# QiL'8

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 May3 comment A question regarding linear disjiontness and the degree of a field extensionYes if $L, K$ are algebraic extensions of $k$ (otherwise, please define linear disjointness). Note that $K'\cap L$ is linearly disjoint with $K$ over $k$, so the compositum $(K'\cap L)K$ has degree over $K$ equal to $[K'\cap L : k]$. But $(K'\cap L)K\subseteq K'$.... (this works in any characteristic). May3 reviewed Close Advice Commutation algebra May3 reviewed Approve suggested edit on Is there any prime $p$ such that $(p-1)!+1=p^m$ May3 comment Inverse images under universally injective morphismsOne can also use Stein factorization : $X\to {\bf Spec}(f_*O_X)\to Y$. It is known that the first map has geometrically connected fibers, and the second has geometrically connected fibers by the universally injective hypothesis. May3 revised Inverse images under universally injective morphismsdeleted 90 characters in body May3 comment Inverse images under universally injective morphismsHeidar Svan: write @QiL'8 instead of QiL in your comment if you want me be aware of your comment. May3 comment Inverse images under universally injective morphisms@HeidarSvan: I am very sorry, I confused the morphism with the map $O_Y\to f_*O_X$. Your initial statement is correct. When $\mathrm{Spec}(f_*O_X)\to Y$ is universally injective, it is purely inseparable (see "Algebraic geometry and arithmetic curves", 5.3.13). For any $y\in Y$, $k(y)\to H^0(X_y, O_{X_y})$ is then purely inseparable (op. cit., 5.3.14). So, when you extend to an algebraic closure $\bar{k}$ of $k(y)$, the global sections of the structure sheaf of $X_{\bar{k}}$ is still a local algebra, hence $X_{\bar{k}}$ is connected. May3 comment Class group of $k[x,y,z,w]/(xy-zw)$@xyzzyz: No problem, I just wanted to give the information for arbitrary fields. It is true that all books on toric varieties I know are written for $\mathbb C$. Now can you find out the rational convex polyhedral defining the variety we are interesting here and give its class group by this way ? May3 comment Class group of $k[x,y,z,w]/(xy-zw)$@xyzzyz:Toric varieties are defined for any field and what you said is valid over any field. May2 comment Class group of $k[x,y,z,w]/(xy-zw)$Suppose $n[Z]$ is a principal divisor defined by a rational function $f$ on $X$. Then $f$ is regular and invertible on $U:=X\setminus Z$. Now compute $O_X(U)^{\star}$ (invertible elements, you should find $k^*y^{\mathbb Z}z^{\mathbb Z}$) and observe that the support of $\mathrm{div}(f)$ is then always reducible, hence different from $Z$. May2 comment If a maximal ideal $\mathfrak m$ is flat, then $\dim_k (\mathfrak m/\mathfrak m^2) \leq 1$Your are welcome @Louis. May2 comment The genus of a curve with a group structureif your morphism to $\mathbb P^1$ is injective, over an algebraically closed field of characteristic $0$, it will be an isomorphism because the degree of the morphism will be one. See this question math.stackexchange.com/questions/341281. The easiest proof that $\mathbb P^1$ is not an algebraic group I know is again the answer given by Piotr. May1 comment Intersection of powers of maximal ideals@YACP: thanks, I will have a look when I will back to home. May1 comment Intersection of powers of maximal idealscontinued: for smooth subvarieties, the proof can be easier because locally a smooth subvariety is defined by a part of a system of coordinates, so you can imagine the ambient variety is an affine space with coordinates $t_1, \dots, t_n$ and $\mathfrak p$ generated by $t_1, \dots, t_d$ for some $d\le n$. Then computation in this case is easy. May1 comment Intersection of powers of maximal ideals@Karl: this should be well-known. I just did some computations by myself. For example, in the case $s=2$, if $f$ belongs to the intersection of $m^2$, then its differential $df$ vanishes at all points $V(\mathfrak p)$. Using the case $s=1$ and the fact that cotangent sheaf of the ambient space is locally free, we see that $df$ has coefficients in $\mathfrak p$. "Integrating $df$" then implies that $f\in \mathfrak p^2$. Higher powers are more complicated especially in positive characteristics. Note that locally complete intersection subvarieties include smooth subvarieties... May1 comment Intersection of powers of maximal ideals@YACP: ah sorry, I was referring to your comment. I don't have acccess to Math. Ann. articles for the moment. May1 comment If a maximal ideal $\mathfrak m$ is flat, then $\dim_k (\mathfrak m/\mathfrak m^2) \leq 1$@YACP: thanks for your kind words ! I am kind of burn out at this moment, there are also some other reasons I prefer not to comment on (!) here. May1 comment If a maximal ideal $\mathfrak m$ is flat, then $\dim_k (\mathfrak m/\mathfrak m^2) \leq 1$@Louis: sorry I don't have a link. Try to find the book in your library, the proof I alluded to is in Prop. 3G, p. 21. Projectivity is not needed. And yes, a submodule of a flat module is not necessarily flat. But the proof doesn't rely on this assumption. May1 comment If a maximal ideal $\mathfrak m$ is flat, then $\dim_k (\mathfrak m/\mathfrak m^2) \leq 1$@YACP: I am in comment only mode since a few weeks. May1 comment The genus of a curve with a group structureHow do you prove the finiteness of Aut(C) using just Riemann-Roch ? Your proof of $P^1$ having no group structure is incomplete. You can't pull-back a group law by $f$ (moreover $f$ is never injective).