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A vector space $V$ of dimension $n$ has an associated determinant line $Det(V)$.
An element of $Det(V)$ is represented as a (formal limear combination) of expresstions of the form $v_1 \wedge v_2 \wedge \ldots \wedge v_n$, subject to the usual multilinearity and antisymmetry relations.

I'm wondering what is analog of the above fact/construction in the world of super vector spaces.

Let $V$ be a supervector space of dimension $n|m$. Then there is a line $Ber(V)$ called the Berezinian of $V$, that behaves like a super-determinant.

Here's a naive description of the Berezinian: for $V=V_0\oplus V_1$, it is given by $$Ber(V)=Det(V_0)\otimes Det(V_1)^*.$$ That's clearly not a good description of $Ber(V)$, as it relies on the decomposition of $V$ into even and odd parts, which is not a $GL(V)$-invariant thing to do.

I want to make sure that I don't get non-invariant answers. To ensure that, I'll do things in families (and thus make the question more complicated $-$ sorry for that):

Let $\Lambda=\mathbb R(\theta_1,\ldots,\theta_n)$ \Lambda=\Lambda(\theta_1,\ldots,\theta_n)$be a Grassmann algebra (=exterior algebra) on$n$variables, and let$V$be a$Spec(\Lambda)$-parametrized family of super vector spaces, i.e., a super vector bundle$V\to Spec(\Lambda)$. How can I describe concretely a section of the associated line bundle$Ber(V)\to Spec(\Lambda)$? For those who don't like the above language, I can translate into algebra. Let$\Lambda=\mathbb R(\theta_1,\ldots,\theta_n)$, \Lambda=\Lambda(\theta_1,\ldots,\theta_n)$, and let $V$ be a free $\Lambda$-module on $n$ even generators and $m$ odd generators. How can I describe concretely an even element of the rank one $\Lambda$-module $Ber(V)$?

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A vector space $V$ of dimension $n$ has an associated determinant line $Det(V)$.
An element of $Det(V)$ is represented as a (formal limear combination) of expresstions of the form $v_1 \wedge v_2 \wedge \ldots \wedge v_n$, subject to the usual multilinearity and antisymmetry relations.

I'm wondering what is analog of the above fact/construction in the world of super vector spaces.

Let $V$ be a supervector space of dimension $n|m$. Then there is a line $Ber(V)$ called the Berezinian of $V$, that behaves like a super-determinant.

Here's a naive description of the Berezinian: for $V=V_0\oplus V_1$, it is given by $$Ber(V)=Det(V_0)\otimes Det(V_2)^*.$$ Det(V_1)^*.$$That's clearly not a good description of Ber(V), as it relies on the decomposition of V into even and odd parts, which is not a GL(V)-invariant thing to do. I want to make sure that I don't get non-invariant answers. To ensure that, I'll do things in families (and thus make the question more complicated - sorry for that): Let \Lambda=\mathbb R(\theta_1,\ldots,\theta_n) be a Grassmann algebra, and let V be a Spec(\Lambda)-parametrized family of super vector spaces, i.e., a super vector bundle V\to Spec(\Lambda). How can I describe concretely a section of the associated line bundle Ber(V)\to Spec(\Lambda)? For those who don't like the above language, I can translate into algebra. Let \Lambda=\mathbb R(\theta_1,\ldots,\theta_n), and let V be a free \Lambda-module on n even generators and m odd generators. How can I describe concretely an even element of the rank one \Lambda-module Ber(V)? 2 added 1 characters in body A vector space V of dimension n has an associated determinant line Det(V). An element of Det(V) is represented as a (formal limear combination) of expresstions of the form v_1 \wedge v_2 \wedge \ldots \wedge v_n, subject to the usual multilinearity and antisymmetry relations. I'm wondering what is analog of the above fact/construction in the world of super vector spaces. Let V be a supervector space of dimension n|m. Then there is a line Ber(V) called the Berezinian of V, that behaves like a super-determinant. Here's a naive description of the Berezinian: for V=V_0\oplus V_1, it is given by$$Ber(V)=Det(V_0)\oplus $Ber(V)=Det(V_0)\otimes Det(V_2)^*.$$That's clearly not a good description of$Ber(V)$, as it relies on the decomposition of$V$into even and odd parts, which is not a$GL(V)$-invariant thing to do. I want to make sure that I don't get non-invariant answers. To ensure that, I'll do things in families (and thus make the question more complicated$-$sorry for that): Let$\Lambda=\mathbb R(\theta_1,\ldots,\theta_n)$be a Grassmann algebra, and let$V$be a$Spec(\Lambda)$-parametrized family of super vector spaces, i.e., a super vector bundle$V\to Spec(\Lambda)$. How can I describe concretely a section of the associated line bundle$Ber(V)\to Spec(\Lambda)$? For those who don't like the above language, I can translate into algebra. Let$\Lambda=\mathbb R(\theta_1,\ldots,\theta_n)$, and let$V$be a free$\Lambda$-module on$n$even generators and$m$odd generators. How can I describe concretely an even element of the rank one$\Lambda$-module$Ber(V)\$?

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