Finite symmetric tensor categories with the Chevalley property in characteristic 2
aa r X i v : . [ m a t h . QA ] N ov FINITE SYMMETRIC TENSOR CATEGORIES WITH THECHEVALLEY PROPERTY IN CHARACTERISTIC PAVEL ETINGOF AND SHLOMO GELAKI
Dedicated to Nicol´as Andruskiewitsch for his 60th birthday
Abstract.
We prove an analog of Deligne’s theorem for finite symmetric ten-sor categories C with the Chevalley property over an algebraically closed field k of characteristic 2. Namely, we prove that every such category C admits asymmetric fiber functor to the symmetric tensor category D of representationsof the triangular Hopf algebra ( k [d] / (d ) , ⊗ ⊗ d). Equivalently, we provethat there exists a unique finite group scheme G in D such that C is symmet-ric tensor equivalent to Rep D ( G ). Finally, we compute the group H ( A, K )of equivalence classes of twists for the group algebra K [ A ] of a finite abelian p -group A over an arbitrary field K of characteristic p >
0, and the Sweedlercohomology groups H i Sw ( O ( A ) , K ), i ≥
1, of the function algebra O ( A ) of A . Introduction
The main objective of this paper is to classify finite symmetric tensor categorieswith the Chevalley property over an algebraically closed field k of characteristic 2.This completes the classification of finite integral symmetric tensor categories withthe Chevalley property over an algebraically closed field of characteristic p > p > p = 2 , α be the Frobenius kernel of the additive group G a . Then kα = k [d] / (d )with d primitive. Let D := Rep( α , ⊗ ⊗ d) be the symmetric tensor category offinite dimensional representations of the triangular Hopf algebra k [d] / (d ) equippedwith the R -matrix 1 ⊗ ⊗ d . Recall [V] that an object in D is a finitedimensional k -vector space V together with a linear map d : V → V satisfyingd = 0. In particular, D has two indecomposable objects, namely, the unit object(i.e., the vector space k with d = 0), and the two dimensional vector space k withd the strictly upper triangular matrix E .Recall that a finite group scheme in D is, by definition, a finite dimensional cocommutative Hopf algebra H in D . In particular, this means that d : H → H isa derivation of H satisfying d = 0, and∆( h ) = (1 ⊗ ⊗ d)(∆( h )) , h ∈ H. We can now state our main result (compare with [O, Conjecture 1.3]).
Date : December 3, 2019.
Key words and phrases.
Symmetric tensor categories, Chevalley property, quasi-Hopf algebras,associators, Sweedler cohomology, finite group schemes. D may be considered as a non-semisimple analog in characteristic 2 of the category sVec ofsupervector spaces, see [V]. Theorem 1.1.
Let C be a finite symmetric tensor category with the Chevalleyproperty over an algebraically closed field k of characteristic . Then C admits asymmetric fiber functor to D . Thus, there exists a unique finite group scheme G in D such that C is symmetric tensor equivalent to the category Rep D ( G ) of finitedimensional representations of G which are compatible with the action of π ( D ) . Remark 1.2.
Theorem 1.1 answers [BE, Question 1.2] for finite symmetric tensorcategories with the Chevalley property over k , and we expect it to hold for everyfinite symmetric integral tensor category over k .Finally, we note that the arguments used to prove [EG2, Theorem 1.1] andTheorem 1.1 in fact prove a stronger result (see Theorem 2.21).The organization of the paper is as follows. Section 2 is devoted to the proof ofTheorem 1.1. In Section 3 we compute the group H ( A, K ) of equivalence classesof twists for the group algebra K [ A ] of a finite abelian p -group A over an arbitraryfield K of characteristic p > H i Sw ( O ( A ) , K ) forevery i ≥ Acknowledgments.
P. E. was partially supported by NSF grant DMS 1502244.S. G. is grateful to the University of Michigan and MIT for their warm hospitality.2.
The proof of Theorem 1.1
All constructions in this section are done over an algebraically closed field k of characteristic 2 unless otherwise is explicitly stated. To lighten notation, wesometimes write 1 for 1 ⊗ ⊗ ⊗ = Vec). Thus by [EO, Theorem2.6], C is symmetric tensor equivalent to Rep( H, R,
Φ) for some finite dimensionaltriangular quasi-Hopf algebra (
H, R,
Φ) with the Chevalley property over k . Thus,we have to prove the following theorem. Theorem 2.1.
