I am trying to see the structure of $\bigwedge^{2}_{\mathbb{Z}}(A)$ where $A$ is a local integral domain with small residue field.
Let $A$ be a local integral domain with maximal ideal $M$, residue field $k$ and its group of units $A^{\times}$. Now we have that $A$ is an $A^{\times}$-module where the action is multiplication by a square i.e. $u\cdot a= u^{2}a$ for all $u\in A^{\times}$. Then if we consider $\bigwedge^{2}_{\mathbb{Z}}(A)$, we have this induces an action of an $A^{\times}$-module on $\bigwedge^{2}_{\mathbb{Z}}(A)$ , which is given by
$$u\cdot(a\wedge b)=u^{2}a\wedge u^{2}b$$.
I am considering the next quotient
$$\left(\bigwedge^{2}_{\mathbb{Z}}(A)\right)_{A^{\times}}= \frac{\bigwedge^{2}_{\mathbb{Z}}(A) }{I}$$
where $I=\langle u^{2}a\wedge u^{2}b - a\wedge b\; |\; u\in A^{\times}\; a,b\in A\rangle$ .
Since $A$ is a local integral domain I am trying to figure out whether the quotient is zero or not. My first idea is checking if the ideal $I$ has an unit and if that depends on the size of the residue field $k$.
I was trying some examples of local integral domains $A$ to see if there is something.
For $A=\mathbb{Z}_{p}$ ($p$-adic integers) then, if $u\in A^{\times}\cap\mathbb{Z}$, I have that the ideal $I$ contains elements of the form $(u^{4}-1)(a\wedge b)$. Thus if $u^{4}-1\in A^{\times}$, then the quotient is zero.
If $p\geq 7$, there exist $u\in A^{\times}\cap\mathbb{Z}$ such that $u^{4}-1$ is a unit in $A^{\times}$(take $u=2$). If $p\leq5$ there is not $u\in A^{\times}\cap \mathbb{Z}$ such that $u^{4}-1$ is a unit, since I get that $u^{4}-1\in M$. However I can not conclude that there are not units in $I$.
For $A=\mathbb{Z}_{(p)}$ (localization at $p$). Similarly to the $p$-adic integers, if $u=\frac{x}{y}\in A^{\times}$, then I have to see whether or not $x^{4}-y^{4}$ is a unit in $\mathbb{Z}_{(p)}$. That only happens when $p\geq 7$ .
For that reason, I am trying to check others examples of local integral domains where the size of residue fields is less or equal to $5$. For example the following fields $A=\mathbb{F}_{2}$, $\mathbb{F}_{3}$, $\mathbb{F}_{4}$ in those cases through a straightforward calculation, it is easy to check that $I=0$ and but in some cases, the quotient is not zero.
So any idea or hint might be helpful or where can I read about this in order to give some ideas about this problem?
Thank you for your time!
