Finite generation of the cohomology rings of some pointed Hopf algebras

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1 Finite generation of the cohomology rings of some pointed Hopf algebras Van C. Nguyen Hood College, Frederick MD joint work with Xingting Wang and Sarah Witherspoon Maurice Auslander Distinguished Lectures and International Conference April 25 30, 2018

2 Setting & Motivation Let k be a field and H be a finite-dimensional Hopf algebra over k. The cohomology of H is H (H, k) := n 0 Ext n H(k, k).

3 Setting & Motivation Let k be a field and H be a finite-dimensional Hopf algebra over k. The cohomology of H is H (H, k) := n 0 Ext n H(k, k). Conjecture (Etingof-Ostrik 04) For any finite-dimensional Hopf algebra H, H (H, k) is finitely generated.

4 Setting & Motivation Let k be a field and H be a finite-dimensional Hopf algebra over k. The cohomology of H is H (H, k) := n 0 Ext n H(k, k). Conjecture (Etingof-Ostrik 04) For any finite-dimensional Hopf algebra H, H (H, k) is finitely generated. GOAL: Study the finite generation of H (H, k), for some pointed Hopf algebras.

5 Finite generation of cohomology ring F.g. Cohomology Conjecture Applications: Quillen s stratification theorem, modular representation theory, support variety theory, algebraic geometry, commutative algebra, some homological conjectures Partial Results: finite group algebras over pos. char., finite group schemes over pos. char., Lusztig s small quantum group over C, Drinfeld double of Frob. kernels of finite alg. groups, certain pointed Hopf algebras Remarks: H (H, k) is a graded-commutative ring. H (H, k) is a finitely generated k-algebra H (H, k) is left (or right) Noetherian H ev (H, k) is Noetherian and H (H, k) is a f.g. module over H ev (H, k).

6 Finite generation of cohomology ring F.g. Cohomology Conjecture Applications: Quillen s stratification theorem, modular representation theory, support variety theory, algebraic geometry, commutative algebra, some homological conjectures Partial Results: finite group algebras over pos. char., finite group schemes over pos. char., Lusztig s small quantum group over C, Drinfeld double of Frob. kernels of finite alg. groups, certain pointed Hopf algebras Remarks: H (H, k) is a graded-commutative ring. H (H, k) is a finitely generated k-algebra H (H, k) is left (or right) Noetherian H ev (H, k) is Noetherian and H (H, k) is a f.g. module over H ev (H, k).

7 Finite generation of cohomology ring F.g. Cohomology Conjecture Applications: Quillen s stratification theorem, modular representation theory, support variety theory, algebraic geometry, commutative algebra, some homological conjectures Partial Results: finite group algebras over pos. char., finite group schemes over pos. char., Lusztig s small quantum group over C, Drinfeld double of Frob. kernels of finite alg. groups, certain pointed Hopf algebras Remarks: H (H, k) is a graded-commutative ring. H (H, k) is a finitely generated k-algebra H (H, k) is left (or right) Noetherian H ev (H, k) is Noetherian and H (H, k) is a f.g. module over H ev (H, k).

8 Finite generation of cohomology ring F.g. Cohomology Conjecture Applications: Quillen s stratification theorem, modular representation theory, support variety theory, algebraic geometry, commutative algebra, some homological conjectures Partial Results: finite group algebras over pos. char., finite group schemes over pos. char., Lusztig s small quantum group over C, Drinfeld double of Frob. kernels of finite alg. groups, certain pointed Hopf algebras Remarks: H (H, k) is a graded-commutative ring. H (H, k) is a finitely generated k-algebra H (H, k) is left (or right) Noetherian H ev (H, k) is Noetherian and H (H, k) is a f.g. module over H ev (H, k).

9 Preliminary Ingredients Definition A Hopf algebra H over a field k is a k-vector space which is an algebra (m, u) a coalgebra (, ε) together with an antipode map S : H H.

10 Preliminary Ingredients Definition A Hopf algebra H over a field k is a k-vector space which is an algebra (m, u) a coalgebra (, ε) together with an antipode map S : H H. Example group algebra kg, polynomial rings k[x 1, x 2,..., x n ], universal enveloping algebra U(g) of a Lie algebra g, etc.

11 Today s Object: p 3 -dim pointed Hopf algebras Let k = k with char(k) = p > 2 and H be a p 3 -dim pointed Hopf algebra. The classification of H is done by (V.C. Nguyen-X. Wang 16).

