I've been working through Auslander, Reiten, and Smalø's Representation Theory of Artin Algebras, and I've gotten stuck on Exercise III.8(c). The larger exercise has me consider the following quiver $Q$:
and the following two quotients of its path algebra over a field $k$: $$ \Lambda_1 := kQ/\left\langle \beta\alpha - \gamma^2, \alpha\beta\right\rangle \\ \Lambda_2 := kQ/\left\langle \beta\alpha - \gamma^2, \gamma^4, \alpha\beta - \alpha\gamma\beta\right\rangle $$ The exercise then asks me to show that these algebras are isomorphic if $\operatorname{char}(k)\neq 2$, but not isomorphic if $\operatorname{char}(k) = 2$. I can write down an isomorphism $\Lambda_1\to \Lambda_2$ by sending $$ \alpha\mapsto \alpha \\ \beta\mapsto \beta - \gamma\beta \\ \gamma\mapsto \gamma - \gamma^2/2 $$ and indeed, this doesn't work in characteristic 2. However, I'm not sure how to approach showing that these algebras are actually not isomorphic in characteristic 2. I can write down bases for the two algebras independent of the characteristic, by using the $\beta\alpha = \gamma^2$ relation to write every element in the form $[\alpha]\gamma^n[\beta]$ and then using the other relations to kill off elements in this form. This leaves $$ \{e_1, \beta, \gamma\beta, \alpha\gamma\beta, e_2, \gamma, \gamma^2, \gamma^3, \alpha, \alpha\gamma\} \text{ for } \Lambda_1 \\ \{e_1, \beta, \gamma\beta, \alpha\beta, e_2, \gamma, \gamma^2, \gamma^3, \alpha, \alpha\gamma\} \text{ for } \Lambda_2 $$ So I can't use the dimension to distinguish them. I also tried writing down projective resolutions of the simple modules, and while the matrices involved were slightly different I couldn't see any essential distinction between them. In particular, both algebras have infinite global dimension and all of the Ext groups between the simple modules behave in the same way, seemingly regardless of characteristic.
Are there other simple invariants which would allow me to show that these algebras aren't isomorphic in characteristic 2?