aa r X i v : . [ m a t h . R A ] F e b A NOTE ON TWISTED GROUP RINGS ANDSEMILINEARIZATION
THOMAS BRAZELTON
Abstract.
In this short note, we construct a right adjoint to the functor whichassociates to a ring R equipped with a group action its twisted group ring . Thisright adjoint admits an interpretation as semilinearization , in that it sends an R -module to the group of semilinear R -module automorphisms of the module. Asan immediate corollary, we provide a novel proof of the classical observation thatmodules over a twisted group ring are modules over the base ring together with asemilinear action. Introduction
Twisted group rings, or skew group rings , are classical algebraic objects which providea way to incorporate a group action on a ring into the multiplication on a group ring.These rings are classical enough to evade any attempt to pin down their origin, butthey appear as early as the 1970’s, where Handelman, Lawrence, and Schelter beganto establish their theory [HLS78]. The classical question of which properties of ringsand groups extend to properties on the group ring has been a fruitful direction ofresearch (see for example [Lam66, Appendix 2]), and the same can be said for twistedgroup rings — early examples include the theses of Chen [Che78] and Park [Par78],as well as [FM78, Ost79].Twisted group rings are ubiquitous in modern algebra, appearing in fields as diverseas the representation theory of Lie algebras [McC75] to Heisenberg categorification[RS17]. In equivariant homotopy theory, the algebraic K -theory of twisted grouprings arises when taking fixed points of the equivariant algebraic K -theory spectrum[Mer17]. Upcoming work of the author computes the homotopy groups of this spec-trum as Mackey functors of K -groups of twisted group rings [Bra21], which servedas the impetus for the work here.This note provides a concise introduction to twisted group rings. In particular, wesee that the association of a twisted group ring to a group action is functorial, andadmits a right adjoint. Our main result is as follows. wisted group rings and semilinearization Thomas Brazelton Theorem 1.1. (As Theorem 3.5) For any ring R , there is an adjunction of slicecategories Grp / Aut( R ) ⇄ R/ Ring , where the left adjoint sends a group homomorphism G → Aut( R ) to its twistedgroup ring.We refer to the right adjoint as semilinearization ; if R → End Ab ( M ) determinesan R -module, the right adjoint sends it to the group of semilinear R -module auto-morphisms of M . As an immediate corollary of the above theorem, we recover theclassical observation that modules over a twisted group ring are modules over thebase ring equipped with a semilinear action (Corollary 3.6). A remark on related literature 1.2.
In the work of Fechete and Fechete, theauthors present a right adjoint to the functor (id
Grp / Aut) → Ring from the commacategory of groups over automorphisms of rings, associating to a group action σ : G → Aut( R ) the group ring R σ [ G ] [FF09]. Our approach is morally different, as weview the construction of a twisted group ring over R as naturally valued in R -algebras.In particular our right adjoint is quite different than the one constructed by Fecheteand Fechete, and we interpret it in the context of semilinear module automorphisms.As another point, we would like to draw the attention of the reader to excellentexposition found in the unpublished thesis of Edward Poon on categorical aspects oftwisted group rings [Poo16].2. Semilinear G -actions Definition 2.1.
Let R be a ring, and φ ∈ Aut
Ring ( R ) a ring automorphism of R .We define a φ - semilinear R -module homomorphism f : M → N to be a functionsatisfying(1) f ( m + m ′ ) = f ( m ) + f ( m ′ ) for all m, m ′ ∈ M ,(2) f ( rm ) = φ ( r ) · f ( m ) for all r ∈ R and m ∈ M .In particular, semilinear isomorphisms are precisely those maps which are underlainby bijections.The collection of R -modules with φ -semilinear maps does not form a category unless φ is trivial (we may see that there is no good notion of identity morphism). We may,however, create a category by considering all semilinear maps, that is, all morphismswhich are φ -semilinear for some choice of φ . We see that identity morphisms are id R -semilinear, and that morphisms compose; given f : M → N which is φ -semilinear and g : N → P which is ω -semilinear, then gf is ( ω ◦ φ )-semilinear. We let Mod semi ( R )2 wisted group rings and semilinearization Thomas Brazelton denote this category. Note that Mod ( R ) ⊆ Mod semi ( R ) is the subcategory of id R -semilinear morphisms. Remark 2.2.
There is always a forgetful functor
Mod semi ( R ) → Aut
Ring ( R ) , given by sending a φ -semilinear morphism f to φ . In particular, this induces a grouphomomorphism U : Aut Mod semi ( R ) ( M ) → Aut
Ring ( R ) for any M ∈ Mod ( R ). Definition 2.3.