12 Today s Object: p 3 -dim pointed Hopf algebras Let k = k with char(k) = p > 2 and H be a p 3 -dim pointed Hopf algebra. The classification of H is done by (V.C. Nguyen-X. Wang 16). We are interested in the case when H 0 = kc q = g, grh = B(V )#kc q, q is divisible by p (more general), where V = kx ky is kc q -module.

13 Today s Object: p 3 -dim pointed Hopf algebras Let k = k with char(k) = p > 2 and H be a p 3 -dim pointed Hopf algebra. The classification of H is done by (V.C. Nguyen-X. Wang 16). We are interested in the case when H 0 = kc q = g, grh = B(V )#kc q, q is divisible by p (more general), where V = kx ky is kc q -module. B(V ) is a rank two Nichols algebra of Jordan type over C q. B(V ) = k x, y /(x p, y p, yx xy 1 2 x 2 ). with action g x = x and g y = x + y.

14 Today s Object: p 3 -dim pointed Hopf algebras Let k = k with char(k) = p > 2 and H be a p 3 -dim pointed Hopf algebra. The classification of H is done by (V.C. Nguyen-X. Wang 16). We are interested in the case when H 0 = kc q = g, grh = B(V )#kc q, q is divisible by p (more general), where V = kx ky is kc q -module. B(V ) is a rank two Nichols algebra of Jordan type over C q. B(V ) = k x, y /(x p, y p, yx xy 1 2 x 2 ). with action g x = x and g y = x + y. B(V ) Hopf alg in YD Cq C q bosonization grh = B(V )#kc q lifting pointed Hopf algebras H

15 Today s Object: two Hopf algebras Let k = k with char(k) = p > 2 and w = g 1. Consider the following Hopf algebras over k: 1 The p 2 q-dim bosonization grh = B(V )#kc q is isomorphic to k w, x, y subject to w q, x p, y p, yx xy 1 2 x 2, xw wx, yw wy wx x.

16 Today s Object: two Hopf algebras Let k = k with char(k) = p > 2 and w = g 1. Consider the following Hopf algebras over k: 1 The p 2 q-dim bosonization grh = B(V )#kc q is isomorphic to k w, x, y subject to w q, x p, y p, yx xy 1 2 x 2, xw wx, yw wy wx x. 2 The 27-dim liftings in p = q = 3 are H = H(ɛ, µ, τ) = k w, x, y subject to w 3 = 0, x 3 = ɛx, y 3 = ɛy 2 (µɛ τ µ 2 )y, yw wy = wx + x (µ ɛ)(w 2 + w), xw wx = ɛ(w 2 + w), yx xy = x 2 + (µ + ɛ)x + ɛy τ(w 2 w), with ɛ {0, 1} and τ, µ k.

17 Today s Object: two Hopf algebras Let k = k with char(k) = p > 2 and w = g 1. Consider the following Hopf algebras over k: 1 The p 2 q-dim bosonization grh = B(V )#kc q is isomorphic to k w, x, y subject to w q, x p, y p, yx xy 1 2 x 2, xw wx, yw wy wx x. 2 The 27-dim liftings in p = q = 3 are H = H(ɛ, µ, τ) = k w, x, y subject to w 3 = 0, x 3 = ɛx, y 3 = ɛy 2 (µɛ τ µ 2 )y, yw wy = wx + x (µ ɛ)(w 2 + w), xw wx = ɛ(w 2 + w), yx xy = x 2 + (µ + ɛ)x + ɛy τ(w 2 w), with ɛ {0, 1} and τ, µ k. Main Results (N-Wang-Witherspoon 17) The cohomology rings of B(V )#kc q and of H(ɛ, µ, τ) are finitely generated.

18 Strategy: May spectral sequence & permanent cocycles Take H as B(V )#kc q or H(ɛ, µ, τ) (p = q = 3). Assign lexicographic order on monomials in w, x, y with w < x < y.

19 Strategy: May spectral sequence & permanent cocycles Take H as B(V )#kc q or H(ɛ, µ, τ) (p = q = 3). Assign lexicographic order on monomials in w, x, y with w < x < y. N-filtration on H, grh = k[w, x, y]/(w q, x p, y p ).

20 Strategy: May spectral sequence & permanent cocycles Take H as B(V )#kc q or H(ɛ, µ, τ) (p = q = 3). Assign lexicographic order on monomials in w, x, y with w < x < y. N-filtration on H, grh = k[w, x, y]/(w q, x p, y p ). H (grh, k) = ( k 3) k [ξ w, ξ x, ξ y ], deg(ξ i ) = 2.

