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May
27
awarded  Popular Question
May
27
asked Maximal ideals in polynomial rings over a field
May
27
comment Sheaf cohomology in non-commutative setup
You can deduce it from the case where $A = \mathbb{Z}$. See the last paragraph here.
May
27
comment Sheaf cohomology in non-commutative setup
For any sheaf $A$ of rings whatsoever (commutative or not), the category of $A$-modules has enough injectives.
May
26
comment Simple question on exact sequences.
Certain element-chasing methods are valid in general abelian categories. See Ch. VIII in Categories for the working mathematician.
May
26
comment Why do we believe the Church-Turing Thesis?
There are also known models of hypercomputation, e.g. infinite-time Turing machines. But perhaps you might regard them as (too) unrealistic.
May
26
comment Simple question on exact sequences.
The $\operatorname{im} = \operatorname{ker}$ thing is understood to mean "isomorphic as subobjects of the relevant object".
May
26
comment How to establish an isomorphism between these two tensor products?
The universal property is the straightforward way! You can work with the generators if you like, but bear in mind that you have to respect the relations as well.
May
26
answered Nonsingular affine curve which is not unmixed
May
26
comment Nonsingular affine curve which is not unmixed
What is your definition of curve? I would say that a subvariety that has a point as an irreducible component is not a curve, simply because a point is 0-dimensional.
May
24
comment Spheres in different dimension are not homotopy equivalent
Well, the spheres (for $n > 1$) are all simply-connected, so there aren't any interesting covering spaces...
May
24
comment Questions about epimorphisms and projectives in functor categories
See this question.
May
24
answered Does taking the direct limit of chain complexes commute with taking homology?
May
24
comment Questions about epimorphisms and projectives in functor categories
I meant each $@_i$ has a left adjoint. Of course, this is because each $@_i$ is representable, as you observed.
May
23
comment Questions about epimorphisms and projectives in functor categories
Each representable presheaf $\mathcal{I}(i, -)$ is free because it occurs as the image of $1$ under the left adjoint of the evaluation-at-$i$ functor. (This, in some sense, is the content of the Yoneda lemma.) That they are projective is easily shown to be a consequence of this, exactly like how free modules are projective. For the canonical projective covering of a presheaf $P$, just take the collection of all morphisms from any representable presheaf to $P$, and then take the amalgamation of all those.
May
23
answered Analogy between prime numbers and singleton sets?
May
23
revised Questions about epimorphisms and projectives in functor categories
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May
23
answered Questions about epimorphisms and projectives in functor categories
May
23
comment a group is not the union of two proper subgroups - how to internalize this into other categories?
My previous comments about toposes with enough points were incorrect; please disregard.
May
23
revised a group is not the union of two proper subgroups - how to internalize this into other categories?
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