Let R be a ring. A semilinear G -action on an R -module M is agroup homomorphism G → Aut
Mod semi ( R ) ( M ).Such a semilinear action determines a unique G -ring structure on R by post-compositionwith the natural homomorphism U : Aut Mod semi ( R ) ( M ) → Aut
Ring ( R ). In practice, wewill care about the case where R is already equipped with a G -ring structure viasome group homomorphism θ : G → Aut
Ring ( R ). In this setting, the definition of asemilinear G -action is given by an appropriate choice of lift of θ along U . Definition 2.4.
Let R be a G -ring. A semilinear G -action on an R -module M is agroup homomorphism f : G → Aut
Mod semi ( R ) ( M ) making the diagram commute G Aut
Mod semi ( R ) ( M ) Aut Ring ( R ) , f θU where θ is the G -action on R . Explicitly, this is the data of a θ g -semilinear modulehomomorphism f ( g ) for every g ∈ G so that f ( e ) is the identity and f ( gh ) = f ( g ) f ( h )is θ gh -semilinear.3. Twisted group rings and semilinearization
Given an R -module M with multiplication χ : R → End Ab ( M ), we can functorially de-termine its group of semilinear R -module automorphisms. We define semi R (End Ab ( M ))to be the following group:semi R (End Ab ( M )) := (cid:8) ( σ, φ ) ∈ End Ab ( M ) × × Aut
Ring ( R ) | σ ◦ χ ( r ) = χ ( φ ( r )) · σ ∀ r ∈ R (cid:9) = [ φ ∈ Aut
Ring ( R ) { φ -semilinear R -module automorphisms of M } = Aut Mod semi ( R ) ( M ) . wisted group rings and semilinearization Thomas Brazelton Briefly forgetting that End Ab ( M ) is an endomorphism ring of an abelian group, wecan replicate the construction above for an arbitrary ring S , provided that S comeswith of a ring homomorphism from R . This motivates the following definition. Definition 3.1.
Let R be a ring, and suppose that f : R → S is an object of theslice category R/ Ring . Then we define the group of semilinear inner automorphismsunder R assemi R ( S ) := (cid:8) ( s, φ ) ∈ S × × Aut
Ring ( R ) | sf ( r ) = f ( φ ( r )) s for each r ∈ R (cid:9) . One may easily verify that this is a group, where multiplication occurs diagonally as( s, φ ) · ( s ′ , ψ ) := ( ss ′ , φψ ). We remark that semi R ( S ) has a forgetful group homomor-phism to Aut Ring ( R ). Proposition 3.2.
For a ring R , there is a functor of slice categoriessemi R : R/ Ring → Grp / Aut
Ring ( R ) , which we call semilinearization . Proof.
Suppose we have a morphism in R/ Ring of the form
RS T. f gh
Then we define semi R ( h ) : semi R ( S ) → semi R ( T )( s, φ ) ( h ( s ) , φ ) . To verify that ( h ( s ) , φ ) ∈ semi R ( T ), we see that h ( s ) g ( r ) = h ( s ) h ( f ( r )) = h ( sf ( r )) = h ( f φ ( r ) s ) = g ( φ ( r )) h ( s ) , for any r ∈ R . It is clear that semi R ( h ) is a group homomorphism, and moreoverthat it commutes with the forgetful maps from semi R ( S ) and semi R ( T ) to Aut Ring ( R ).It is straightforward to check that semi R ( − ) preserves identities, composition, andassociativity, and thus defines a functor. (cid:3) Definition 3.3.
Let R be a ring, and let θ : H → Aut
Ring ( R ) be an element of theslice category Grp / Aut
Ring ( R ). The twisted group ring R θ [ H ] has the same elementsas the group ring R [ H ], but the multiplication is twisted by θ in the following way:( r h ) · ( r h ) := r θ h ( r ) h h , where we understand that this definition extends additively. The ring R θ [ H ] comesequipped with a natural ring homomorphism R → R θ [ H ] sending r to r H .4 wisted group rings and semilinearization Thomas Brazelton Proposition 3.4.
The assignment of a twisted group ring to a group action assem-bles into a functor twist R : Grp / Aut
Ring ( R ) → R/ Ring , which we refer to as twistification . Proof.
To define twist R on morphisms, suppose we have a morphism in Grp / Aut
Ring ( R ): G K
Aut
Ring ( R ) , fθ ψ meaning that θ g ( r ) = ψ f ( g ) ( r ) for any r ∈ R . Then there is a function F : R θ [ G ] → R ψ [ H ] sending rg rf ( g ). We verify that F is a ring homomorphism by observingthat F (( r g ) · ( r g )) = F ( r θ g ( r ) g g ) = r ψ f ( g ) ( r ) f ( g ) f ( g )= ( r f ( g )) · ( r f ( g )) = F ( r g ) · F ( r g ) . It is immediate to check this assignment is functorial. (cid:3)
Theorem 3.5.