Let ( H, R, Φ) be a finite dimensional triangular quasi-Hopf algebrawith the Chevalley property over k . Then ( H, R, Φ) is pseudotwist equivalent toa triangular Hopf algebra with R -matrix ⊗ d for some d ∈ P ( H ) such that d = 0 . We will prove Theorem 2.1 in several steps.2.1. gr( H ) . Let (
H, R,
Φ) be a finite dimensional triangular quasi-Hopf algebrawith the Chevalley property over k . Let I := Rad( H ) be the Jacobson radical of H .Since I is a quasi-Hopf ideal of H , the associated graded algebra gr( H ) = L r ≥ H [ r ]has a natural structure of a graded triangular quasi-Hopf algebra with some R -matrix R ∈ H [0] ⊗ and associator Φ ∈ H [0] ⊗ (see, e.g., [EG2, 2.2]). Proposition 2.2. [EG2, Proposition 3.2]
The following hold: (1) H [0] is semisimple. (2) ( H [0] , R , Φ ) is a triangular quasi-Hopf subalgebra of (gr( H ) , R , Φ ) . YMMETRIC TENSOR CATEGORIES WITH THE CHEVALLEY PROPERTY 3 (3) Rep( H [0] , R , Φ ) is symmetric tensor equivalent to Rep( G ) for some finitesemisimple group scheme G over k . (4) (gr( H ) , R , Φ ) is pseudotwist equivalent to a graded triangular Hopf algebrawith R -matrix ⊗ , whose degree -component is ( kG, ⊗ . (cid:3) Corollary 2.3. [EG2, Corollary 3.3]
Let ( H, R, Φ) be a finite dimensional trian-gular quasi-Hopf algebra with the Chevalley property over k . Then gr( H ) is pseu-dotwist equivalent to k G for some finite group scheme G over k containing G as aclosed subgroup scheme. (cid:3) Remark 2.4.
By Nagata’s theorem (see, e.g, [A, p.223]), we have G = Γ ⋉ P D ,where Γ is a finite group of odd order and P is a finite abelian 2-group. Hence, wehave kG = k Γ ⋉ O ( P ).Let Γ := G / G ◦ . Then Γ is a finite constant group of odd order, and we have G = Γ ⋉ G ◦ . Thus, we have O ( G ) = O (Γ) ⊗ O ( G ◦ ) as algebras.By the results of this subsection, we may assume without loss of generality inthe proof of Theorem 2.1 that R = 1+ terms of higher degree.2.2. Trivializing R . Let V be a k -vector space, and let τ : V ⊗ → V ⊗ be theflip map. Recall that ∧ V := Im(id + τ ) ⊂ Γ V := Ker(id + τ ) ⊂ V ⊗ ,S V := V ⊗ / ∧ V, V (1) := Γ V / ∧ V, and that V (1) is called the Frobenius twist of V and Γ V the divided second sym-metric power of V . Note that V (1) is the image of the compositionΓ V ֒ → V ⊗ ։ S V. Let π : Γ V → V (1) be the natural surjective map.Let ( H, R,
Φ) be as in the end of Section 2.1.
Proposition 2.5.
The following hold: (1)
Suppose R = 1 + d n − ⊗ d n − modulo terms of degree ≥ n ≥ suchthat d n − ∈ Rad( H ) . Then ( H, R, Φ) can be twisted to a form such that R = 1 + d n ⊗ d n modulo terms of degree ≥ n + 1 , where d n ∈ Rad( H ) and d n − d n − has degree ≥ n/ , by a pseudotwist J n such that J n − hasdegree ≥ n if d n − = 0 , and degree ≥ n + p if deg(d n − ) = p > . (2) If R = 1 then ( H, R, Φ) can be twisted to the form R = 1 + d ⊗ d , where d ∈ Rad( H ) is an element of positive degree. Moreover, if R = 1 + d ′ ⊗ d ′ modulo terms of degree ≥ n , where d ′ ∈ Rad( H ) , then this can be achievedby a pseudotwist J with J − of degree ≥ n if d ′ = 0 , and degree ≥ n + p if d ′ = 0 and has degree p , so that d − d ′ has degree ≥ n/ . (3) If R = 1 + d ⊗ d then d = 0 .Proof. (1) Let R = 1 + d n − ⊗ d n − modulo terms of degree ≥ n , and consider R modulo terms of degree ≥ n + 1. We have R = 1 + d n − ⊗ d n − + e s modulo termsof degree ≥ n + 1, where e s ∈ H ⊗ has degree ≥ n . Let s ∈ gr( H ) ⊗ [ n ] be theleading part of e s . Then s is symmetric because R R = 1 ⊗
1, so s ∈ Γ gr( H )[ n ].Moreover, if t ∈ ∧ gr( H )[ n ] then we can replace s by s + t by twisting.Let v := π ( s ) be the image of s in gr( H ) (1) [ n ] = gr( H )[ n/ (1) (note that thisspace can be nonzero only if n is even). Then we can twist s into the form v ⊗ v by apseudotwist J with J − ≥ n . So we will get R = 1 + d n − ⊗ d n − + e v ⊗ e v PAVEL ETINGOF AND SHLOMO GELAKI modulo terms of degree ≥ n + 1, where e v is a lift of v to H . If d n − = 0, thiscompletes the proof (we can set d n = e v ). Thus, it remains to consider the case whend n − = 0 and has degree p ; so we may assume that n > p (because for n ≤ p , wecan set d n = d n − and J = 1). In this case, let us twist by J = 1 + d n − ⊗ e v (notethat deg( J − ≥ n + p ). Since R R = 1 ⊗
1, we have deg(d n − ) ≥ n/
2, hencedeg(d n − ⊗ d n − e v ) ≥ n/ n/ p = n + p ≥ n + 1 , so twisting by J brings R to the form R = 1 + (d n − + e v ) ⊗ (d n − + e v ) moduloterms of degree ≥ n + 1, i.e., we may take d n = d n − + e v , as desired.(2) Follows immediately from (1). Namely, for the first statement we take d tobe the stable limit of the d m ’s and J to be the product of the J m ’s, and for thesecond statement we take d n − = d ′ , d to be the stable limit of the d m ’s, and J tobe the product of the J m ’s for m ≥ n .(3) Follows from the identity R R = 1 ⊗ (cid:3) Thus, from now on we may assume that R = 1 + d ⊗ d for some d ∈ Rad( H )with d = 0 (but in general d is not a primitive element yet, as we have not madeΦ = 1). Remark 2.6.
