Finite generation of the cohomology rings of some pointed Hopf algebras

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

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).

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.

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).

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.

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.

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).

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.

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.

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.

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!

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 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 ).

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

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.

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.

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.

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

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.

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}.

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

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}.

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

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

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.

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?

References D. J. Anick, On the homology of associative algebras, Trans. Amer. Math. Soc. 296 (1986), no. 2, 641 659. S. Cojocaru and V. Ufnarovski, BERGMAN under MS-DOS and Anick s resolution, Discrete Math. Theoretical Comp. Sci. 1 (1997), 139 147. E. Friedlander and A. Suslin, Cohomology of finite group schemes over a field, Invent. Math., 127 (1997), no. 2, 209 270. J. P. May, The cohomology of restricted Lie algebras and of Hopf algebras, J. Algebra 3 (1966), 123 146. V. C. Nguyen and X. Wang, Pointed p 3 -dimensional Hopf algebras in positive characteristic, to appear in Alg. Colloq., arxiv:1609.03952. A. V. Shepler and S. Witherspoon, Resolutions for twisted tensor products, preprint, arxiv:1610.00583. Thank You!