Twistification and semilinearization define an adjunctiontwist R : Grp / Aut
Ring ( R ) ⇆ R/ Ring : semi R . The final section of this note is dedicated to the proof of Theorem 3.5. We willinclude one corollary of the natural bijection associated to this adjunction.
Corollary 3.6. (Modules over a twisted group ring are R -modules with semilinear G -action) Let R be a G -ring. Then for any R -module End Ab ( M ) ∈ R/ Ring , there isa natural isomorphismHom R/ Ring ( R θ [ G ] , End Ab ( M )) ∼ = Hom Grp / Aut
Ring ( R ) (cid:0) G, Aut
Mod semi ( R ) ( M ) (cid:1) . The left side describes extensions of the R -module structure on M to an R θ [ G ]-module structure, while the right side describes the possible semilinear G -actions on M compatible with the G -action on R .4. Proof of Theorem 3.5
In order to verify the natural bijection for the adjunction in Theorem 3.5, we willtreat the bijection and naturality separately for ease of reading.5 wisted group rings and semilinearization Thomas Brazelton
Proposition 4.1.
For any ring homomorphism χ : R → S and group homomorphism θ : G → Aut
Ring ( R ) there is a bijectionΠ : Hom R/ Ring ( R θ [ G ] , S ) ∼ −→ Hom
Grp / Aut
Ring ( R ) ( G, semi R ( S )) , given by sending a ring homomorphism f : R θ [ G ] → S to the group homomorphism g ( f (1 R g ) , θ g ). Proof.
We first check that ( f (1 R g ) , θ g ) ∈ semi R ( S ) for any g ∈ G . That is, we mustsee that f (1 R g ) χ ( r ) is equal to χ ( θ g ( r )) f (1 R g ) for any r ∈ R . Recall that f is ahomomorphism under R , meaning that f ( re G ) = χ ( r ) for any r ∈ R . Thus f (1 R g ) χ ( r ) = f (1 R g ) f ( re G ) = f ((1 R g ) · ( re G )) = f (1 R θ g ( r ) ge G )= f ( θ g ( r ) g ) = f (( θ g ( r ) e G ) · (1 R g ))= χ ( θ g ( r )) f (1 R g ) . We observe that the function g ( f (1 R g ) , θ g ) is a group homomorphism since θ gg ′ = θ g θ g ′ , and f (1 R gg ′ ) = f (1 R g ) f (1 R g ′ ). Finally, we remark that Π( f ) is a grouphomomorphism over Aut Ring ( R ), so Π is well-defined.Next we check that Π is injective. Suppose f and h are two ring homomorphisms R θ [ G ] → S under R with the property that Π( f ) = Π( h ). By definition, we see that f ( re G ) = χ ( r ) = h ( re G ) for each r ∈ R . For any r ∈ R and g ∈ G , we have that( re G ) · (1 R g ) = rθ e G (1 R ) e G g = rg . Thus f ( rg ) = f ( re G ) f (1 R g ), so it suffices for us tocheck that f (1 R g ) = h (1 R g ) for each g ∈ G . This is clearly true when Π( f ) = Π( h ).So f and h agree, and hence Π is injective.Finally we verify that Π is surjective. As semi R ( S ) is by definition a subset of theproduct S × × Aut
Ring ( R ), it comes equipped with natural projection maps S × semi R ( S ) Aut Ring ( R ) , π π s which are group homomorphisms as semi R ( S ) is endowed with diagonal multiplica-tion. Suppose that α : G → semi R ( S ) is any group homomorphism over Aut Ring ( R ).Then π s ( α ( g )) = θ g , so α is really determined by the data of πα : G → S × . Weobserve that(2) ( πα ( g )) χ ( r ) = χ ( θ g ( r ))( πα ( g )) for any r ∈ R, since α ( g ) = ( πα ( g ) , θ g ) is an element of semi R ( S ).Our goal is to concoct a ring homomorphism f : R θ [ G ] → S in R/ Ring with theproperty that Π( f ) = α . Such an f must satisfy(1) f ( re G ) = χ ( r ) since f is a morphism under R wisted group rings and semilinearization Thomas Brazelton (2) f (1 R g ) = πα ( g ) since Π( f ) = α .As we notice that f ( rg ) = f (( re G ) · (1 R g )) for any r and g , it makes sense to define f to be the function f : R θ [ G ] → Srg χ ( r ) πα ( g ) . If we can verify that f is a ring homomorphism under R , then Π( f ) = α , andwe will have verified that Π is surjective. Clearly f preserves multiplicative andadditive identities, and preserves addition by definition, so it suffices to check it ismultiplicative. Let r , r ∈ R , and g , g ∈ G . Then f (( r g )( r g )) = f ( r θ g ( r ) g g ) = χ ( r θ g ( r )) πα ( g g )= χ ( r ) χ ( θ g ( r )) πα ( g ) πα ( g ) . By Equation 2, we see that the above is equal to χ ( r ) ( χ ( θ g ( r )) πα ( g )) πα ( g ) = ( χ ( r ) πα ( g )) ( χ ( r ) πα ( g )) = f ( r g ) f ( r g ) . (cid:3) Proposition 4.3.