21 Strategy: May spectral sequence & permanent cocycles Take H as B(V )#kc q or H(ɛ, µ, τ) (p = q = 3). Assign lexicographic order on monomials in w, x, y with w < x < y. N-filtration on H, grh = k[w, x, y]/(w q, x p, y p ). H (grh, k) = ( k 3) k [ξ w, ξ x, ξ y ], deg(ξ i ) = 2. (May 66) May spectral sequence E, 1 with respect to the cup product. = H (grh, k) = E, = gr H (H, k).

22 Strategy: May spectral sequence & permanent cocycles Take H as B(V )#kc q or H(ɛ, µ, τ) (p = q = 3). Assign lexicographic order on monomials in w, x, y with w < x < y. N-filtration on H, grh = k[w, x, y]/(w q, x p, y p ). H (grh, k) = ( k 3) k [ξ w, ξ x, ξ y ], deg(ξ i ) = 2. (May 66) May spectral sequence E, 1 with respect to the cup product. Lemma (Friedlander-Suslin 97) = H (grh, k) = E, = gr H (H, k). If ξ w, ξ x, ξ y are permanent cocyles (meaning they survive at E -page), then gr H (H, k) and H (H, k) are noetherian over k[ξ w, ξ x, ξ y ]. Consequently, H (H, k) is finitely generated as a k-algebra. = Need to find such permanent cocycles!

23 Tool 1: Twisted tensor product resolution (Shepler-Witherspoon 16) Let A and B be associative k-algebras. A twisting map τ : B A A B is a bijective k-linear map that respects the identity and multiplication maps of A and of B.

24 Tool 1: Twisted tensor product resolution (Shepler-Witherspoon 16) Let A and B be associative k-algebras. A twisting map τ : B A A B is a bijective k-linear map that respects the identity and multiplication maps of A and of B. The twisted tensor product algebra A τ B is the vector space A B together with multiplication m τ given by such a twisting map τ.

25 Tool 1: Twisted tensor product resolution (Shepler-Witherspoon 16) Let A and B be associative k-algebras. A twisting map τ : B A A B is a bijective k-linear map that respects the identity and multiplication maps of A and of B. The twisted tensor product algebra A τ B is the vector space A B together with multiplication m τ given by such a twisting map τ. Let P (M) be an A-projective resolution of M and P (N) be a B-projective resolution of N

26 Tool 1: Twisted tensor product resolution (Shepler-Witherspoon 16) Let A and B be associative k-algebras. A twisting map τ : B A A B is a bijective k-linear map that respects the identity and multiplication maps of A and of B. The twisted tensor product algebra A τ B is the vector space A B together with multiplication m τ given by such a twisting map τ. Let P (M) be an A-projective resolution of M and P (N) be a B-projective resolution of N = Construct a projective resolution Y of A τ B-modules from P (M) and P (N)??

27 Tool 1: Twisted tensor product resolution (Shepler-Witherspoon 16) Let A and B be associative k-algebras. A twisting map τ : B A A B is a bijective k-linear map that respects the identity and multiplication maps of A and of B. The twisted tensor product algebra A τ B is the vector space A B together with multiplication m τ given by such a twisting map τ. Let P (M) be an A-projective resolution of M and P (N) be a B-projective resolution of N = Construct a projective resolution Y of A τ B-modules from P (M) and P (N)?? Twisted tensor product resolution: (Shepler-Witherspoon 16) introduced some compatibility conditions that are sufficient for constructing a projective resolution Y = Tot(P (M) P (N)) of M N as a module over A τ B.

28 Twisted tensor product resolutions over B(V )#kc q H = B(V )#kc q = (A τ B) µ C. A = k[x]/(x p ), B = k[y]/(y p ), C = k[w]/(w q ).

29 Twisted tensor product resolutions over B(V )#kc q H = B(V )#kc q = (A τ B) µ C. A = k[x]/(x p ), B = k[y]/(y p ), C = k[w]/(w q ). The twisting map τ : B A A B: τ(y r x l ) = r t=0 ( ) r l(l + 1)(l + 2) (l + t 1) t 2 t x l+t y r t.

