# Equivariant Sheaf: Explanation on Stalks

I have a question about the explanation of the data defining a so called equivariant sheaf $$F$$ on a scheme X from wiki: https://en.wikipedia.org/wiki/Equivariant_sheaf. Let denote by $$\sigma: G \times_S X \to X$$ an action of a group scheme $$G$$ on $$X$$ . Then a $$O_X$$-module $$F$$ is called equivariant if there exist in isomorphism $$\phi: \sigma^* F \simeq p_2^*F$$ of $$\mathcal{O}_{G \times_S X}$$-modules and additionally the "cocycle" condition $$p_{23}^* \phi \circ (1_G \times \sigma)^* \phi = (m \times 1_X)^* \phi$$ is satisfied where $$p_{23}, 1_G \times \sigma, m \times 1_X$$ a maps between $$G \times G \times X$$ and $$G \times X$$.

FOLLOWING EXPLANATION I DON'T UNDERSTAND: Then there is said that the cocycle condition tells that on level of stalks the isomorphism $$F_{gh \cdot x} \simeq F_x$$ is the same as $$F_{g \cdot h \cdot x} \simeq F_{h \cdot x} \simeq F_x$$; namely it reflects the associativity. (*)

What I don't understand is why the induced isomorphism $$(m \times 1_X)^* \phi$$ provides a map $$F_{gh \cdot x} \to F_x$$ on the level of stalks?

Namely I don't see why $$F_{gh \cdot x}$$ and $$F_x$$ are correct domain and codomain of this map $$(m \times 1_X)^* \phi$$.

Indeed, the induced isomorphism $$(m \times 1_X)^* \phi$$ is a sheaf iso $$(m \times 1_X)^*\sigma^* F= (\sigma \circ (m \times 1_X))^*F \to (m \times 1_X)^*p_2^*F=(p_2 \circ (m \times 1_X))^*F.$$

Fix a point $$(g,h,x) \in G \times G \times X$$.

Then $$\sigma \circ (m \times 1_X)(g,h,x)= gh \cdot x$$ and $$p_2 \circ (m \times 1_X)(g,h,x)= x$$

It's not clear to me why the stalks at $$(g,h,x)$$ of the domain and codomain are given by $$((\sigma \circ (m \times 1_X))^*F)_{(g,h,x)}= F_{gh \cdot x}$$ and $$((p_2 \circ (m \times 1_X))^*F)_{(g,h,x)}= F_x$$ as stated in (*)?

I learned that generally for a morphism $$f:X \to Y$$ and a sheaf $$F$$ on $$Y$$ we have following formula for the stalk in $$z \in X$$ of the pullback sheaf:

$$(f^*F)_z= O_{X,z} \otimes_{O_{Y,f(z)}} F_{f(z)}.$$

Now we apply this formula it to our situation, therefore $$f=\sigma \circ (m \times 1_X)$$ and $$z=(g,h,z)$$ then we obtain

$$((\sigma \circ (m \times 1_X))^*F)_{(g,h,x)}= O_{G \times G \times X,(g,h,x)} \otimes_{O_{X,gh \cdot x}} F_{gh \cdot x}$$

But (*) says that it should equal $$F_{gh \cdot x}$$. Why? Or where is the error in my reasonings? What happens with the left "factor"?

• The wikipedia article does not claim that the stalk of a sheaf on $G\times G\times X$ equals the stalk of a sheaf on $X$. The wikipedia article claims that the cocycle condition, formulated in terms of sheaves on $G\times G\times X$, implies a stalk condition for sheaves on $X$. – Jason Starr Mar 27 '19 at 12:55
• @JasonStarr: how does the argument that this stalk condition (*) for sheaves on $X$ comes from the cocycle condition work? I don't think that here one can just "neglect the left factor" to obtain this. Is this done by a "composition" argument to come back to $X$ or is it a bit deeper application of descent? – KarlPeter Mar 28 '19 at 0:37
• The morphism $\sigma$ induces local homomorphisms $\sigma_{g,x}^*:\mathcal{O}_{X_R,g\cdot x} \to \mathcal{O}_{X_R,x}$ for every local $\mathcal{O}_S$-algebra $(R,\mathfrak{m},k)$ and $g,x\in G(k)$. Associativity of $\sigma$ implies $\sigma^*_{h,x}\circ \sigma^*_{g,h\cdot x}$ equals $\sigma^*_{gh,x}$ for every $g,h\in G(k)$, $x\in X(k)$. You can reduce (*) to this associativity. Use the cocycle condition to extend the $G$-group action from $X$ to the scheme $X_F$ whose underlying topological space equals $X$ and with structure sheaf $\mathcal{O}_X\oplus F\cdot \epsilon$, $\epsilon^2=0$. – Jason Starr Mar 28 '19 at 9:26