The function Π is a natural bijection.
Proof.
Let j : G → K be a group homomorphism over Aut( R ) and let h : S → T be a ring homomorphism under R . For naturality, it suffices to verify that the leftsquare commutes if and only if the right square commutes:(4) R θ [ G ] SR ψ [ K ] T twist R ( j ) µ hλ ⇐⇒ G semi R ( S ) K semi R ( T ) . j Π( µ ) semi R ( h )Π( λ ) As we are asking for commutativity in a slice category, we know that the left handsquare already commutes on certain objects of R θ [ G ], namely those arriving thestructure map from R , which are elements of the form re G . Thus h ( µ ( re G )) = λ (twist R ( j )( re G )) = λ ( rj ( e G )) = λ ( re K ) , for all r ∈ R . Since this is a diagram of ring homomorphisms, the question ofcommutativity reduces to a question of commutativity on the elements in R θ [ G ] ofthe form 1 R g for g ∈ G .A directly analogous statement is true about the right square in Equation 4. As thisis a diagram over Aut Ring ( R ), we do not need to worry about compatibility withprojection maps π s : semi R ( S ) → Aut
Ring ( R ). Thus we can restrict our attention to7 wisted group rings and semilinearization Thomas Brazelton π semi R ( S ) ⊆ S × . Phrased differently, it suffices to verify that the following diagramcommutes G S × K T × . π Π( µ ) j h | S × π Π( λ ) As Π( µ )( g ) = ( µ (1 R g ) , θ g ), we see that π Π( µ ) = µ (1 R − ), and similarly π Π( λ ) = λ (1 R − ). Therefore the above diagram can be rewritten as G S × K T × . µ (1 R − ) j hλ (1 R − ) The commutativity of this diagram is equivalent to the condition that hµ and λj agree on elements of the form 1 R g , which as we have already seen is equivalent tothe commutativity of the left square in Equation 4. (cid:3) Acknowledgements
The author would like to thank Mona Merling for guidance throughout this note,and Maxine Calle for helpful edits. The author is supported by an NSF GraduateResearch Fellowship (DGE-1845298).
References [Bra21] Thomas Brazelton. Homotopy groups of equivariant algebraic K-theory.
In progress , 2021.1[Che78] Nan-Hung Chen.
Global dimension of skew group rings . ProQuest LLC, Ann Arbor, MI,1978. Thesis (Ph.D.)–Rutgers The State University of New Jersey - New Brunswick. 1[FF09] I. Fechete and D. Fechete. Some categorial aspects of the skew group rings.
An. Univ.Oradea Fasc. Mat. , 16:197–207, 2009. 2[FM78] Joe W. Fisher and Susan Montgomery. Semiprime skew group rings.
J. Algebra , 52(1):241–247, 1978. 1[HLS78] David Handelman, John Lawrence, and William Schelter. Skew group rings.
Houston J.Math. , 4(2):175–198, 1978. 1[Lam66] Joachim Lambek.
Lectures on rings and modules . With an appendix by Ian G. Connell.Blaisdell Publishing Co. Ginn and Co., Waltham, Mass.-Toronto, Ont.-London, 1966. 1[McC75] J. C. McConnell. Representations of solvable Lie algebras. II. Twisted group rings.
Ann.Sci. ´Ecole Norm. Sup. (4) , 8(2):157–178, 1975. 1[Mer17] Mona Merling. Equivariant algebraic K-theory of G -rings. Math. Z. , 285(3-4):1205–1248,2017. 1 wisted group rings and semilinearization Thomas Brazelton [Ost79] James Osterburg. The coefficient ring of the skew group ring. Czechoslovak Math. J. ,29(104)(1):144–147, 1979. 1[Par78] Jae Keol Park.
Artinian skew group rings and semiprime twisted group rings . ProQuestLLC, Ann Arbor, MI, 1978. Thesis (Ph.D.)–University of Cincinnati. 1[Poo16] Edward Poon.
SKEW GROUP RINGS , 2016. https://alistairsavage.ca/pubs/Poon-Skew_group_rings.pdf .2[RS17] Daniele Rosso and Alistair Savage. A general approach to Heisenberg categorification viawreath product algebras.
Math. Z. , 286(1-2):603–655, 2017. 1, 286(1-2):603–655, 2017. 1