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Jul
25
comment Is a projective system of finite etale covers of a scheme S the same as its limit?
It hinges entirely on the first "equation". Then everything else is just general nonsense.
Jul
25
comment Finitely generated ideal in Boolean ring; how do we motivate the generator?
It's probably easier to see what's going on if you think in terms of the logical "and" and "or" operators.
Jul
25
comment Question on the definition of sheaves.
Rather, one must define the sheaf on a basis for the topology.
Jul
24
comment Tensor product with an L on top
It doesn't explain the first formula. That is not the derived tensor product.
Jul
24
comment Tensor product with an L on top
$\overset{\mathbf{L}}{\otimes}$ is the left derived tensor product. You can think of it as a refinement of $\mathrm{Tor}_*$.
Jul
23
comment What are local homomorphisms, geometrically?
@zcn So, it appears to me that your proposition says the following: $A \to A / I$ reflects units if and only if the closed subset $V (I) \subseteq \operatorname{Spec} A$ contains all the closed points (of $\operatorname{Spec} A$). In particular, if $A \to B$ reflects units, then the scheme-theoretic image of $\operatorname{Spec} B \to \operatorname{Spec} A$ contains all closed points.
Jul
23
comment What are local homomorphisms, geometrically?
However, do you have an explicit counterexample where there is a non-principal open subscheme of $Y$ through which $X \to Y$ factors? I was hoping that $X \to Y$ would be left orthogonal to arbitrary open immersions.
Jul
23
comment What are local homomorphisms, geometrically?
Well spotted! I thought that it was some kind of surjectivity condition, but I couldn't quite figure out how.
Jul
23
comment How do we get a simplicial homology functor?
Replacing $\mathbf{Ab}$ with its skeleton only solves the issue of objects being defined up to isomorphism. You still have to worry about functoriality.
Jul
22
comment Asterisk Notation
To be honest, I was half-expecting a follow-up question of, "what is the direct image functor?"
Jul
22
comment Asterisk Notation
It is the direct image functor.
Jul
22
comment Is the Axiom of Choice necessary to prove $\mathbb R \approx \mathcal P(\omega)$?
The order you define is called the lexicographic order. It is not a well-ordering. (Try finding an infinite descending chain!)
Jul
22
comment Why are duals in a rigid/autonomous category unique up to unique isomorphism?
Yes. The same applies to limits, colimits, adjoints, etc.
Jul
22
comment Why are duals in a rigid/autonomous category unique up to unique isomorphism?
That's not the point. A dual for $X$ is a triple $(Y, \epsilon, \eta)$ such that etc.; and for any two duals, there is a unique isomorphism connecting them that is compatible with all the data. That in no way says anything about the automorphisms of $Y$ as a bare object.
Jul
22
comment Why are duals in a rigid/autonomous category unique up to unique isomorphism?
When we say "unique up to unique isomorphism", it must be understood in the right sense: namely, that there is a unique isomorphism of structured objects. For a much easier example, see here.
Jul
21
comment The cone minus its apex deformation retracts onto its basis
Isn't $C (X) \setminus \{ P \}$ just homeomorphic to $X \times [0, 1)$?
Jul
21
comment Question on calculating hypercohomology
You won't really find the necessary background on quasicoherent sheaves in Weibel. Try Hartshorne, or any algebraic geometry textbook.
Jul
21
comment Question on calculating hypercohomology
If you don't have the background, then why are you trying to do this? First things first: the functor $F$ you are supposed to be thinking about is the global sections functor. Second: cohomology of quasicoherent sheaves on affine schemes is trivial, so hypercohomology reduces to ordinary cohomology in this case.
Jul
21
comment Two definitions of functions
The second notion, modulo the evident notion of equivalence, is properly called a function class.
Jul
20
comment What kind of points are there in a finite type $k$-scheme?
The generic point is usually not open. Think of an algebraic curve, for example.