30 Twisted tensor product resolutions over B(V )#kc q H = B(V )#kc q = (A τ B) µ C. A = k[x]/(x p ), B = k[y]/(y p ), C = k[w]/(w q ). The twisting map τ : B A A B: τ(y r x l ) = r t=0 ( ) r l(l + 1)(l + 2) (l + t 1) t 2 t x l+t y r t. Let P A (k) : x p 1 A x A xp 1 A x A ε k 0, P B (k) : y p 1 B y B y p 1 B y B ε k 0.

31 Twisted tensor product resolutions over B(V )#kc q H = B(V )#kc q = (A τ B) µ C. A = k[x]/(x p ), B = k[y]/(y p ), C = k[w]/(w q ). The twisting map τ : B A A B: τ(y r x l ) = r t=0 ( ) r l(l + 1)(l + 2) (l + t 1) t 2 t x l+t y r t. Let P A (k) : x p 1 A x A xp 1 A x A ε k 0, P B (k) : y p 1 B y B y p 1 B y B ε k 0. The total complex: K := Tot(P A (k) P B (k)) with differential d n = (d i 1 + ( 1) i d j ). i+j=n

32 Twisted tensor product resolutions over B(V )#kc q H = B(V )#kc q = (A τ B) µ C. A = k[x]/(x p ), B = k[y]/(y p ), C = k[w]/(w q ). The twisting map τ : B A A B: τ(y r x l ) = r t=0 ( ) r l(l + 1)(l + 2) (l + t 1) t 2 t x l+t y r t. Let P A (k) : x p 1 A x A xp 1 A x A ε k 0, P B (k) : y p 1 B y B y p 1 B y B ε k 0. The total complex: K := Tot(P A (k) P B (k)) with differential d n = (d i 1 + ( 1) i d j ). i+j=n (K, d) is a resolution of k over A τ B. H (K ) = H (A B, k) = H (A, k) H (B, k).

33 Twisted tensor product resolutions over B(V )#kc q K n = Pi A (k) Pj B (k) = i+j=n H (K ) = H (A τ B, k). i+j=n (A τ B) φ ij.

34 Twisted tensor product resolutions over B(V )#kc q K n = Pi A (k) Pj B (k) = i+j=n H (K ) = H (A τ B, k). K is C q = g -equivariant. i+j=n (A τ B) φ ij.

35 Twisted tensor product resolutions over B(V )#kc q K n = Pi A (k) Pj B (k) = i+j=n H (K ) = H (A τ B, k). K is C q = g -equivariant. i+j=n (A τ B) φ ij. q 1 P kcq (g 1) ( s=0 (k) : kc q kc g s ) q kc q (g 1) kc q ε k 0. Twisted tensor resolution Y := Tot(K P kcq (k)) with twisted chain map µ : kc q K K kc q given by the C q -action.

36 Twisted tensor product resolutions over B(V )#kc q K n = Pi A (k) Pj B (k) = i+j=n H (K ) = H (A τ B, k). K is C q = g -equivariant. i+j=n (A τ B) φ ij. q 1 P kcq (g 1) ( s=0 (k) : kc q kc g s ) q kc q (g 1) kc q ε k 0. Twisted tensor resolution Y := Tot(K P kcq (k)) with twisted chain map µ : kc q K K kc q given by the C q -action. Described all H n (B(V )#kc q, k) as k-vector space. Found ξ w, ξ x, ξ y H 2 (B(V )#kc q, k), needed permanent cocycles.

37 Twisted tensor product resolutions over B(V )#kc q K n = Pi A (k) Pj B (k) = i+j=n H (K ) = H (A τ B, k). K is C q = g -equivariant. i+j=n (A τ B) φ ij. q 1 P kcq (g 1) ( s=0 (k) : kc q kc g s ) q kc q (g 1) kc q ε k 0. Twisted tensor resolution Y := Tot(K P kcq (k)) with twisted chain map µ : kc q K K kc q given by the C q -action. Described all H n (B(V )#kc q, k) as k-vector space. Found ξ w, ξ x, ξ y H 2 (B(V )#kc q, k), needed permanent cocycles. Theorem (N-Wang-Witherspoon 17) H (B(V )#kc q, k) is finitely generated as a k-algebra.

38 Tool 2: Anick resolution (Anick 86) H = H(ɛ, µ, τ) = T (V )/(I ).

39 Tool 2: Anick resolution (Anick 86) H = H(ɛ, µ, τ) = T (V )/(I ). B = {w, x, y}, a basis of V with the ordering w < x < y.

