term $Spec(\Gamma_Y \mathcal{O}_Y)$, we can find the $\mathcal{O}_Y$ from the linked page ringed space, but I can’t see $\Gamma_Y$ defined anywhere.
is taking Hom space in the wrong category, no?
I fixed it.
Thanks.
I tried to liven up functor of points with an illustration. Is there a standard example of a non-representable functor of points?
The functor of points of a non-affine scheme is not representable by an affine scheme (of course). For instance, $\mathbb{P}^n$. Or do you mean a (pre)sheaf that isn’t representable by any scheme?
Right. I guess I was just asking for a simple non-affine scheme.
1 to 10 of 10