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Jun
14
awarded  Notable Question
Jun
13
answered Every $n$-dimensional variety is birationally equivalent to a hypersurface in $\mathbb{A}^{n+1}.$
Jun
13
comment A commutative ring $A$ is an algebra over a field $F$ if and only if $A$ contains (an isomorphic copy of) $F$ as a subring
The headline claim is false: the trivial ring is an $F$-algebra. But that is the only problem. For the second problem you don't have to take $R = M$, but what you suggested works too.
Jun
13
comment Every $n$-dimensional variety is birationally equivalent to a hypersurface in $\mathbb{A}^{n+1}.$
That's where the separability hypothesis comes in – you need to be able to know that $k (X)$ is generated by one element over your transcendence basis.
Jun
13
comment Every $n$-dimensional variety is birationally equivalent to a hypersurface in $\mathbb{A}^{n+1}.$
You probably should assume some separability hypothesis. But here's a hint: take a transcendence basis for the function field...
Jun
13
answered Monads on Set and their strength
Jun
13
revised which of the following is/are algebraic over rationals
added 4 characters in body
Jun
13
comment Definition of the Coproduct of Categories?
The coproduct of ordinary categories differs from the coproduct of $\mathbf{Ab}$-enriched categories, which also differs from the coproduct of additive categories.
Jun
12
comment Meaning of quote: “model theory = algebraic geometry - fields”?
Of course. But the methods and definitions used then are substantially different compared to model theory, which is much closer to the classical/naïve conception of algebraic geometry.
Jun
12
comment Motivating the definition of right derived functors in the context of derived categories.
Isn't the second thing a left derived functor?
Jun
12
comment Meaning of quote: “model theory = algebraic geometry - fields”?
More accurately, beyond the case of algebraically closed fields, which is a really nice theory...
Jun
12
comment Is this equality in a double category true?
It's true in a 2-category. Do you really need the double category version?
Jun
12
answered Sites for étale $G$-spaces
Jun
11
comment Hartshorne proof of adjunction formula proposition II.8.20
Perhaps you should explicitly mention that exterior powers are preserved by base change (and hence localisation).
Jun
11
comment Example of a Noetherian module that is not a ring?
@JulianKuelshammer My comment is not an answer to the main question. In fact, none of these comments are.
Jun
11
comment '$R$-rational points,' where $R$ is an arbitrary ring
However, when $R$ is a field, this is not to be confused with the notion of $R$-rational points...
Jun
11
answered Hartshorne Lemma(II 6.1)
Jun
11
comment What is the difference between $\ell$-adic cohomology and cohomology with coefficient in $Z_\ell$?
Étale cohomology with coefficients in an infinite constant sheaf is sometimes not what you expect. For example, in $H^1$, under good conditions, for a constant abelian group $A$, $H^1_{\text{ét}} (X, A) \cong \mathrm{Hom}(\pi_1^{\text{ét}} (X), A)$, where the RHS is the group of continuous homomorphisms. Since $\pi_1^{\text{ét}} (X)$ is profinite, this will be trivial if every finite subgroup of $A$ is trivial.
Jun
11
revised What is the difference between $\ell$-adic cohomology and cohomology with coefficient in $Z_\ell$?
added 162 characters in body; edited tags
Jun
11
comment When a divisor on an algebraic curve is canonical
Riemann and Roch are two people, so it's just the "Riemann–Roch theorem".