40 Tool 2: Anick resolution (Anick 86) H = H(ɛ, µ, τ) = T (V )/(I ). B = {w, x, y}, a basis of V with the ordering w < x < y. T = {tips}, reduced words (monomials) whose all proper subwords are irreducible; T relations (I ).

41 Tool 2: Anick resolution (Anick 86) H = H(ɛ, µ, τ) = T (V )/(I ). B = {w, x, y}, a basis of V with the ordering w < x < y. T = {tips}, reduced words (monomials) whose all proper subwords are irreducible; T relations (I ). R = {all proper prefixes (left factors) of the tips}.

42 Tool 2: Anick resolution (Anick 86) H = H(ɛ, µ, τ) = T (V )/(I ). B = {w, x, y}, a basis of V with the ordering w < x < y. T = {tips}, reduced words (monomials) whose all proper subwords are irreducible; T relations (I ). R = {all proper prefixes (left factors) of the tips}. (Cojocaru-Ufnarovski 97): Quiver Q = Q(B, T ): Vertices: {1} R. Arrows: 1 v for v B and all f g for f, g R where gf uniquely contains a tip as a prefix.

43 Tool 2: Anick resolution (Anick 86) H = H(ɛ, µ, τ) = T (V )/(I ). B = {w, x, y}, a basis of V with the ordering w < x < y. T = {tips}, reduced words (monomials) whose all proper subwords are irreducible; T relations (I ). R = {all proper prefixes (left factors) of the tips}. (Cojocaru-Ufnarovski 97): Quiver Q = Q(B, T ): Vertices: {1} R. Arrows: 1 v for v B and all f g for f, g R where gf uniquely contains a tip as a prefix. In each homological degree n of the Anick resolution, define a free basis C n = {all paths of length n starting from 1 in Q}.

44 Tool 2: Anick resolution (Anick 86) H = H(ɛ, µ, τ) = T (V )/(I ). B = {w, x, y}, a basis of V with the ordering w < x < y. T = {tips}, reduced words (monomials) whose all proper subwords are irreducible; T relations (I ). R = {all proper prefixes (left factors) of the tips}. (Cojocaru-Ufnarovski 97): Quiver Q = Q(B, T ): Vertices: {1} R. Arrows: 1 v for v B and all f g for f, g R where gf uniquely contains a tip as a prefix. In each homological degree n of the Anick resolution, define a free basis C n = {all paths of length n starting from 1 in Q}. The differentials d are defined recursively, with a simultaneous recursive definition of a contracting homotopy s: d 3 s 2 H kc 2 d 2 s 1 H kc 1 d 1 s 0 ε H k 0. η

45 Anick resolution for some liftings H = H(ɛ, µ, τ) Example (H = H(ɛ, µ, τ) = k w, x, y subject to) w 3 = 0, x 3 = ɛx, y 3 = ɛy 2 (µɛ τ µ 2 )y, yw wy = wx + x (µ ɛ)(w 2 + w), xw wx = ɛ(w 2 + w), yx xy = x 2 + (µ + ɛ)x + ɛy τ(w 2 w), with ɛ {0, 1} and τ, µ k. 1 w x y w 2 x 2 y 2

46 Anick resolution for some liftings H = H(ɛ, µ, τ) 1 w x y w 2 x 2 y 2 C 1 = {w, x, y} = B, C 2 = {w 3, x 3, y 3, xw, yw, yx} = T, C 3 = {w 3+1, x 3+1, y 3+1, xw 3, yw 3, yx 3, x 3 w, y 3 w, y 3 x, yxw}.

47 Anick resolution for some liftings H = H(ɛ, µ, τ) 1 w x y w 2 x 2 y 2 C 1 = {w, x, y} = B, C 2 = {w 3, x 3, y 3, xw, yw, yx} = T, C 3 = {w 3+1, x 3+1, y 3+1, xw 3, yw 3, yx 3, x 3 w, y 3 w, y 3 x, yxw}. Define differentials, d 3 s 2 H kc 2 d 2 s 1 H kc 1 d 1 s 0 ε H k 0. η

48 Anick resolution for some liftings H = H(ɛ, µ, τ) 1 w x y w 2 x 2 y 2 Define differentials, C 1 = {w, x, y} = B, C 2 = {w 3, x 3, y 3, xw, yw, yx} = T, C 3 = {w 3+1, x 3+1, y 3+1, xw 3, yw 3, yx 3, x 3 w, y 3 w, y 3 x, yxw}. d 3 s 2 H kc 2 d 2 s 1 H kc 1 = Found ξ w, ξ x, ξ y H 2 (H(ɛ, µ, τ), k), needed permanent cocycles. d 1 s 0 ε H k 0. η

