Let $\mathsf{CNL}$ be the category of complete noetherian local rings with (fixed) residue field $\mathbb{F}$. The morphisms are local homomorphisms fixing the residue field. The full subcategory of artinian objects (i,e, artinian complete local rings residue field $\mathbb{F}$) is denoted by $\mathsf{Ar}$.
A homomorphism $f: A \rightarrow B$ in $\mathsf{Ar}$ is called a small extension or small homomorphism, if $f$ is surjective and its kernel $\ker f = (t)$ is a principal ideal, which is annihilated by $\mathfrak{m}_A$, the maximal ideal of $A$. (i.e. $\mathfrak{m}_A (t) = (0)$.)
Given such a small extension, we can obtain an exact sequence $$ 0 \rightarrow tA \rightarrow A \xrightarrow{f} B \rightarrow 0. $$
My Question 1: How to prove (or disprove) the following claim: For ANY object $A$ in $\mathsf{Ar}$, there exists a small extension $f: A \rightarrow A_0$ for some $A_0 \in \mathsf{Ar}$ with $\ell(A) > \ell(A_0)$, where $\ell(R)$ is the length of the artinian ring $R$.
Motivation: The claim is used in proofs by induction in many contexts, such as the Appendix A on Page 106 of this one : Galois Deformations and Hecke Curves by B. Mazur.
My question 2: Does every surjective homomorphisms in $\mathsf{Ar}$ factors as the composition of small homomorphisms? I think this is related to the construction of Question 1 but not sure about this.
Thank you all for anwering or commenting this post and sorry for any mistakes!
For question 1: the only restriction seems to be that $A \neq \mathbb{F}$. $A$ is artinian, so, if $\mathfrak{m}$ is its maximal ideal, there is a minimal integer $p \geq 2$ such that $\mathfrak{m}^p=\{0\}$. Take some nonzero $t \in \mathfrak{m}^{p-1}$ and define $A_0=A/tA$.
Question 2 asserts that any local surjection $s: A \rightarrow B$ of complete Noetherian Artinian local rings with residue field $F$ is a product of small extensions.
So consider such a surjection $s: A \rightarrow B$. Let’s show that if $s$ isn’t an isomorphism, then it factors through a small extension $A \rightarrow C$. Indeed, let $I$ be the kernel of $s$, and let $t \in I$ be nonzero (else it’s an empty product). There is $p \geq 1$ minimal such that $t\mathfrak{m}^p=\{0\}$, so choose $q \in t\mathfrak{m}^{p-1}$ nonzero and take $C=A/(q)$.
Now, repeating this with $C \rightarrow B$, we know that there is a sequence of small extensions $A \rightarrow C_1 \rightarrow C_2 \rightarrow C_3 \rightarrow \ldots \rightarrow C_n$ that is a factor of $s$. In particular (I hope I’m not using it correctly) if $B$ wasn’t reached at some point, $\ell(A) \geq n+\ell(B)$ and we get a contradiction if $n$ is too large. So the statement should be true.