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Saal Hardali
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Where $f_+(x,\theta) = \Sigma_{n \ge 0} a_n x^{p^n} + \alpha \theta, f_-(x,\theta) = \Sigma_{n \ge 0} b_n x^{p^n} + \beta \theta$$f_+(x,\theta) = \Sigma_{n \ge 0} a_n x^{p^n} + \alpha \theta, f_-(x,\theta) = \Sigma_{n \ge 0} b_n x^{p^n} + \beta \theta$. We claimed that the correct "strictness" condition is $gr(df) = Id$ where $gr$ is the associated graded w.r.t. the $\mathbb{Z}/2$-filtration. In coordinates this condition translates to:

Where $f_+(x,\theta) = \Sigma_{n \ge 0} a_n x^{p^n} + \alpha \theta, f_-(x,\theta) = \Sigma_{n \ge 0} b_n x^{p^n} + \beta \theta$. We claimed that the correct "strictness" condition is $gr(df) = Id$ where $gr$ is the associated graded w.r.t. the $\mathbb{Z}/2$-filtration. In coordinates this condition translates to:

Where $f_+(x,\theta) = \Sigma_{n \ge 0} a_n x^{p^n} + \alpha \theta, f_-(x,\theta) = \Sigma_{n \ge 0} b_n x^{p^n} + \beta \theta$. We claimed that the correct "strictness" condition is $gr(df) = Id$ where $gr$ is the associated graded w.r.t. the $\mathbb{Z}/2$-filtration. In coordinates this condition translates to:

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Saal Hardali
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$$(f_+(x,\theta), f_-(x,\theta) : Nil(R)^{\text{even}}_x \oplus Nil(R)^{\text{odd}}_{\theta} \to Nil(R)^{\text{even}} \oplus Nil(R)^{\text{odd}} $$$$(f_+(x,\theta), f_-(x,\theta)) : Nil(R)^{\text{even}}_x \oplus Nil(R)^{\text{odd}}_{\theta} \to Nil(R)^{\text{even}} \oplus Nil(R)^{\text{odd}} $$

This is a group for composition. Abstractly this functor can be defined as the functor of strict automorphisms of the $\mathbb{Z}/2$-filtered formal additive super group (where strict now means the same thing as before but we are now only enforcing it on the associated graded):

$$(f_+(x,\theta), f_-(x,\theta) : Nil(R)^{\text{even}}_x \oplus Nil(R)^{\text{odd}}_{\theta} \to Nil(R)^{\text{even}} \oplus Nil(R)^{\text{odd}} $$

This is a group for composition. Abstractly this functor can be defined as the functor of strict automorphisms of the $\mathbb{Z}/2$-filtered formal super group (where strict now means the same thing as before but we are now only enforcing it on the associated graded):

$$(f_+(x,\theta), f_-(x,\theta)) : Nil(R)^{\text{even}}_x \oplus Nil(R)^{\text{odd}}_{\theta} \to Nil(R)^{\text{even}} \oplus Nil(R)^{\text{odd}} $$

This is a group for composition. Abstractly this functor can be defined as the functor of strict automorphisms of the $\mathbb{Z}/2$-filtered formal additive super group (where strict now means the same thing as before but we are now only enforcing it on the associated graded):

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Saal Hardali
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Where $Nil(R)$ is the nilpotent radical of $R$. This is of course the functor of points of $Spec H^{\ast}(B \mathbb{Z}/p)$ by the above discussion.

Similarly to the previous case we can define the functor of points of the automorphisms of $\hat{\mathbb{G}}^{1|1}_{a}$ as

Where $Nil(R)$ is the nilpotent radical of $R$. Similarly to the previous case we can define the functor of points of the automorphisms of $\hat{\mathbb{G}}^{1|1}_{a}$ as

Where $Nil(R)$ is the nilpotent radical of $R$. This is of course the functor of points of $Spec H^{\ast}(B \mathbb{Z}/p)$ by the above discussion.

Similarly to the previous case we can define the functor of points of the automorphisms of $\hat{\mathbb{G}}^{1|1}_{a}$ as

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