49 Anick resolution for some liftings H = H(ɛ, µ, τ) 1 w x y w 2 x 2 y 2 Define differentials, C 1 = {w, x, y} = B, C 2 = {w 3, x 3, y 3, xw, yw, yx} = T, C 3 = {w 3+1, x 3+1, y 3+1, xw 3, yw 3, yx 3, x 3 w, y 3 w, y 3 x, yxw}. d 3 s 2 H kc 2 d 2 s 1 H kc 1 = Found ξ w, ξ x, ξ y H 2 (H(ɛ, µ, τ), k), needed permanent cocycles. Theorem (N-Wang-Witherspoon 17) H (H(ɛ, µ, τ), k) is finitely generated as a k-algebra. d 1 s 0 ε H k 0. η

50 Anick resolution for some liftings H = H(ɛ, µ, τ) d 2 (1 w 3 ) = w 2 w, d 2 (1 x 3 ) = x 2 x ɛ x, d 2 (1 y 3 ) = y 2 y + ɛy y + (µɛ τ µ 2 ) y, d 2 (1 xw) = x w w x ɛw w ɛ w, d 2 (1 yw) = y w w y w x 1 x + (µ ɛ)w w + (µ ɛ) w, d 2 (1 yx) = y x x y + x x (µ + ɛ) x ɛ y + τw w τ w. d 3 (1 w 4 ) = w w 3, d 3 (1 x 4 ) = x x 3, d 3 (1 y 4 ) = y y 3, d 3 (1 xw 3 ) = x w 3 w 2 xw, d 3 (1 x 3 w) = x 2 xw + w x 3 + ɛwx xw + ɛx xw + ɛw xw, d 3 (1 yw 3 ) = y w 3 w 2 yw + w 2 xw + w xw, d 3 (1 yxw) = y xw x yw + w yx + ɛw yw + x xw + (µ + ɛ)w xw, d 3 (1 y 3 w) = y 2 yw + w y 3 + wy yx + wx yx + (ɛ µ)wy yw +(µ ɛ)wx yw τw 2 yw + y yx (ɛ + µ)y yw +τw 2 xw + x yx + (µ ɛ)x yw + (µ 2 ɛµ)w yw + τw xw, d 3 (1 yx 3 ) = y x 3 x 2 yx + τwx xw + ɛx yx τx xw + ɛτw xw, d 3 (1 y 3 x) = y 2 yx + x y 3 xy yx τwx yw τwy yw +τw 2 yx + τwx xw + ɛτw 2 yw + (ɛτ + µτ)w 2 xw +µy yx + τy yw µx yx + τx xw +τw yx + (ɛτ + µτ)w yw + ɛτw xw.

51 Recap The Menu F.g. of cohomology rings of (Hopf) algebras? p 2 q-dim Hopf algebra B(V )#kc q 27-dim Hopf algebra H(ɛ, µ, τ) Tool 1: Twisted tensor product resolutions Tool 2: Anick resolutions Other algebras? Other tools?

52 References D. J. Anick, On the homology of associative algebras, Trans. Amer. Math. Soc. 296 (1986), no. 2, S. Cojocaru and V. Ufnarovski, BERGMAN under MS-DOS and Anick s resolution, Discrete Math. Theoretical Comp. Sci. 1 (1997), E. Friedlander and A. Suslin, Cohomology of finite group schemes over a field, Invent. Math., 127 (1997), no. 2, J. P. May, The cohomology of restricted Lie algebras and of Hopf algebras, J. Algebra 3 (1966), V. C. Nguyen and X. Wang, Pointed p 3 -dimensional Hopf algebras in positive characteristic, to appear in Alg. Colloq., arxiv: A. V. Shepler and S. Witherspoon, Resolutions for twisted tensor products, preprint, arxiv: Thank You!

arxiv: v1 [math.ra] 19 Oct 2017

arxiv: v1 [math.ra] 19 Oct 2017 FINITE GENERATION OF SOME COHOMOLOGY RINGS VIA TWISTED TENSOR PRODUCT AND ANICK RESOLUTIONS VAN C. NGUYEN, XINGTING WANG, AND SARAH WITHERSPOON arxiv:1710.07141v1 [math.ra] 19 Oct 2017 Abstract. Over a