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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 …

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}( …

Let $K$ be a field, and $F_K$ be the fraction field of the polynomial ring $R_K$ in $n^2$ indeterminates $X_{11},X_{12},...,X_{nn}$ over $K$. Now set $A = (X_{ij})_{i,j} \in M_n (F_K)$, and let $\chi_ …

Let $0 \leq m \leq n$ be integers. The group $S_n$ of permutations acts on the ring $\mathbb{Z}[X_1,\dots,X_n]$ by permuting the coordinates, with fixed subring $\mathbb{Z}[\sigma_1,\dots,\sigma_n]$, …

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 ring and consider the ring generated by $A$ and $kn^2$ indeterminates $X_{lij}$ (with $1 \leq l \leq k $ and $1 \leq i,j \leq n$). If $M_l$ is the matrix $( X_{lij} )_{i,j}$, then one can …

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 …

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 …

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 …

Let $S$ be some (fixed) subset of $\mathbb{Z} [X_1, \dots , X_n]$ which contains only homogeneous polynomials, and if $F$ is a field, let $X(F)$ be the set of $ x \in P^{n-1}(F)$ such that $f (x) = 0$ …

Consider a commutative ring $A= ( \mathbb{R}^n , + , \times) $, where $+$ is the usual one. Assume further that $ \times $ is continuous (with respect to the usual topology). Let $H$ be the set of non …

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 …

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 …

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 …