Odd Verma's Theorem

We formulate several basic properties of Verma supermodules over regular symmetrizable Kac--Moody Lie superalgebras, exhibiting $\mathfrak{gl}(1|1)$-nature as revealed through changing Borel subalgebras. We investigate variants of Verma modules obtained by changing Borel subalgebras, which enable us to realize the principal block of $\mathfrak{gl}(1|1)$ as an extension-closed abelian subcategory of category $\mathcal{O}$. This phenomenon is precisely formulated in terms of semibricks. On the other hand, by applying the exchange property of odd reflections, we describe compositions of homomorphisms between Verma modules associated with different Borel subalgebras that share the same character. As an application, we refine existing results on the associated varieties and projective dimensions of Verma modules.