论文标题
关于超级Koszul综合体和Berezinian的注释
A Note on Super Koszul Complex and the Berezinian
论文作者
论文摘要
我们构建了一个免费的超级共同的$ a $ a $ -module $ v $等级$ p | q $的超级Koszul综合体，并证明其同源性集中在一个学位上，并产生了$ a $的精确分辨率。然后，我们研究了超级Koszul复合物的双重，并表明其同源性也集中在一个度上，同构为$π^{p+q} a $，而$π$均改变了奇偶校验。最后，我们表明，鉴于$ v $的自动形态，超级科苏尔综合体双重偶然的同源类别的诱导转变是由自动态的berezinian乘法给出的，从而将这个同源性小组与berezinian模块相关联。
We construct the super Koszul complex of a free supercommutative $A$-module $V$ of rank $p|q$ and prove that its homology is concentrated in a single degree and it yields an exact resolution of $A$. We then study the dual of the super Koszul complex and show that its homology is concentrated in a single degree as well and isomorphic to $Π^{p+q} A$, with $Π$ the parity changing functor. Finally, we show that, given an automorphism of $V$, the induced transformation on the only non-trivial homology class of the dual of the super Koszul complex is given by the multiplication by the Berezinian of the automorphism, thus relating this homology group with the Berezinian module of $V$.
