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Consider the real number field $\mathbb{R}$. Apply the following short exact sequence $$ 0 \to \mu_{2} \to \mathbb{G}_{m} \xrightarrow{(\cdot)^2} \mathbb{G}_{m} \to 0 $$ to $X = \operatorname{Spec} \mathbb{R}$. We obtain that $H_{\text{ét}}^{2}(X, \mu_{2}) = \mathbb{Z}/2\mathbb{Z}$. There is a one-to-one correspondence between a banded $\mu_{2}$-gerbe and an element in $H_{\text{ét}}^{2}(X, \mu_{2})$.

The trivial element in $H^{2}(X, \mu_{2})$ corresponds to the classifying stack $B\mu_{2,\mathbb{R}}$ of $\mu_{2}$ over $\mathbb{R}$.

My question is: for the nontrivial element in $H^{2}(X, \mu_{2})$, what is the corresponding $\mu_{2}$-gerbe $Y$? As a quotient stack, what does $Y$ look like?

Mike
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  • include some details on the "one to one correspondence between banded gerbes and comology classes". Then youll get more response. – hm2020 Jun 22 '25 at 10:18
  • You might find more success on mathoverflow. I suspect people who can provide an answer will not be frequenting mse often. – user1515097 Jun 27 '25 at 03:16
  • @Mike - Include a referece to the correspondence I speak of. – hm2020 Jun 29 '25 at 14:42

1 Answers1

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There is an exact sequence of group schemes $$ 0 \to \mu_2 \to SL_2 \to PGL_2 \to 0 $$ which induces an injection $H^1(K, PGL_2) \hookrightarrow H^2(K, \mu_2)$ for any field $K$. Here we are using Hilbert's Theorem 90 for special linear groups: $H^1(K, SL_n) = 0$. The pointed set $H^1(K, PGL_2)$ classifies $PGL_2$-torsors over $\operatorname{Spec}K$. We can interpret this map geometrically as follows. Given a torsor $P \to \operatorname{Spec} K$, there is naturally an action of $SL_2$ on $P$ via composition $SL_2 \to PGL_2$. Then $[P/SL_2]$ is a $\mu_2$-gerbe.

Now $PGL_2$-torsors are also the same as twisted forms of $\mathbb{P}^1$, i.e. plane conics $C \subset \mathbb{P}^2_{K}$. In the case of the nontrivial element of $H^2(\mathbb{R}, \mu_2)$, this corresponds to the unique (up to isomorphism) conic which is not isomorphic to $\mathbb{P}^1_\mathbb{R}$, namely $C = \{x^2 + y^2 + z^2 = 0\}$. The corresponding $PGL_2$-torsor abstractly can be written as the scheme of isomorphisms $P = \mathrm{Isom}(C, \mathbb{P}^1)$. Then $[P/SL_2]$ where $SL_2$ acts on $\mathbb{P}^1$ is the nontrivial $\mu_2$-gerbe over $\operatorname{Spec} \mathbb{R}$

One should be able to describe $P$ explicitly as an affine scheme over $\mathbb{R}$ from the equation of $C$ but I haven't figured out how yet. I believe it is a closed three dimensional subvariety of $\mathbb{A}^5$.

Dori Bejleri
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  • Do you have a reference to a proof of the correspondence between banded gerbes and etale cohomology classes? – hm2020 Jun 29 '25 at 14:41
  • @hm2020 The original reference is Giraud's "Cohomologie non abelienne" but you could find a summary for example in the appendix to Deligne and Milne, Tannakian Categories: https://www.jmilne.org/math/xnotes/tc.html. – Dori Bejleri Jun 29 '25 at 17:22