# Structure of $\bigwedge^{2}_{\mathbb{Z}}(A)$ with $A$ a local integral domain

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.