Proposition 2.5 implies that the degree p of d in Proposition 2.5(2)and its degree p part δ ∈ gr( H )[ p ] (when d = 0) are uniquely determined. Indeed,if ( H, R,
Φ) is pseudotwist equivalent to (
H, R ′ , Φ ′ ) where R = 1 + d ⊗ d and R ′ = 1 + d ′ ⊗ d ′ modulo terms of degree ≥ n , and if d = 0 and has degree p < n/ J can be chosen so that J − ≥ n + p > p , so d ′ − d has degree ≥ p + 1, as desired. In particular, if R = J − J then whenever R is twisted to 1 + d ⊗ d, we must have d = 0. This is the case whenRep( H, R,
Φ) is Tannakian (as follows from Theorem 2.1). However, d itself is notunique (e.g., it can be conjugated by an invertible element x of 1 + Rad( H ), whichresults from applying the coboundary twist attached to x ).2.3. Trivializing Φ . Let (
H, R,
Φ) be a finite dimensional triangular quasi-Hopfalgebra with the Chevalley property over k , where R = 1 + d ⊗ d for some elementd ∈ Rad( H ) with d = 0.By Corollary 2.3, gr( H ) = k [ G ] = L i ≥ k [ G ][ i ], as graded Hopf algebras, forsome finite group scheme G over k . We let m , ε denote the multiplication andcounit maps of O ( G ).If Φ = 1 then d = 0 and ∆(d) = d ⊗ ⊗ d, so we are done. Thus we mayassume that Φ = 1. Consider Φ −
1. If it has degree ℓ then let φ be its projectionto gr( H ) ⊗ [ ℓ ].For every permutation ( i i i ) of (123), we will use φ i i i to denote the 3-tensorobtained by permuting the components of φ accordingly. Lemma 2.7.
The following hold: (1) φ ∈ Z ( O ( G ) , k ) is a normalized Hochschild -cocycle of O ( G ) with coeffi-cients in the trivial module k , i.e., φ ◦ (id ⊗ id ⊗ m ) + φ ◦ ( m ⊗ id ⊗ id) = ε ⊗ φ + φ ◦ (id ⊗ m ⊗ id) + φ ⊗ ε and φ ◦ (id ⊗ id ⊗
1) = φ ◦ (1 ⊗ id ⊗ id) = φ ◦ (id ⊗ ⊗ id) = ε ⊗ ε. (2) Alt( φ ) := φ + φ + φ + φ + φ + φ = 0 . YMMETRIC TENSOR CATEGORIES WITH THE CHEVALLEY PROPERTY 5
Proof. (1) Follows from [EG2, (2.1)-(2.2)] in a straightforward manner.(2) Follows from [EG2, (2.8)] in a straightforward manner. (cid:3)
The case R = 1 ⊗ . In this subsection we will assume that R = 1 ⊗
1, i.e.,d = 0.
Lemma 2.8.
The following hold: (1) φ + φ + φ = 0 = φ + φ + φ . (2) φ = φ . (3) Cyc( φ ) := φ + φ + φ = 0 .Proof. (1) Follows from [EG2, (2.6)-(2.7)] in a straightforward manner.(2) Using (1) and Lemma 2.7(2), we get0 = φ + φ + φ + ( φ + φ + φ )= Alt( φ ) + φ + φ = φ + φ , as claimed.(3) By (2), we have φ = φ . Thus the claim follows from (1). (cid:3) Following [EG2, 2.8] , we set y t := x ∗ t and y ( l ) t := ( x lt ) ∗ , 1 ≤ t ≤ n , 1 ≤ l ≤ r t − y (1) t = y t ), and for every 1 ≤ i, j ≤ n , let(2.1) β j := rj − X l =1 y ( l ) j ⊗ y (2 rj − l ) j . Proposition 2.9.
