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Aug
22
comment In an abelian category,every morphism can be written as composition of epi and mono.
In the meantime, you might to look at proposition 6.3.17 here.
Aug
22
comment In an abelian category,every morphism can be written as composition of epi and mono.
For completeness, you should state the definition of abelian category Weibel uses. Of course, all you have to do is prove that the canonical $\operatorname{Ker} \operatorname{coker} f \to \operatorname{Coker} \operatorname{ker} f$ is an isomorphism. I am sure this has been asked before on this site somewhere...
Aug
21
comment What is $X^{\omega}$ where $X$ is a set?
In this context, a finite sequence is not a sequence "indexed by the nonnegative integers".
Aug
20
comment Exercise on localization as a colimit
Yes, it works. If anything what you need is that $A$ is commutative.
Aug
19
comment Ring of rational power series
If $A$ is a field then $R_A$ should be the subring of $A (t)$ consisting of those rational functions whose denominator does not vanish at 0. Or perhaps you mean to ask about Laurent series?
Aug
19
comment homology commutes with direct sum and product?
Any additive functor preserves finite direct sums/products. In particular, both cokernels and kernels (hence also images and quotients) commute with finite direct sums/products.
Aug
18
comment Cohomology of wedge equals direct sum of cohomologies
Well, now that you've restricted to path-connected spaces, you can establish the claim in degree 0 by hand and in positive degrees by reducing (hah) to reduced cohomology. No?
Aug
18
comment Cohomology of wedge equals direct sum of cohomologies
Strictly speaking direct sums are only defined for modules. Note that the formula is false if $X$ and $Y$ are not (path-)connected.
Aug
18
comment Quotient of locally free sheaf is locally free?
It also works for noetherian affine schemes. At any rate, you can verify by hand (if you like) that $\mathbb{Z} / 2 \mathbb{Z}$ is not locally free.
Aug
16
comment Examples of advancement in mathematics due to war
Do you mean the Cold War?
Aug
16
comment Examples of advancement in mathematics due to war
@AsafKaragila You are thinking of Jean Leray.
Aug
16
comment Looking for a smooth curve that is not rational
The twisted cubic is in fact rational: your parametrisation has a rational inverse, defined away from the singularities $[0 : 0 : 0 : 1]$ and $[1 : 0 : 0 : 0]$ by $[x : y : z : w] \mapsto [y : z]$.
Aug
15
comment Scheme theoretic dual of $\mathbb P^n_k$
A point of the dual projective space is an equivalence class of linear forms on the original vector space, which defines a hyperplane in the usual way.
Aug
15
comment Scheme theoretic dual of $\mathbb P^n_k$
Typically $V$ is finite dimensional. If $\dim V = n + 1$ then $\mathbb{P} (V) \cong \mathbb{P}_k^n$.
Aug
14
comment Stacks versus sheaves with values in categories
Perhaps your confusion stems from the formulation of stacks in terms of fibred categories instead of indexed categories. The definitions are otherwise very much analogous – the key point is that the category of descent data can be defined by a 2-limit.
Aug
13
comment A local homeomorphism between compact, connected, topological spaces
I think the claim is false without the Hausdorff condition.
Aug
12
comment how to lift geometrically integralness using etale(+something else) morphisms
I think it's unlikely, but I am not an expert.
Aug
12
comment how to lift geometrically integralness using etale(+something else) morphisms
$\operatorname{Spec} \mathbb{C} \to \operatorname{Spec} \mathbb{R}$ is finite étale, $\operatorname{Spec} \mathbb{R}$ is geometrically integral as an $\mathbb{R}$-scheme, but $\operatorname{Spec} \mathbb{C}$ is not geometrically integral as an $\mathbb{R}$-scheme.
Aug
12
comment how to lift geometrically integralness using etale(+something else) morphisms
What is your definition of variety? If it's just "reduced and of finite type", then $X \amalg X \to X$ is a counterexample.
Aug
12
comment Every closed (not-necessarily symmetric) monoidal category is canonically self-enriched, right?
It is unusual to consider categories enriched in something other than a symmetric monoidal category in the first place – partly because a lot of the theory becomes awkward (e.g. opposite categories).