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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 …

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 …

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 …

Let $X$ be a smooth projective surface over $\mathbb{C}$. Then there is the exponential sheaf sequence: $$ 0 \rightarrow \mathbb{Z} \rightarrow \mathscr{O}_X \rightarrow \mathscr{O}_X^\times \rightarr …

This question is a follow-up to this question which I asked on MSE. Let $f: X \rightarrow Y$ be a surjective morphism of schemes, and $\mathscr{F}$ a coherent sheaf on $Y$. Are there conditions we c …

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 …

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. …

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 …

In a (necessarily non-Noetherian) integral domain $A$ of (Krull) dimension $1$, is it possible that there is an infinite collection of prime ideals $\mathfrak{p}_i$ such that $\cap_i \mathfrak{p}_i \n …

There is an exercise (p. 88) in Beauville's book Complex Algebraic Surfaces that claims that: For $X$ a smooth complex projective variety, if the Kodaira dimension (defined in this book as the maxi …

A useful criterion for triviality of a line bundle $\mathscr{L}$ on an integral curve $C$ is that the trivial line bundle is the unique line bundle of degree $0$ which admits a global section. This is …

Let $X$ be a geometrically integral (or geometrically reduced and geometrically connected) proper scheme over a finite field $k = \mathbb{F}_q$, so its Picard scheme exists and $\mathrm{Pic}^0_{X/k}$ …

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{ …

Consider a hypersurface $V(f) \subseteq \mathbb{C}^{n+1}$ with an isolated singularity at the origin. If $L := V(f) \cap S^{2n+1}_\epsilon$ is the link of $V(f)$ (with $S^{2n+1}_\epsilon$ a sufficient …