The -cocycle φ is a coboundary.Proof. By Lemma 2.7(1), φ ∈ Z ( O ( G ) , k ) so we can express it in the followingform: φ = X ≤ i In Proposition 2.9 we can choose f ∈ Γ k [ G ] , i.e., we can choose f to be symmetric.Proof. Since φ = φ by Lemma 2.8(2), we have df = d ( f ). This implies that f + f ∈ Z ( O ( G ) , k ) is a 2-cocycle, so it follows from [EG2, Proposition 2.4(2)]that we have f + f = X i a i β i + X i Choose f ∈ ( O ( G ) ∗ ) ⊗ symmetric with the same degree ℓ as φ such that φ = df ,which is possible by Lemma 2.10. Let e f be a symmetric lift of f to H . Then thepseudotwist F := 1 + e f is symmetric, which implies that ( H, , Φ) F = ( H F , , Φ F ),and the pseudotwisted associator Φ F is equal to 1+ terms of degree ≥ ℓ + 1. Bycontinuing this procedure, we will come to a situation where ( H, , Φ) F = ( H F , , F ∈ H ⊗ , as desired. This concludes the proof of Theorem2.1 in the case where R = 1.2.3.2. The case R = 1 + d ⊗ d with d = 0 . In this subsection we will assume that R = 1 + d ⊗ d with d = 0. Suppose d has degree p , and let δ be its projection togr( H )[ p ].The following lemma is the analogue of Lemma 2.8 in this case. Lemma 2.11. The following hold: (1) ∆( δ ) = δ ⊗ ⊗ δ . (2) The degree of ∆(d) − d ⊗ − ⊗ d is ≥ ℓ − p . (3) Let T ∈ gr( H ) ⊗ [ ℓ − p ] be the part of ∆(d) − d ⊗ − ⊗ d of degree exactly ℓ − p (so T = 0 if ℓ ≤ p ). Then we have (a) T ⊗ δ + φ + φ + φ = 0 . (b) δ ⊗ T + φ + φ + φ = 0 . (c) φ + φ = T ⊗ δ + δ ⊗ T . (d) Cyc( φ ) = Cyc( T ⊗ δ ) . (4) T is a symmetric -cocycle.Proof. (1) is clear. (2) and (3) follow immediately from the hexagon relations [EG2,(2.6)-(2.7)] ((3)(d) is obtained by applying Cyc to (3)(a) and using that Alt( φ ) = 0).Also, let Q := T + T . By (3)(c), we have Q ⊗ δ = δ ⊗ Q . Thus both left andright tensorands of Q can only be multiples of δ , i.e., Q is a multiple of δ ⊗ δ . But π ( Q ) = 0, hence Q = 0, proving (4). (cid:3) Proposition 2.12. The -cocycle φ has the form φ = T ⊗ δ + df for some f ∈ k [ G ] ⊗ [ ℓ ] .Proof. By Lemma 2.7(1), φ ∈ Z ( O ( G ) , k ) and we can express it in the followingform: φ = X ≤ i We also have π ( β i ) = y (2 ri − ) i . Thus,(2.6) X i,j a ij y (2 ri − ) i ⊗ y j + π ( T ) ⊗ δ = 0 , which implies that X j a ij y j = a i δ, and X i a i y (2 ri − ) i = π ( T )for some a i ∈ k . Hence, X a i β i + T = dh for some h ∈ k [ G ] (as the left hand side is a symmetric 2-cocycle killed by π , hencea coboundary). So, X a i β i ⊗ δ = T ⊗ δ + d ( h ⊗ δ )(as dδ = 0). This implies that φ = T ⊗ δ + df , where f := f ′ + h ⊗ δ , as desired. (cid:3) Lemma 2.13. In Proposition 2.12 we can choose f ∈ Γ k [ G ] , i.e., we can choose f to be symmetric.Proof. Since φ = φ + T ⊗ δ + δ ⊗ T by Lemma 2.11(3)(c), we have df = d ( f ).Thus, f + f ∈ Z ( O ( G ) , k ) is a 2-cocycle, and we can proceed in exactly the sameway as in the proof of Lemma 2.10 to get to a situation where f is symmetric. (cid:3) Proposition 2.14. The -cocycle T ⊗ T is a coboundary.Proof. Let f be a symmetric element provided by Lemma 2.13, and let e f bea symmetric lift of f to H . Then the pseudotwist F := 1 + e f is symmetric.Thus, ( H, R, Φ) F = ( H F , R F , Φ F ), and Φ F − ≥ ℓ with degree ℓ part T ⊗ δ . Thus, we haveΦ F = 1 + (∆(d) − d ⊗ − ⊗ d) ⊗ δ + U, where U ∈ H ⊗ has degree ≥ ℓ + 1.The pentagon equation [EG2, (2.3)] for Φ F yields that dU has degree ≥ ℓ − p ,and its part of degree 2 ℓ − p is T ⊗ T . This means that U has degree s ≤ ℓ − p .Let u be the leading part of U . If s < ℓ − p then the pentagon equation [EG2,(2.3)] yields that du = 0, and arguing as above we see that u = df , where f issymmetric. Thus, by a gauge transformation, we can make sure that u = 0. Thus,we may assume that s = 2 ℓ − p . In this case [EG2, (2.3)] yields du = T ⊗ T , i.e., T ⊗ T is a coboundary, as claimed. (cid:3) Proposition 2.15. The -cocycle φ is a coboundary.Proof. By [EG2, Proposition 2.4(2)] on the structure of cohomology, π ( T ) = 0.Thus by (2.6), a ij = 0 for all i, j , so φ is a coboundary. (cid:3) We can now proceed as in the case R = 1. Namely, by Proposition 2.15, we have φ = df for some f ∈ ( O ( G ) ∗ ) ⊗ with the same degree ℓ as φ , and by Lemma 2.13,we can choose f to be symmetric. Then letting e f be a symmetric lift of f to H ,we get the symmetric pseudotwist F := 1 + e f , and by this pseudotwist we come to YMMETRIC TENSOR CATEGORIES WITH THE CHEVALLEY PROPERTY 9 the situation where Φ − ≥ ℓ + 1. Thus ∆(d) − d ⊗ − ⊗ d also hasdegree ≥ ℓ + 1.However, unlike in the case R = 1, we are not done yet since the pseudotwist F spoils the R -matrix. Namely, since f is symmetric, R has been brought to the form R = 1 + d ⊗ d + [d ⊗ d , f ] + terms of degree > ℓ − . Thus, we need the following lemma. Lemma 2.16. We can twist further to make sure that R = 1 + d ⊗ d and still Φ − has degree ≥ ℓ + 1 .Proof. Let v := π ( f ). Then f = v ⊗ v + h + h for some h ∈ k [ G ] ⊗ . Thus, bytwisting by the pseudotwist J := 1 + [d ⊗ d , h ] + d v ⊗ v d, we come to the situationwhere Φ − ≥ ℓ + 1, but R = 1 + d ⊗ d + [d , v ] ⊗ [d , v ] + terms of degree > ℓ − . Now, if ℓ < p then ℓ/ p > ℓ , so twisting by J := 1 + d ⊗ [d , v ], we get to asituation when Φ − ≥ ℓ + 1 and R = 1 + d ⊗ d + terms of degree > ℓ − . Now Proposition 2.5 implies that using twists J with J − ≥ ℓ + 1 wecan come to a situation where Φ = 1 modulo degree ≥ ℓ + 1 and R = 1 + d ⊗ d onthe nose, providing the desired induction step.It remains to consider the situation ℓ ≥ p . By twisting by J := 1 + d ⊗ [d , v ],we will get to a situation where Φ − ⊗ W + terms of degree ≥ ℓ + 1 and R = 1 + d ⊗ d + terms of degree > ℓ − W := ∆([d , v ]) + [d , v ] ⊗ ⊗ [d , v ] . If deg( W ) > ℓ − p then we are done with the induction step, so it remains to considerthe case deg( W ) ≤ ℓ − p . In this case the hexagon relations [EG2, (2.6)-(2.7)] yield W = 0. Thus we come to a situation where Φ − ≥ ℓ +1 and R − − d ⊗ dhas degree > ℓ − p . So by Proposition 2.5, by applying twists of degree > ℓ , wecan make sure that R = 1 + d ⊗ d and still Φ − ≥ ℓ + 1, as desired. (cid:3) Thus it follows from the above that by continuing this procedure, we will cometo a situation where ( H, ⊗ d , Φ) F = ( H F , ⊗ d , F ∈ H ⊗ , as desired. This concludes the proof of Theorem2.1 in the case where R = 1 + d ⊗ d.The proofs of Theorems 2.1 and 1.1 are complete. (cid:3) Remark 2.17. Here is another short proof of the case when R is twist equivalentto 1, which uses the result of Coulembier. If R = 1 then the symmetric squareof a representation V is the usual one, so for any injection k → V the inducedmap k → S V is injective. By [C, Theorem C], this implies that the categoryRep( H, , Φ) is locally semisimple. Hence by [C, Proposition 6.2.2], the maximalTannakian subcategory of Rep( H, , Φ) is a Serre subcategory. Since the subcat-egory of Rep( H, , Φ) generated by simple objects is Tannakian, we see that thewhole category Rep( H, , Φ) is Tannakian, which implies the desired statement. Remark 2.18. The case when R = 1 is more subtle, as it is not captured by firstorder deformation theory. Indeed, the category D = Rep( k [d] / (d ) , ⊗ d) hasa nontrivial first order deformation over k [ h ] / ( h ), with the same R -matrix R , butwith ∆(d) = d ⊗ ⊗ d + h d ⊗ d and associator Φ := 1 + h d ⊗ d ⊗ d. Thisdeformation is nontrivial because φ := d ⊗ d ⊗ d is a nontrivial 3-cocycle. However,it does not lift to k [ h ] / ( h ), as the difference between the left hand side and theright hand side of the pentagon equation [EG2, (2.3)] is h d ⊗ .The existence of such deformations is typical. For example, consider the categoryVec( Z /p Z ) in characteristic p > 0. Clearly, it has no nontrivial formal deformations,since H ( Z /p Z , k × ) is trivial. However, it has a nontrivial first order deformation,since H ( Z /p Z , k ) = k . This deformation in fact lifts modulo h i for any i ≤ p , butdoes not lift modulo h p +1 . This is because µ p and α p are “the same” up to order p − p . Corollary 2.19. Let ( H, R ) be a finite dimensional triangular Hopf algebra withthe Chevalley property over k . Then ( H, R ) is twist equivalent to a triangular Hopfalgebra with R -matrix ⊗ d for some d ∈ P ( H ) such that d = 0 .Proof. Applying Theorem 2.1 to ( H, R, 1) yields the existence of a pseudotwist J for H such that ( H, R, J = ( H J , ⊗ d , J = 1,which is equivalent to J being a twist. (cid:3) Corollary 2.20. Let C be a finite symmetric tensor category over k such that FPdim( C ) = 2 . Then C is symmetric tensor equivalent