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You state no question, so I am not sure what you are asking. The argument Eisenbud seemed to have in mind is flawed, as you noticed. However the result is true: Let $I$ be the maximal ideal of $B$, a …

Sure, by $(1)$ and $(4)$. Any integral morphism is affine by definition. If $f$ is an affine morphism with $f_* \mathcal{O}_X = \mathcal{O}_Y$, then $f$ is clearly an isomorphism.

No (in general). Let $A$ be a non-zero ring and let $S = \mathrm{Spec}(A[T]/(T^2))$. Let $M$ be a free $A$-module of infinite rank, viewed as an $A[T]/(T^2)$-module via the section $A[T]/(T^2) \righta …

No (in general). Take $S = \mathrm{Spec}(A)$ and $X_k = \mathrm{Spec}(A[T]/(T^2))$, with affine transition maps given by $T \mapsto f T$ for some $f \in A$. The limit $X$ is the spectrum of $A \oplus …

You can apply the following statement to $X = \mathbb{P}^1_K$ and $L = O(1)$ when $K$ is a separably closed field. Let $L$ be a line bundle on a reduced connected scheme $X$ such that $H^{0}( …

An $n$-tuple $F=(F_1,\dots,F_n)$ is "unimodular" iff $\exists x \in k^n, F(x) \neq 0$ in $k$. Thus an equivalent question is: if $F =(F_1,\dots,F_n) \in k[X_1,\dots,X_n]^n$ satisfies $J(F) = 1$ and …

This is false. The noetherian local ring $R = \mathbb{F}_3[[X,Y]]/(Y^2 - X^3)$ has dimension one, and if $x,y$ are the images of $X,Y$ in $R$ then the sequence $$ R \supseteq (x,y) \supseteq (x^2,y) …

It is true when $R$ is reduced, without $\mathbb{Z}$-torsion. If your blocks are $(M_{i,j})_{1 \leq i,j \leq n}$ and if $$N = \sum_{\sigma \in \mathfrak{S}_n} \epsilon(\sigma) M_{1,\sigma(1)} \dots M …

Huneke only states an inclusion. But you are right, one gets a slightly stronger statement, with weaker assumptions. Namely, for any ring $R$, and any sequence $x_1,\dots,x_n$ a of elements of $R$, co …

For any ring $A$, let $\mathrm{wEt}_A$ be the category of weakly etale $A$-algebras ; it is a cocomplete category. By a theorem of Van der Kallen, the truncated Witt vector functor $$ W_r : \mathrm{w …

Let $f_i \in \mathbb{F}_q[X_1,\dots,X_n]$ have degree at most $d$ for each $i \in [|1,r|]$, and assume that the affine scheme $$ X = \operatorname{Spec} \mathbb F_q [x_1,\ldots,x_n]/(f_1, \ldots, f_r) …

The field $K((X))$ has infinite transcendence degree over $K$ (if $K$ is at most countable, this just follows from a cardinality argument). Thus we can find a countable family $(t_i)_{i \geq 1}$ of el …

No : just take $A = \mathbb{F}_p$. This is the only counterexample : any complete local sub-algebra $\iota_A \colon A \hookrightarrow {\Bbb F}_p[[X]]$, with $A \neq \mathbb{F}_p$, is noetherian of Kr …

Let $A$ be a commutative ring with identity. I denote by $\Delta_{\leq 1}$ the full subcategory of the simplex category $\Delta$ with objects $[0]$ and $[1]$. Let $B_{\cdot} : \Delta_{\leq 1}^{\mathrm …

Many expositions of Popescu's desingularization theorem indicate that an other proof of this theorem can be found in "Cinq exposés sur la désingularisation" by M. Andre, Ecole Polytechnique Feder …