Examples of Equivariant Sheaves under Group action

Question

I feel it very unintuitive to understand what an equivariant sheaf is. In the simplest example, L/K is a finite Galois extension, G=Gal(L/K), G acts on Spec L, what are the equivariant sheaves on L?

Answer 1

I don't think that the example you chose is the simplest one. It might be better to start with say a vector bundle $\mathcal V \to X$ over some base space $X$. Suppose now that $G$ acts on $X$. A $G$-equivariant structure on $\mathcal V$ is a choice of $G$-action on $\mathcal V$ (assuming it exists) preserving the vector bundle structure and compatible with the given $G$-action on $X$.

E.g. if $G$ acts on a space $X$, and $\mathcal V$ is functorially assigned to $X$, e.g. the tangent bundle or cotangent bundle, then $\mathcal V$ will have a natural $G$-equivariant structure.

Another example: if $X$ is $n$-dimensional projective space, with its usual action of $GL_{n+1}$ (which factors through $PGL_{n+1}$) and $\mathcal V$ is the tautological line bundle, then $X$ is naturally $GL_{n+1}$-equivariant. To see this, note that (if we remove the zero section) then the tautological line bundle over $\mathbb P^n$ is just ${\mathbb C}^{n+1}\setminus \{0\} \to ({\mathbb C}^{n+1}\setminus \{0\} )/ {\mathbb C}^{\times}\cong {\mathbb P}^n$, and $GL_{n+1}$-acts compatibly on the members of this diagram.

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Another way to phrase the same structure on $\mathcal V$ is to say that for each $g\in G$, there is a given isomorphism $\alpha_g: \mathcal V \cong g^\*\mathcal V$ such that for $g,h \in G$, one has $h^*(\alpha_g) \circ \alpha_h = \alpha_{g h}$.

This latter condition can then be translated to a corresponding condition on the sheaf of sections of $\mathcal V$, which finally can be translated to a condition on any sheaf.

What this amounts to is that one has an action of $G$ on the sections of the sheaf (let's call it $\mathcal F$) compatible with the $G$-action on $X$. The only thing one has to be careful about is that since the sheaf may not be determined by its global sections, one has to think about sections over arbitrary open subsets $U$, and then one has to take into account the fact that $U$ may not be $G$-invariant.

So putting it all together, one gets for any $U$ open in $S$ an isomorphism $\alpha_{g,U}: \Gamma(U,\mathcal F) \cong \Gamma(gU,\mathcal F)$ (which is the action of the element $g \in G$), compatible with restriction to open subsets, and such that $\alpha_{g,h U} \circ \alpha_{h,U} = \alpha_{g h, U}.$

Caveat: I hope I have my composition formulas correct, but if I've gotten things tangled up, I'm sure they'll be corrected.

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As for your original question, you should specify in what topology you want to consider your sheaves (Zariski or etale are the two possibilites that come to mind), and perhaps also what kind of sheaves you are thinking about (e.g. locally constant etale, coherent, ... ).

E.g. the structure sheaf of Spec $L$ has a natural equivariant $Gal(L/K)$-structure, just given by the $Gal(L/K)$-action on $L$. (But note that the terminology here is much more complicated than the actual facts being discused: Spec $L$ is just a point, and so the $Gal(L/K)$-action is necessarily trivial on this point, and giving a sheaf is just the same as giving an abelian group (or set, or ..., depending on what sort of sheaves we are discussing). So giving a $Gal(L/K)$-equivariant sheaf is just giving an abelian group (or set, or ... ) with a $Gal(L/K)$-action.)

Added: I see that in the time it took me to write this, it has been clarified that you mean etale sheaves. Note that if you are thinking of quasi-coherent $\mathcal O$-modules, then by (Grothendieck's interpretation of) Hilbert's Thm. 90, any sheaf is actually pulled back from the Zariski site, and so one can just think of Zariski sheaves.

One natural requirement in the $\mathcal O$-module context is to ask that the $Gal(L/K)$-equivariant structure on the $\mathcal O$-module to be compatible with the given $Gal(L/K)$-equivariant structure on $\mathcal O$ (the structure sheaf of Spec $L$). It turns out that in this case any such sheaf is just pulled back from Spec $K$, as I will explain.

By what I've already discussed above, to give a quasi-coherent sheaf on Spec $L$ with a $Gal(L/K)$-equivariant structure, compatible with the $\mathcal O$-module structure and the given equivariant structure on $\mathcal O$, we are just talking about $L$-vector spaces with a semi-linear action of $Gal(L/K)$.

By Hilbert's thm. 90 again, any such vector space is in fact spanned by its $Gal(L/K)$-invariants (which are naturally a $K$-linear subspace), and so the $L$-vector space is of the form $L\otimes_K V$, where $V$ is a $K$-vector space, and the $Gal(L/K)$-action is just given by its action on the first factor.

In terms of sheaves, this says that any such sheaf is pulled back from a quasi-coherent sheaf on Spec $K$.