to either Vec( Z / Z ) , Rep( Z / Z ) , Rep( α ) or D .Proof. Follows immediately from Theorem 1.1. (cid:3) Strengthening of [EG2, Theorem 1.1] and Theorem 1.1. The argumentsused in this section and [EG2, Section 3] in fact prove a stronger result. Namely,we have the following theorem. Theorem 2.21. Let E ⊂ C be finite symmetric tensor categories over an alge-braically closed field k with characteristic p > , such that E contains all the simplesof C . The following hold: (1) Suppose p > . If E has a fiber functor to sVec , then so does C . (2) Suppose p = 2 . If E has a fiber functor to Vec , then C has a fiber functorto D . Indeed, in both cases it follows that C is integral, so we have C = Rep( H, R, Φ) forsome finite dimensional triangular quasi-Hopf algebra over k . Now the argumentsare exactly the same, except the radical of H should be replaced by the annihilatorof E inside C , which is a nilpotent quasi-Hopf ideal of H since E contains all thesimples of C .3. Twists and Sweedler cohomology for finite abelian p -groups In this section we let K be an arbitrary field of characteristic p > 0, and F q bea finite field of characteristic p > YMMETRIC TENSOR CATEGORIES WITH THE CHEVALLEY PROPERTY 11 Truncated Witt vectors. Let W n ( K ) be the ring of truncated Witt vectorsof length n with coefficients in K . Recall that W n ( K ) = K n as a set, with nontrivialaddition and multiplication given, e.g., in [L, VI, p.330-332]. Example 3.1. We have the following:(1) W ( K ) = K as rings.(2) The addition and multiplication in W ( K ) are given as follows( x , x ) + ( y , y ) = x + y , x + y + p − X i =1 i (cid:18) p − i − (cid:19) x i y p − i ! and ( x , x )( y , y ) = ( x y , y p x + y x p ) . (3) W n ( F p ) = Z /p n Z for every n ≥ x := ( x , . . . , x n − ) ∈ W n ( K ), let F ( x ) = ( x p , . . . , x pn − ). (Note that if n > F ( x ) = x p .) Recall that F : W n ( K ) → W n ( K ) is a ring homomorphism, andwe have an additive homomorphism P : W n ( K ) → W n ( K ) , x F ( x ) − x. The kernel of P is the cyclic group W n ( F p ) = Z /p n Z . Lemma 3.2. The following hold: (1) If K is perfect then W n ( K ) / P ( W n ( K )) is a free Z /p n Z -module. (2) W n ( F q ) / P ( W n ( F q )) ∼ = Z /p n Z .Proof. (1) First note that since K is perfect, we have W n ( K ) /p s W n ( K ) ∼ = W s ( K )for every 0 ≤ s ≤ n .Secondly, let a ∈ W n ( K ) be an element such that its image a in K is notin P ( K ). We claim that the order of a in W n ( K ) / P ( W n ( K )) is p n . Indeed,suppose s < n is such that p s a = 0 in W n ( K ) / P ( W n ( K )), i.e., p s a = P ( y )for some y ∈ W n ( K ). Then P ( y ) = 0 in W n ( K ) /p s W n ( K ) = W s ( K ). Thus y = k ∈ Z /p s Z ⊆ W n ( K ) /p s W n ( K ) (as ker( P ) = Z /p n Z ), so y = k + p s z for someinteger k and z ∈ W n ( K ). But then p s a = P ( y ) = P ( p s z ), so if z is the image of z in K then a = P ( z ), which is a contradiction.Finally, take a ∈ W n ( K ) such that p n − a = 0 in W n ( K ) / P ( W n ( K )), and con-sider its image a in K . We have shown that a must be in P ( K ), i.e., a = x p − x for some x in K . Let x := ( x , , . . . , ∈ W n ( K ). We have a − P ( x ) = py for some y ∈ W n ( K ) (again using that K is perfect). Thus a = py in W n ( K ) / P ( W n ( K )),proving freeness.(2) Since the kernel of P : W n ( F q ) → W n ( F q ) is Z /p n Z , it follows that thecokernel of P has order p n . Thus W n ( F q ) / P ( W n ( F q )) is abelian of order p n , sothe claim follows from Part (1). (cid:3) Remark 3.3. If K is not perfect then for instance W ( K ) is not a free Z /p Z -module. Indeed, take an element (0 , a ) in W ( K ), where a ∈ K is not a p thpower. Then p (0 , a ) = (0 , , a ) = 0, but (0 , a ) = p ( x, y ) for any x, y , since p ( x, y ) = (0 , x p ). Twists for abelian groups and torsors. Recall that an interesting invari-ant of a tensor category C over K is the group of tensor structures on the identityfunctor of C (i.e., the group of isomorphism classes of tensor autoequivalences of C which act trivially on the underlying abelian category) up to an isomorphism[Da, BC]. This group is called the second invariant (or lazy) cohomology group of C and denoted by H ( C , K ).In particular, if C := Rep K ( A ) is the representation category of a finite abeliangroup A then H ( A, K ) := H ( C , K ) is the group of gauge equivalence classesof twists for the Hopf algebra K [ A ] [EG1]. Lemma 3.4. Let A be a finite abelian group. We have a canonical group isomor-phism H ( A, K ) ∼ = Hom( G, A ) , where G := Aut( K/K ) = Gal( K s /K ) . Proof. Let J be a twist for K [ A ], and consider the twisted K -algebra ( K [ A ] J ) ∗ . Ob-serve that (up to K -algebra isomorphism) this algebra depends only on [ J ]. Since by[AEGN, Theorem 6.5] every twist for K [ A ] is trivial, it follows that ( K [ A ] J ) ∗ ⊗ K K and Fun( A, K ) are isomorphic as K -algebras. Thus, ( K [ A ] J ) ∗ is a semisimple com-mutative K -algebra. Furthermore, ( K [ A ] J ) ∗ is an A -algebra, which is isomorphicto the regular representation of A as an A -module. Thus ( K [ A ] J ) ∗ is an A -torsor.Conversely, suppose B is an A -torsor, i.e., a commutative semisimple K -algebrawith an A -action such that B ⊗ K K ∼ = Fun( A, K ). By Wedderburn theorem, B decomposes uniquely into a direct sum of field extensions L i of K : B = L i L i .Since the space of A -invariants in B is 1-dimensional, A acts transitively on the setof fields L i . Let H ⊆ A be the stabilizer of L := L . Clearly L is a cyclic extensionof K with Galois group H . Then it is well known that L ∼ = ( K [ H ] ∗ ) J for a unique(up to gauge equivalence) Hopf 2-cocycle J for K [ H ] ∗ . Viewing J as a twist for K [ H ] (hence for K [ A ]), it is easy to see that the class [ J ] is uniquely determinedby the isomorphism class of the A -torsor B .Finally we note that A -torsors form an abelian group under the product rule( B , B ) ( B ⊗ B ) A , where a ∈ A acts on B by a and on B by a − , and that( K [ A ] IJ ) ∗ ∼ = (( K [ A ] I ) ∗ ⊗ ( K [ A ] J ) ∗ ) A (see, e.g., [AEGN, Remark 3.12]).It now follows from the above that the group H ( A, K ) is canonically isomor-phic to the group of A -torsors over K . Since the latter is canonically isomorphic tothe Galois cohomology group H ( G, A ) = Hom( G, A ), the claim follows. (cid:3) Invariant cohomology of abelian groups. Let A be a finite abelian groupof exponent dividing p n . Let G be as in Section 3.2, and let G n be its maximalabelian quotient of exponent dividing p n . Then Hom( G, A ) = Hom( G n , A ). Thusby Lemma 3.4, we have a canonical group isomorphism(3.1) H ( A, K ) ∼ = Hom( G n , A ) . Theorem 3.5. Let A be a finite abelian group of exponent dividing p n . Then thefollowing hold: (1) We have a canonical group isomorphism H ( A, K ) ∼ = Hom( A ∨ , W n ( K ) / P ( W n ( K ))) , where A ∨ := Hom( A, Z /p n Z ) . When considering Hom from a profinite group, as usual it means continuous homomorphisms. K s is the separable closure of K . YMMETRIC TENSOR CATEGORIES WITH THE CHEVALLEY PROPERTY 13 (2) If moreover K is perfect then we have a canonical group isomorphism H ( A, K ) ∼ = A ⊗ Z /p n Z ( W n ( K ) / P ( W n ( K ))) . Proof. (1) Recall that Artin-Schreier-Witt theory provides a canonical group iso-morphism G n ∼ = −→ Hom( W n ( K ) / P ( W n ( K )) , Z /p n Z )(see, e.g., [L, VI, p.330–332]). Thus we get from (3.1) a canonical group isomor-phism H ( A, K ) ∼ = Hom(Hom( W n ( K ) / P ( W n ( K )) , Z /p n Z ) , A ) . The claim follows now from the fact that Hom( B ∨ , A ) = Hom( A ∨ , B ) for every B .(2) By Lemma 3.2(1), W n ( K ) / P ( W n ( K )) is a free Z /p n Z -module. Thereforethe groupHom( A ∨ , W n ( K ) / P ( W n ( K ))) ∼ = Hom(Hom( W n ( K ) / P ( W n ( K )) , Z /p n Z ) , A )is the same as the group A ⊗ Z /p n Z ( W n ( K ) / P ( W n ( K ))), as desired. (cid:3) Corollary 3.6. We have a group isomorphism H ( Z /p n Z , K ) ∼ = W n ( K ) / P ( W n ( K )) . In particular, we have a group isomorphism H ( Z /p n Z , F q ) ∼ = Z /p n Z . Proof. By Theorem 3.5(1), H ( Z /p n Z , F q ) ∼ = W n ( F q ) / P ( W n ( F q )), so the secondclaim follows from Lemma 3.2(2). (cid:3) Remark 3.7. (1) Theorem 3.5(1) implies that if K is algebraically closed then H ( A, K ) = 0, which agrees with [EG2, Proposition 5.7] for i = 2.