Search Results

These notes, from a course Fargues taught at Chicago and transcribed by Sean Howe, are very nice and make a very strong effort to motivate this conjecture and the surrounding theory by analogy with 'h …

Does anyone know an example of an integral scheme $X$ over a field $k$ such that $X_{\overline{k}}$ is connected but reducible? Does it make a difference if $k$ is perfect, or if we ask for $X_{\overl …

Let $G$ be a reductive group scheme over some base $X$ and $P \subseteq G$ a parabolic subgroup. To a $P$-torsor $\mathscr{E}_P$, we may associate a $G$-torsor $\mathscr{E} = G \times^P \mathscr{E}_P$ …

Thanks to Laurent Moret-Bailly for pointing out that I missed a crucial hypothesis! Now I can construct the quasi-inverse, which I'll record below in case some future person is confused by the same pr …

You should read the introduction to Bhargav Bhatt's lecture notes on prismatic cohomology: available here. This is a new cohomology theory introduced by Bhatt-Scholze (closely related to prior work by …

As pointed out in the comments, this is false for general bases. Let $k$ be a field, $S = \mathrm{Spec}(k[t])$, let $\overline{X} = \mathbb{P}^1 \times_{\mathrm{Spec} k} S$ with projection $\overline{ …

Xinwen Zhu has fantastic notes on all sorts of affine Grassmannians from the point of view of algebraic geometry: see here. (You can take your base field to be $\mathbf{C}$ everywhere, and some of the …

First, note that the category of finite length modules on a noetherian local ring $(A, \mathfrak{m})$ is equivalent to the direct limit of the categories of finitely generated modules on $A/\mathfrak{ …

Yes, this is true. A group scheme over a field is smooth if and only if it is geometrically reduced, so the hypotheses ensure that $N$ is smooth. You can even allow $G$ and $G'$ to be arbitrary group …

The Beauville-Laszlo theorem holds in much greater generality - see Tag 0BNI on the Stacks Project. Let $A$ be any ring and let $f\in A$ be a non-zero divisor. Then the category of $f$-torsion free $ …

Let $K$ be a field of characteristic $0$, let $A = K[[t_1, \ldots, t_n]]$ be a power series ring over $K$, and let $V$ be a free $A$-module. Let $\nabla \colon V \rightarrow V \otimes_A \Omega^1_{A/K} …

You can see pretty easily that the angle Großencharacter appearing in Hecke's equidistribution theorem cannot arise as the Großencharacter associated to a CM elliptic curve just by thinking about $\in …

First of all, there's no need to use flat cohomology here. By Theorem III.3.9 in Milne's Etale Cohomology, the canonical map $H^i_{\mathrm{et}}(X, G) \rightarrow H^i_{\mathrm{fppf}}(X, G)$ is an isomo …

Let $\mathscr{X}$ be a smooth proper DM stack over a field $k$ (perhaps assumed to be separably closed and/or of char. $0$) and let $\pi \colon \mathscr{X} \rightarrow X$ be its coarse moduli space. …

I'm curious if it is possible to formulate cup products and prove that they exist in a general way which would subsume a lot of examples: e.g. group cohomology, sheaf cohomology for sheaves on topolog …