Equivariant sheaf

<h2 id="notes-on-the-definition">Notes on the definition</h2>
On the stalk level, the cocycle condition says that the isomorphism 
 
 
 
 
 F
 
 g
 h
 ⋅
 x
 
 
 ≃
 
 F
 
 x
 
 
 
 
 {\displaystyle F_{gh\cdot x}\simeq F_{x}}
 
 is the same as the composition 
 
 
 
 
 F
 
 g
 ⋅
 h
 ⋅
 x
 
 
 ≃
 
 F
 
 h
 ⋅
 x
 
 
 ≃
 
 F
 
 x
 
 
 
 
 {\displaystyle F_{g\cdot h\cdot x}\simeq F_{h\cdot x}\simeq F_{x}}
 
; i.e., the associativity of the group action. The condition that the unit of the group acts as the identity is also a consequence: apply 
 
 
 
 (
 e
 ×
 e
 ×
 1
 
 )
 
 ∗
 
 
 ,
 e
 :
 S
 →
 G
 
 
 {\displaystyle (e\times e\times 1)^{*},e:S\to G}
 
 to both sides to get 
 
 
 
 (
 e
 ×
 1
 
 )
 
 ∗
 
 
 ϕ
 ∘
 (
 e
 ×
 1
 
 )
 
 ∗
 
 
 ϕ
 =
 (
 e
 ×
 1
 
 )
 
 ∗
 
 
 ϕ
 
 
 {\displaystyle (e\times 1)^{*}\phi \circ (e\times 1)^{*}\phi =(e\times 1)^{*}\phi }
 
 and so 
 
 
 
 (
 e
 ×
 1
 
 )
 
 ∗
 
 
 ϕ
 
 
 {\displaystyle (e\times 1)^{*}\phi }
 
 is the identity.
Note that 
 
 
 
 ϕ
 
 
 {\displaystyle \phi }
 
 is an additional data; it is "a lift" of the action of G on X to the sheaf F. Moreover, when G is a connected algebraic group, F an invertible sheaf and X is reduced, the cocycle condition is automatic: any isomorphism 
 
 
 
 
 σ
 
 ∗
 
 
 F
 ≃
 
 p
 
 2
 
 
 ∗
 
 
 F
 
 
 {\displaystyle \sigma ^{*}F\simeq p_{2}^{*}F}
 
 automatically satisfies the cocycle condition.<a class="footnote-ref" id="fnref:3" href="#fn:3">3</a>
If the action of G is free, then the notion of an equivariant sheaf simplifies to a sheaf on the quotient X/G, because of the <a href="/facts/Descent_along_torsors/mQsTxno0">descent along torsors</a>.
By <a href="/facts/Yoneda%27s_lemma/j04vXd6x">Yoneda's lemma</a>, to give the structure of an equivariant sheaf to an 
 
 
 
 
 
 
 O
 
 
 
 X
 
 
 
 
 {\displaystyle {\mathcal {O}}_{X}}
 
-module F is the same as to give group homomorphisms for rings R over 
 
 
 
 S
 
 
 {\displaystyle S}
 
,

G
 (
 R
 )
 →
 Aut
 ⁡
 (
 X
 
 ×
 
 S
 
 
 Spec
 ⁡
 R
 ,
 F
 
 ⊗
 
 S
 
 
 R
 )
 
 
 {\displaystyle G(R)\to \operatorname {Aut} (X\times _{S}\operatorname {Spec} R,F\otimes _{S}R)}
 
.<a class="footnote-ref" id="fnref:4" href="#fn:4">4</a>
There is also a definition of equivariant sheaves in terms of <a href="/facts/Simplicial_sheaves/YIbmQrU7">simplicial sheaves</a>. Alternatively, one can define an equivariant sheaf to be an equivariant object in the <a href="/facts/Category_(mathematics)/7WzxUThf">category</a> of, say, <a href="/facts/Coherent_sheaf/h2NoUN7e">coherent sheaves</a>.

