Timeline for What algebraic condition corresponds to injectivity of a morphism of varieties?
Current License: CC BY-SA 4.0
12 events
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| Dec 10, 2021 at 23:23 | comment | added | Johan | An answer is: for all $a \in A$ there is an $n > 0$ such that $(a \otimes 1 - 1 \otimes a)^n = 0$ in $A \otimes_B A$. | |
| Dec 9, 2021 at 11:04 | comment | added | Laurent Moret-Bailly | @CarlosEsparza: Right (except it is "not mono"). Concerning epimorphisms of rings, you may have a look at this seminar. | |
| Dec 9, 2021 at 8:18 | comment | added | Carlos Esparza | @LaurentMoret-Bailly And the way to see that it is not epi in the category of schemes is to consider $f, g: \mathbb{A}^1 \to \operatorname{Spec} k[\epsilon]/\epsilon^2$ sending $t \mapsto \epsilon$ and $t \mapsto 0$. | |
| Dec 9, 2021 at 8:02 | comment | added | Laurent Moret-Bailly | @CarlosEsparza Sorry, I was not quite clear: we have an injective morphism of reduced affine $\mathbb{C}$-schemes which is not a monomorphism of affine $\mathbb{C}$-schemes (but, as you say, it is a monomorphism in the category of reduced $\mathbb{C}$-schemes, affine or not). | |
| Dec 9, 2021 at 2:21 | comment | added | Carlos Esparza | @LaurentMoret-Bailly are you saying that the normalization of the cusp is not a monomorphism (in the category of reduced affine $\mathbb{C}$-schemes)? It seems like monomorphism and injectivity should be equivalent because morphisms of varieties are determined by their map on topological spaces (so injective => monomorphism) and monomorphism have to be injective (otherwise you have a contradiction by using two different maps from $\operatorname{Spec} \mathbb{C}$). | |
| Dec 8, 2021 at 22:35 | comment | added | Tom Goodwillie | @YCor: If $K\to L$ is a field extension adjoining a p-th root to a characteristic p field, then this is epi in reduced rings but not in rings. (The kernel of $L\otimes_K L\to L$ is nilpotent.) | |
| Dec 8, 2021 at 18:19 | history | edited | Carlos Esparza | CC BY-SA 4.0 |
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| Dec 8, 2021 at 13:44 | comment | added | Laurent Moret-Bailly | An epimorphism of rings corresponds to a monomorphism of affine schemes (in the category of affine schemes or, equivalently, of all schemes). Even for reduced affine schemes, this is a stronger condition than injectivity, as the normalization of the cusp demonstrates. | |
| Dec 8, 2021 at 12:19 | comment | added | YCor | I was actually unable to check that an epimorphism in the category of reduced commutative rings is also an epimorphism in the category of all commutative rings. | |
| Dec 8, 2021 at 10:17 | comment | added | Piotr Achinger | Epimorphism in the category of rings? | |
| Dec 8, 2021 at 10:00 | history | edited | Carlos Esparza | CC BY-SA 4.0 |
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| Dec 8, 2021 at 9:41 | history | asked | Carlos Esparza | CC BY-SA 4.0 |