2
votes
1answer
275 views

Frobenius base change of etale maps

Let $A$ be a characteristic $p>0$ commutative ring. Let $B$ be a finitely presented etale $A$ algebra i.e. $$ B=A[x_1,\ldots,x_n]/(f_1,\ldots,f_n), $$ with $det(\frac{\partial f_i}{\partial ...
2
votes
1answer
297 views

About subspaces of $F$-spaces

A topological space $X$ is an $F$-space, if Every finitely generated ideal in the ring of all continuous functions on $X$,denoted by $C(X)$, is principal. The text "Rings of continuous functions" ...
1
vote
0answers
352 views

elementary exact sequence of normal sheaves

Let $Z \subset Y \subset \mathbb{A^n}$ be a smooth subvarieties of $\mathbb{A^n}$. I'm trying to show that there is an exact sequence of normal bundles. $0 \rightarrow N_{Z/Y} \rightarrow N_{Z} ...
4
votes
1answer
425 views

Localizability of differential operators a la Grothendieck

Hello, Maybe this question is trivial, so sorry Let $A$ be a (comm. with 1) $k$-algebra, where $k$ is a ring (comm. with 1). Then we can define the module of differential operators $D^{\leq n} ...
1
vote
1answer
275 views

Presentation of finite modules with null annihilator

Let $R$ be a noetherian local ring and let $M$ be a finite $R$-module. Assume that the annihilator of $M$ is zero. Consider a minimal presentation of M as follows: ...
2
votes
2answers
308 views

vectors with entries from a finite ring

I've been working recently with vectors over finite fields, but I was hoping to work in a more general setting and consider vectors over finite commutative rings. The question I had is as follows: if ...
12
votes
4answers
5k views

Atiyah-MacDonald, exercise 2.11

Let A be a commutative ring with 1 not equal to 0. (The ring A is not necessarily a domain, and is not necessarily noetherian.) Assume we have an injective map of free A-modules A^m -> A^n. Must we ...