<h2 id="linearized-line-bundles">Linearized line bundles</h2>
A structure of an equivariant sheaf on an invertible sheaf or a line bundle is also called a linearization.
Let X be a complete variety over an algebraically closed field acted by a connected reductive group G and L an invertible sheaf on it. If X is normal, then some tensor power 
 
 
 
 
 L
 
 n
 
 
 
 
 {\displaystyle L^{n}}
 
 of L is linearizable.<a class="footnote-ref" id="fnref:5" href="#fn:5">5</a>
Also, if L is very ample and linearized, then there is a G-linear closed immersion from X to 
 
 
 
 
 
 P
 
 
 N
 
 
 
 
 {\displaystyle \mathbf {P} ^{N}}
 
 such that 
 
 
 
 
 
 
 O
 
 
 
 
 
 P
 
 
 N
 
 
 
 
 (
 1
 )
 
 
 {\displaystyle {\mathcal {O}}_{\mathbf {P} ^{N}}(1)}
 
 is linearized and the linearization on L is induced by that of 
 
 
 
 
 
 
 O
 
 
 
 
 
 P
 
 
 N
 
 
 
 
 (
 1
 )
 
 
 {\displaystyle {\mathcal {O}}_{\mathbf {P} ^{N}}(1)}
 
.<a class="footnote-ref" id="fnref:6" href="#fn:6">6</a>
Tensor products and the inverses of linearized invertible sheaves are again linearized in the natural way. Thus, the isomorphism classes of the linearized invertible sheaves on a scheme X form an abelian group. There is a homomorphism to the <a href="/facts/Picard_group/lWH0CoWQ">Picard group</a> of X which forgets the linearization; this homomorphism is neither injective nor surjective in general, and its kernel can be identified with the isomorphism classes of linearizations of the trivial line bundle. 
See Example 2.16 of <a href="https://www-fourier.ujf-grenoble.fr/~mbrion/linearization">[1]</a> for an example of a variety for which most line bundles are not linearizable.

<h2 id="dual-action-on-sections-of-equivariant-sheaves">Dual action on sections of equivariant sheaves</h2>
Given an algebraic group G and a G-equivariant sheaf F on X over a field k, let 
 
 
 
 V
 =
 Γ
 (
 X
 ,
 F
 )
 
 
 {\displaystyle V=\Gamma (X,F)}
 
 be the space of global sections. It then admits the structure of a G-module; i.e., V is a <a href="/facts/Linear_representation/3ydLtaGH">linear representation</a> of G as follows. Writing 
 
 
 
 σ
 :
 G
 ×
 X
 →
 X
 
 
 {\displaystyle \sigma :G\times X\to X}
 
 for the group action, for each g in G and v in V, let

π
        (
        g
        )
        v
        =
        (
        φ
        ∘
        
          σ
          
            ∗
          
        
        )
        (
        v
        )
        (
        
          g
          
            −
            1
          
        
        )
      
    
    {\displaystyle \pi (g)v=(\varphi \circ \sigma ^{*})(v)(g^{-1})}

where 
 
 
 
 
 σ
 
 ∗
 
 
 :
 V
 →
 Γ
 (
 G
 ×
 X
 ,
 
 σ
 
 ∗
 
 
 F
 )
 
 
 {\displaystyle \sigma ^{*}:V\to \Gamma (G\times X,\sigma ^{*}F)}
 
 and 
 
 
 
 φ
 :
 Γ
 (
 G
 ×
 X
 ,
 
 σ
 
 ∗
 
 
 F
 )
 
 
 →
 ∼
 
 
 Γ
 (
 G
 ×
 X
 ,
 
 p
 
 2
 
 
 ∗
 
 
 F
 )
 =
 k
 [
 G
 ]
 
 ⊗
 
 k
 
 
 V
 
 
 {\displaystyle \varphi :\Gamma (G\times X,\sigma ^{*}F){\overset {\sim }{\to }}\Gamma (G\times X,p_{2}^{*}F)=k[G]\otimes _{k}V}
 
 is the isomorphism given by the equivariant-sheaf structure on F. The cocycle condition then ensures that 
 
 
 
 π
 :
 G
 →
 G
 L
 (
 V
 )
 
 
 {\displaystyle \pi :G\to GL(V)}
 
 is a group homomorphism (i.e., 
 
 
 
 π
 
 
 {\displaystyle \pi }
 
 is a representation.)
Example: take 
 
 
 
 X
 =
 G
 ,
 F
 =
 
 
 
 O
 
 
 
 G
 
 
 
 
 {\displaystyle X=G,F={\mathcal {O}}_{G}}
 
 and 
 
 
 
 σ
 =
 
 
 {\displaystyle \sigma =}
 
 the action of G on itself. Then 
 
 
 
 V
 =
 k
 [
 G
 ]
 
 
 {\displaystyle V=k[G]}
 
, 
 
 
 
 (
 φ
 ∘
 
 σ
 
 ∗
 
 
 )
 (
 f
 )
 (
 g
 ,
 h
 )
 =
 f
 (
 g
 h
 )
 