I am just posting my comment as an answer. For a scheme $$S$$, one definition of a group $$S$$-scheme is a datum of $$S$$-schemes, $$(\pi:G\to S, m:G\times_S G\to G, i:G\to G, e:S\to G),$$ of an $$S$$-scheme $$G$$ and $$S$$-morphisms $$m$$, $$i$$, and $$e$$ such that for every $$S$$-scheme $$T$$, the induced datum of sets, $$(G(T),m(T):G(T) \times G(T) \to G(T), i(T):G(T) \to G(T), e(T): \{\text{Id}_T\} \to G(T)),$$ is a group with its group operation, with its group inverse, and with its specified group identity element. In particular, setting $$T$$ equal to $$G\times_S G\times_S G$$ with its three projections to $$G$$, associativity of the group operation implies equality of the compositions $$G\times_S G\times_S G \xrightarrow{\text{Id}_G \times m} G\times_S G \xrightarrow{m} G, \ \ \ \ G\times_S G\times_S G \xrightarrow{m\times_S \text{Id}_G} G\times_S G \xrightarrow{m} G.$$ Similarly, the following compositions are equal, $$G\xrightarrow{\pi} S \xrightarrow{e} G, \ \ G \xrightarrow{\Gamma_i} G\times_S G \xrightarrow{m} G,$$ and the following compositions are equal, $$G\xrightarrow{\text{Id}_G} G, \ \ G \xrightarrow{\Gamma_{\pi\circ e}} G\times_S G \xrightarrow{m} G.$$ Conversely, if each of these pairs of compositions are equal, then each datum of sets above is a group.

In the same way, for an $$S$$-scheme $$\rho:X\to S,$$ for an $$S$$-morphism, $$\sigma:G\times_S X \to X,$$ for every $$S$$-scheme $$T$$, the induced datum of sets, $$\sigma(T):G(T)\times X(T) \to X(T),$$ satisfies the axioms for the action of the group $$G(T)$$ on the set $$X(T)$$ if and only if the following compositions are equal, $$G\times_S G \times_S X \xrightarrow{m\times\text{Id}_X} G\times_S X \xrightarrow{\sigma} X, \ \ G\times_S G \times_S X \xrightarrow{\text{Id}_G \times \sigma} G\times_S X \xrightarrow{\sigma} X,$$ $$X\xrightarrow{\text{Id}_X} X, \ \ X\xrightarrow{\Gamma_{e\circ \rho}}G\times_S X \xrightarrow{\sigma} X.$$

Now consider the special case that $$S$$ equals $$\text{Spec}\ R$$ for a local ring $$(R,\mathfrak{m},k)$$, e.g., $$R$$ might equal the residue field $$k$$ with maximal ideal $$\mathfrak{m}=\{0\}$$. For every $$g\in G(S)$$, there is an induced isomorphism of $$S$$-schemes, $$\sigma_g:X \xrightarrow{\Gamma_{g\circ \rho}}G\times_S X \xrightarrow{\sigma} X.$$ For every point $$x$$ of $$X$$, denote the image point $$\sigma_g(x)$$ by $$g\cdot x$$. Then this isomorphism of schemes induces an isomorphism of stalks, $$\sigma_{g,x}^*:\mathcal{O}_{X,g\cdot x} \to \mathcal{O}_{X,x}.$$ For every pair $$(g,h)\in G(S)\times G(S)$$, the first equality of compositions in the previous paragraph implies that $$\sigma_{gh}$$ equals $$\sigma_g\circ \sigma_h$$. Thus, for every point $$x$$ of $$X$$, also $$\sigma_{gh,x}^*$$ equals $$\sigma_{h,x}^*\circ \sigma_{g,h\cdot x}^*$$.