(2) Theorem 3.5(1) was obtained by Guillot [G] for p = 2 and n = 1.3.4. Sweedler cohomology of algebras of functions on abelian groups. Let A be a finite abelian group, and let O ( A ) be the Hopf algebra of functions on A with values in K . Recall that H ( A, K ) coincides with the second Sweedlercohomology group H ( O ( A ) , K ) with coefficients in K . Theorem 3.8. Let A be a finite abelian group of exponent dividing p n . Then theSweedler cohomology of O ( A ) with coefficients in K is as follows: (1) H ( O ( A ) , K ) = A . (2) H ( O ( A ) , K ) = Hom( A ∨ , W n ( K ) / P ( W n ( K ))) . (3) H i Sw ( O ( A ) , K ) = 0 for every i ≥ .Proof. (1) is clear and (2) is Theorem 3.5(1). To prove (3) consider the normalizedcomplex computing H i Sw ( O ( A ) , K ): C ( K ) → C ( K ) → C ( K ) → · · · , where C i is the algebraic group such that for any field L , C i ( L ) = ( L [ A ] ⊗ i ) × isthe group of invertible elements a in L [ A ] ⊗ i with ε ( a ) = 1. Then C i is a connectedcommutative unipotent algebraic group over K (i.e., an iterated extension of G a ).Now fix n ≥ 2. Since by [EG2, Proposition 5.7], H n Sw ( O ( A ) , K ) = H n +1Sw ( O ( A ) , K ) = 0 , we have a short exact sequence0 → C n − /D n − → C n → D n +1 → , where D i ⊆ C i is the kernel of the differential map d : C i → C i +1 . Thus we havean exact sequence0 → ( C n − /D n − )( K ) → C n ( K ) → D n +1 ( K ) → H ( K, C n − /D n − ) , where H ( K, C n − /D n − ) := H (Gal( K/K ) , ( C n − /D n − )( K ))is the Galois cohomology group. But since C n − /D n − is an iterated extension of G a , and H ( K, G a ) = 0, the Galois cohomology group H ( K, C n − /D n − ) van-ishes. Thus we have a short exact sequence0 → ( C n − /D n − )( K ) → C n ( K ) → D n +1 ( K ) → , which implies that H n +1Sw ( O ( A ) , K ) = D n +1 ( K ) /d ( C n ( K )) = 0, as claimed. (cid:3) References [A] E. Abe. Hopf algebras. Translated from the Japanese by Hisae Kinoshita and HirokoTanaka. Cambridge Tracts in Mathematics, . Cambridge University Press, Cambridge-New York, 1980. xii+284 pp.[AEGN] E. Aljadeff, P. Etingof, S. Gelaki and D. Nikshych. On twisting of finite-dimensionalHopf algebras, Journal of Algebra (2002), 484–501.[BC] J. Bichon and G. Carnovale. Lazy cohomology: an analogue of the Schur multiplier forarbitrary Hopf algebras. J. Pure Appl. Algebra (2006), no. 3, 627–665.[BE] D. Benson and P. Etingof. Symmetric tensor categories in characteristic 2. arXiv:1807.05549 .[C] K. Coulembier. Tannakian categories in positive characteristic. arXiv:1812.02452 .[Da] A. Davydov. Twisting of monoidal structures. Preprint of MPI, MPI/95–123 (1995), arXiv:q- alg/9703001 .[Dr] V. Drinfeld. Quasi-Hopf algebras. (Russian) Algebra i Analiz (1989), no. 6, 114–148;translation in Leningrad Math. J. (1990), no. 6, 1419–1457.[EG1] P. Etingof and S. Gelaki. Invariant Hopf 2-cocycles for affine algebraic groups. Interna-tional Mathematics Research Notices (2017). arXiv:1707.08672 .[EG2] P. Etingof and S. Gelaki. Finite symmetric integral tensor categories with the Chevalleyproperty. International Mathematics Research Notices , to appear. arXiv:1901.00528 .[EO] P. Etingof and V. Ostrik. Finite tensor categories. Moscow Mathematical Journal (3)(2004), 627–654.[EGNO] P. Etingof, S. Gelaki, D. Nikshych and V. Ostrik. Tensor categories. AMS MathematicalSurveys and Monographs book series (2015), 362 pp.[G] P. Guillot. Examples of Sweedler cohomology in positive characteristic. Comm. Alg. (2015), no. 5, 2174–2200.[J] J. Jantzen. Representations of algebraic groups. Second edition. Mathematical Surveysand Monographs . AMS, Providence, RI, 2003. xiv+576 pp.[L] S. Lang. Algebra. Graduate Text in Mathematics (2002). Springer-Verlag, New York.[O] V. Ostrik. On symmetric fusion categories in positive characteristic. arXiv:1503.01492 .[S] M. Sweedler. Cohomology of algebras over Hopf algebra. Trans. AMS (1968).[V] S. Venkatesh. Hilbert basis theorem and finite generation of invariants in symmetricfusion categories in positive characteristic. arXiv:1507.05142 .[W] W. Waterhouse. Introduction to affine group schemes. Graduate Texts in Mathematics . Springer-Verlag, New York-Berlin, 1979. xi+164 pp. Department of Mathematics, Massachusetts Institute of Technology, Cambridge,MA 02139, USA E-mail address : [email protected] Department of Mathematics, Iowa State University, Ames, IA 50011, USA E-mail address ::