 
 {\displaystyle (\varphi \circ \sigma ^{*})(f)(g,h)=f(gh)}
 
 and

(
 π
 (
 g
 )
 f
 )
 (
 h
 )
 =
 f
 (
 
 g
 
 −
 1
 
 
 h
 )
 
 
 {\displaystyle (\pi (g)f)(h)=f(g^{-1}h)}
 
,
meaning 
 
 
 
 π
 
 
 {\displaystyle \pi }
 
 is the <a href="/facts/Left_regular_representation/dBND2lRj">left regular representation</a> of G.
The representation 
 
 
 
 π
 
 
 {\displaystyle \pi }
 
 defined above is a <a href="/facts/Rational_representation/WNfkDD1G">rational representation</a>: for each vector v in V, there is a finite-dimensional G-submodule of V that contains v.<a class="footnote-ref" id="fnref:7" href="#fn:7">7</a>

<h2 id="equivariant-vector-bundle">Equivariant vector bundle</h2>
A definition is simpler for a vector bundle (i.e., a variety corresponding to a <a href="/facts/Locally_free_sheaf/h2NoUN7e">locally free sheaf</a> of constant rank). We say a vector bundle E on an algebraic variety X acted by an algebraic group G is equivariant if G acts fiberwise: i.e., 
 
 
 
 g
 :
 
 E
 
 x
 
 
 →
 
 E
 
 g
 x
 
 
 
 
 {\displaystyle g:E_{x}\to E_{gx}}
 
 is a "linear" isomorphism of vector spaces.<a class="footnote-ref" id="fnref:8" href="#fn:8">8</a> In other words, an equivariant vector bundle is a pair consisting of a vector bundle and the lifting of the action 
 
 
 
 G
 ×
 X
 →
 X
 
 
 {\displaystyle G\times X\to X}
 
 to that of 
 
 
 
 G
 ×
 E
 →
 E
 
 
 {\displaystyle G\times E\to E}
 
 so that the projection 
 
 
 
 E
 →
 X
 
 
 {\displaystyle E\to X}
 
 is equivariant.
Just like in the non-equivariant setting, one can define an <a href="/facts/Equivariant_characteristic_class/CvsVQcg6">equivariant characteristic class</a> of an equivariant vector bundle.

<h2 id="examples">Examples</h2>
<ul><li>The <a href="/facts/Tangent_bundle/dq8zW9Kb">tangent bundle</a> of a manifold or a smooth variety is an equivariant vector bundle.</li>
<li>The sheaf of <a href="/facts/Equivariant_differential_form/VJnPVK5l">equivariant differential forms</a>.</li>
<li>Let G be a semisimple algebraic group, and λ:H→C a character on a maximal torus H. It extends to a Borel subgroup λ:B→C, giving a one dimensional representation Wλ of B. Then GxWλ is a trivial vector bundle over G on which B acts. The quotient Lλ=GxBWλ by the action of B is a line bundle over the flag variety G/B. Note that G→G/B is a B bundle, so this is just an example of the associated bundle construction. The <a href="/facts/Borel%E2%80%93Weil%E2%80%93Bott_theorem/Y4BXqu5I">Borel–Weil–Bott theorem</a> says that all representations of G arise as the cohomologies of such line bundles.</li>
<li>If X=Spec(A) is an affine scheme, a <a href="/facts/Group-scheme_action/jVo9ubJf">Gm-action</a> on X is the same thing as a Z <a href="/facts/Graded_ring/AIUjkSaP">grading</a> on A. Similarly, a Gm equivariant quasicoherent sheaf on X is the same thing as a Z graded A module.</li></ul>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Equivariant_algebraic_K-theory/IwjhsE9V">Equivariant algebraic K-theory</a></li>
<li><a href="/facts/Equivariant_bundle/Dvfcso44">Equivariant bundle</a></li>
<li><a href="/facts/Equivariant_cohomology/CvsVQcg6">Equivariant cohomology</a></li>
<li><a href="/facts/Quotient_stack/lwme8cqw">Quotient stack</a></li></ul>
<h2 id="notes">Notes</h2>

