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Jun
23
comment A Topology such that the continuous functions are exactly the polynomials
If every polynomial function $f : K \to K$ is continuous and $\{ 0 \}$ is closed, then the topology on $K$ must be finer than the Zariski topology. However, the Zariski topology does not have the desired property: for example, the absolute value function $\mathbb{Q} \to \mathbb{Q}$ is Zariski-continuous.
Jun
22
comment Construction(s) of new integral domains from “old ones”
The tensor product of two integral domains over an algebraically closed field is again an integral domain: see here.
Jun
21
comment Why is the category of coherent sheaves not grothendieck?
You can check it by applying $\mathrm{Hom}_A (-, A)$.
Jun
21
comment On the direct sum of rings
$A$ is not a ring, though.
Jun
20
answered Why is the category of coherent sheaves not grothendieck?
Jun
20
comment Every $n$-dimensional variety is birationally equivalent to a hypersurface in $\mathbb{A}^{n+1}.$
It may not be monic but I think it doesn't matter. You can also change variables to make it monic, as in the proof of Noether normalisation.
Jun
20
comment A generalization of abelian categories including Grp
Yes, it does, though one has to be more careful with the statement. See here.
Jun
20
answered A generalization of abelian categories including Grp
Jun
20
comment Two definitions of equivariant sheaves
Well, as stated literally, there is not even a continuity condition in (a).
Jun
19
answered Is there a universal property for the ultraproduct?
Jun
19
comment Number of generators of the maximal ideals in polynomial rings over a field
It essentially follows from the isomorphism right at the end: use the correspondence theorem for ideals of a ring.
Jun
18
comment What do I call a covariant functor which is a filtered colimit of representable functors?
Well, the category of flat functors $\mathcal{C} \to \mathbf{Set}$ is actually equivalent to the opposite of the category of pro-objects in $\mathcal{C}$, so it's not entirely unreasonable.
Jun
18
answered What do I call a covariant functor which is a filtered colimit of representable functors?
Jun
18
accepted Quasicoherent ideal sheaves on open subschemes
Jun
17
comment Is there a universal property for the ultraproduct?
Actually, I was thinking about this one, but I wasn't able to find it just now.
Jun
17
comment Is there a universal property for the ultraproduct?
I recall a question on MO. It's a colimit over a certain diagram derived from the filter.
Jun
17
comment Why is there apparently no general notion of structure-homomorphism?
Sometimes there is no obvious notion of homomorphism. For instance, what's the appropriate notion of morphism for Banach spaces?
Jun
16
revised How to find a minimal polynomial (field theory)
added 1167 characters in body
Jun
16
answered How to find a minimal polynomial (field theory)
Jun
16
comment Every $n$-dimensional variety is birationally equivalent to a hypersurface in $\mathbb{A}^{n+1}.$
If the conclusion holds, then $K (X)$ must admit a transcendence basis over which it is simply generated – separability is sufficient but not necessary. So for a counterexample, I suppose one has to find a field that is not simply generated over any transcendence basis. Perhaps something like the variety $\{ x^p - z, y^p - w \} \subset \mathbb{A}^4$ over $\overline{\mathbb{F}_p}$?