Finally, for a fixed group $$S$$-scheme $$G$$, the $$S$$-schemes together with an $$S$$-action by $$G$$ form a category (with a forgetful functor to the category of $$S$$-schemes). The $$G$$-equivariant morphisms are defined in the usual way. For every $$S$$-scheme $$X$$, for every quasi-coherent $$\mathcal{O}_X$$-module $$\mathcal{F}$$, there is an associated $$S$$-scheme $$X_\mathcal{F}$$ with a closed immersion and a retraction, both of which are universal homeomorphisms, $$j_{\mathcal{F}}:X\hookrightarrow X_{\mathcal{F}}, \ \ r_{\mathcal{F}}:X_\mathcal{F} \to X,$$ with $$j^{-1}\mathcal{O}_{X_\mathcal{F}}$$ such that the structure sheaf of $$X_{\mathcal{F}}$$ equals the commutative, unital $$\mathcal{O}_X$$-algebra $$\mathcal{O}_X \xrightarrow{\text{Id}\oplus 0} \mathcal{O}_X\oplus \mathcal{F}\cdot \epsilon \xrightarrow{(\text{Id},0)} \mathcal{O}_X.$$ For every $$X$$-scheme $$t:T\to X,$$ the lifts of $$t$$ to an $$X$$-morphism $$\widetilde{t}:T\to X_{\mathcal{F}}$$ are naturally equivalent to the $$\mathcal{O}_T$$-module homomorphisms $$\theta:t^*\mathcal{F}\to \mathcal{O}_T,$$ such that $$\text{Image}(\theta)\cdot \text{Image}(\theta)$$ is the zero ideal sheaf in $$\mathcal{O}_T$$.

The $$G$$-linearizations of $$\mathcal{F}$$ are equivalent to the lifts of the $$G$$-action on $$X$$ to a $$G$$-action on $$X_{\mathcal{F}}$$ such that both the closed immersion and the retraction are $$G$$-equivariant. Indeed, chasing universal properties, the fiber product $$G\times_S X_{\mathcal{F}}$$ as a scheme with a morphism to $$G\times_S X$$ is equivalent to $$(G\times_S X)_{\text{pr}_2^*\mathcal{F}}$$. On the other hand, the pullback of $$X_{\mathcal{F}}$$ by $$\sigma:G\times_S X \to X$$ as a scheme with a projections to $$G\times_S X$$ is equivalent to $$(G\times_S X)_{\sigma^*\mathcal{F}}$$. Thus, a pair of morphisms from $$G\times_S X_{\mathcal{F}}$$ to $$G\times_S X$$ and to $$X_{\mathcal{F}}$$ whose compsitions with $$\sigma$$ and with $$r_{\mathcal{F}}$$ commute is equivalent to a morphism from $$(G\times_S X)_{\text{pr}_2^*\mathcal{F}}$$ to $$(G\times_S X)_{\sigma^*\mathcal{F}}$$. Compatibility with closed immersions forces this morphism to arise from a morphism of quasi-coherent sheaves $$\phi:\sigma^*\mathcal{F} \to \text{pr}_2^*\mathcal{F}.$$ The axioms from a group action hold if and only if $$\phi$$ satisfies the usual axioms for a $$G$$-linearization.

Finally, for the lifted $$G$$-action $$\sigma_\phi$$ on $$X_{\mathcal{F}}$$ associated to a $$G$$-linearization $$\phi$$, the maps of stalks $$(\sigma_\phi)_{g,x}$$ are local homomorphisms $$\mathcal{O}_{X,g\cdot x} \oplus \mathcal{F}_{g\cdot x}\cdot \epsilon \xrightarrow{\cong} \mathcal{O}_{X,x} \oplus \mathcal{F}_x \cdot \epsilon.$$ For these local homomorphisms, the associativity identity $$(\sigma_{\phi})_{gh,x}^* = (\sigma_{\phi})_{h,x}^*\circ (\sigma_{\phi})_{g,h\cdot x}^*,$$ gives the associativity in (*).

• Thank you a lot for your detailed answer. Two points are still unclear: The first one is how does the compatibility with closed immersions contribute to the conclusion that the morphism $(G\times_S X)_{\text{pr}_2^*\mathcal{F}} \to (G\times_S X)_{\sigma^*\mathcal{F}}$ arises from a morphism of quasi-coherent sheaves $\phi:\sigma^*\mathcal{F} \to \text{pr}_2^*\mathcal{F}$? And secondsly concerning the "epsilon"-argument providing (*): Why in the induced iso $\mathcal{F}_{g\cdot x}\cdot \epsilon \xrightarrow{\cong} \mathcal{F}_x \cdot \epsilon$ we can "forget" the epsilon? – KarlPeter Mar 28 '19 at 23:53
• The factor $\epsilon$ is a placeholder that reminds us of the algebra structure. The stalk at $x$ of the sheaf of $\mathcal{O}_X$-algebras equals $\mathcal{O}_{X,x}\oplus \mathcal{F}_x\cdot \epsilon$. As a module over the stalk $\mathcal{O}_{X,x}$, this is just the direct sum of modules: $\epsilon$ plays no role. Regarding the closed immersions, please write this out for yourself. – Jason Starr Mar 29 '19 at 7:01
• Could you give a reference where the universal property/ the natural equivalence of $X_{\mathcal{F}}$ you pointed out is worked out detailed – KarlPeter Sep 19 '19 at 1:08