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Let $f\colon X \rightarrow S$ be a semistable curve of genus $g \ge 0$. Being a semistable curve means that $f$ is a morphism of schemes such that $f$ is proper, flat, and of finite presentation; Th …

Let $X$ be a normal Noetherian algebraic space and $\mathscr{L}$ a line bundle on $X$. If $X$ is a scheme, then there is locally principal Weil divisor on $X$ that gives rise to $\mathscr{L}$. Is the …

Let $S$ be a scheme and $A$ an abelian $S$-scheme, i.e., $A \rightarrow S$ is a proper smooth $S$-group scheme whose fibers are $g$-dimensional abelian varieties. Suppose that one has a fiberwise dens …

My question is about the proof of Example 8 in section 9.2 of the book "Neron models." There we have a semistable curve $X$ over an algebraically closed field $K$ and we let $\pi\colon \widetilde{X} \ …

I am trying to understand the first sentence of the proof of 9.1/5 in "Neron models." There we have a proper curve $X$ over a field $K$ and a line bundle $\mathscr{L}$ on $X$. Our ultimate goal is to …

My question is about the proof of 8.4/2 in "Neron models." The claim is that if $f\colon X \rightarrow S$ is a proper flat morphism of finite presentation such that $H^2(X_s, \mathscr{O}_{X_s}) = 0$ f …

Let $R$ be a discrete valuation ring with a uniformizer $\pi$ and $(A, \mathfrak{m}_A)$ a local $R$-algebra that is essentially of finite type (i.e., is a localization of a finite type $R$-algebra), s …

Let $R$ be a Noetherian ring and let $X$ be a finite type, separated $R$-scheme that is normal and integral. Let $x_1, \dotsc, x_n \in X$ be points of height $1$. Does there exist an open affine $U \s …

Let $S$ be a locally noetherian scheme and let $G$ be a separated and smooth $S$-group algebraic space of finite presentation. Does there exist an open sub-(group algebraic space) $G^0 \subset G$ char …

My question is about the proof of Lemma D.3 on p. 147 of the book "Neron models" by Bosch, Lutkebohmert, and Raynaud. Namely, towards the end of that proof there is the sentence "That $\varphi$ satisf …

Let $f\colon X \rightarrow Y$ be a proper morphism with $Y$ Noetherian (and even affine, if you wish), and let $\mathscr{A} = \bigoplus_{n \ge 0} \mathscr{A}_n$ be a quasi-coherent graded $\mathscr{ …

Recall that a discrete valuation ring $R$ is excellent if the extension $\widehat{K}/K$ is separable, where $\widehat{R}$ is the completion of $R$ (with respect to the maximal ideal), $K = \mathrm{Fra …

Let $f\colon X \rightarrow S$ be a proper morphism of schemes. Is the cohomology group $H^2(S, f_* \mathbb{G}_m)$ the same regardless of whether it is computed in the etale or the fppf topology? And i …

Let $X$ be a scheme and let $\mathscr{I} \subset \mathscr{O}_X$ be a quasi-coherent sheaf of ideals. Suppose that for each $x \in X$, the stalk $\mathscr{I}_x$ is generated by an element $f_x \in \mat …

I am trying to understand Example 7.5/9 from the book "Neron models". There one has a discrete valuation ring $R'$ that is the localization of $\mathbb{Z}[\zeta_p]$ at $p$, so that the absolute ramifi …