<ul><li>J. Bernstein, V. Lunts, "Equivariant sheaves and functors," Springer Lecture Notes in Math. 1578 (1994).</li>
<li>Mumford, David; Fogarty, John; Kirwan, Frances (1994). <a href="https://books.google.com/books?id=dFlv3zn_2-gC">Geometric Invariant Theory</a>. Berlin: Springer-Verlag. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-3-540-56963-3. <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=1304906">1304906</a>.</li>
<li>Gaitsgory, D. (2005). <a href="https://web.archive.org/web/20150122111026/http://www.math.harvard.edu/~gaitsgde/267y/catO.pdf">"Geometric Representation theory, Math 267y, Fall 2005"</a> (PDF). Archived from <a href="http://www.math.harvard.edu/~gaitsgde/267y/catO.pdf">the original</a> (PDF) on 22 January 2015.</li>
<li>Thomason, R.W. (1987). "Algebraic K-theory of group scheme actions". In Browser, William (ed.). Algebraic topology and algebraic K-theory : proceedings of a conference, October 24-28, 1983 at Princeton University, dedicated to John C. Moore on his 60th birthday. Vol. 113. Princeton, N.J.: Princeton University Press. p. 539-563. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 9780691084268.</li></ul>
<h2 id="external-links">External links</h2>
<ul><li><a href="https://math.stackexchange.com/q/264796">Equivariant sheaves</a></li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">MFK 1994, Ch 1. § 3. Definition 1.6. - Mumford, David; Fogarty, John; Kirwan, Frances (1994). Geometric Invariant Theory. Berlin: Springer-Verlag. ISBN 978-3-540-56963-3. MR 1304906. <a href="https://books.google.com/books?id=dFlv3zn_2-gC" target="_blank">https://books.google.com/books?id=dFlv3zn_2-gC</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">Gaitsgory 2005, § 6. - Gaitsgory, D. (2005). "Geometric Representation theory, Math 267y, Fall 2005" (PDF). Archived from the original (PDF) on 22 January 2015. <a href="https://web.archive.org/web/20150122111026/http://www.math.harvard.edu/~gaitsgde/267y/catO.pdf" target="_blank">https://web.archive.org/web/20150122111026/http://www.math.harvard.edu/~gaitsgde/267y/catO.pdf</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">MFK 1994, end of the proof of Ch. 1, § 3., Proposition 1.5. - Mumford, David; Fogarty, John; Kirwan, Frances (1994). Geometric Invariant Theory. Berlin: Springer-Verlag. ISBN 978-3-540-56963-3. MR 1304906. <a href="https://books.google.com/books?id=dFlv3zn_2-gC" target="_blank">https://books.google.com/books?id=dFlv3zn_2-gC</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">Thomason 1987, § 1.2. - Thomason, R.W. (1987). "Algebraic K-theory of group scheme actions". In Browser, William (ed.). Algebraic topology and algebraic K-theory : proceedings of a conference, October 24-28, 1983 at Princeton University, dedicated to John C. Moore on his 60th birthday. Vol. 113. Princeton, N.J.: Princeton University Press. p. 539-563. ISBN 9780691084268. <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn:5">MFK 1994, Ch 1. § 3. Corollary 1.6. - Mumford, David; Fogarty, John; Kirwan, Frances (1994). Geometric Invariant Theory. Berlin: Springer-Verlag. ISBN 978-3-540-56963-3. MR 1304906. <a href="https://books.google.com/books?id=dFlv3zn_2-gC" target="_blank">https://books.google.com/books?id=dFlv3zn_2-gC</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
<li id="fn:6">MFK 1994, Ch 1. § 3. Proposition 1.7. - Mumford, David; Fogarty, John; Kirwan, Frances (1994). Geometric Invariant Theory. Berlin: Springer-Verlag. ISBN 978-3-540-56963-3. MR 1304906. <a href="https://books.google.com/books?id=dFlv3zn_2-gC" target="_blank">https://books.google.com/books?id=dFlv3zn_2-gC</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></li>
<li id="fn:7">MFK 1994, Ch. 1. § 1. the lemma just after Definition 1.3. - Mumford, David; Fogarty, John; Kirwan, Frances (1994). Geometric Invariant Theory. Berlin: Springer-Verlag. ISBN 978-3-540-56963-3. MR 1304906. <a href="https://books.google.com/books?id=dFlv3zn_2-gC" target="_blank">https://books.google.com/books?id=dFlv3zn_2-gC</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></li>
<li id="fn:8">If E is viewed as a sheaf, then g needs to be replaced by 
 
 
 
 
 g
 
 −
 1
 
 
 
 
 {\displaystyle g^{-1}}
 
. <a href="#fnref:8" class="footnote-back-ref">↩</a></li>
</ol>

Equivariant sheaf open-in-new

Equivariant sheaf