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Jun
24
comment Does the nerve of a category preserve directed colimits?
(However, not every compact category is finite.)
Jun
23
comment Is a filtered category necessarily (essentially) small?
Yes, that is a small subtlety. It is similar to the pitfall of saying that $[\mathcal{C}^\mathrm{op}, \mathbf{Set}]$ is the free colimit-completion of $\mathcal{C}$ – this is true when $\mathcal{C}$ is essentially small but can fail when $\mathcal{C}$ is not essentially small.
Jun
23
comment Is a filtered category necessarily (essentially) small?
That is the standard proof as far as I know. Bear in mind that many results we have about filtered colimits are, strictly speaking, for colimits over small filtered diagrams – so your generalisation may be less useful than at first sight.
Jun
23
answered Is a filtered category necessarily (essentially) small?
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.