Could White win this rook-endgame? [Event "FIDE Women's World Chess Championship"]
[Site "Sochi RUS"]
[Date "2015.03.23"]
[Round "3.1"]
[White "Marie Sebag"]
[Black "Natalia Pogonina"]
[Result "1-0"]
[ECO "C65"]
[WhiteElo "2482"]
[BlackElo "2456"]
[PlyCount "177"]
[EventDate "2015.03.17"]
[FEN ""]

1. e4 e5 2. Nf3 Nc6 3. Bb5 Nf6 4. d3 Bc5 5. O-O Nd4 6. Nxd4 Bxd4 7. c3 Bb6 8.
Na3 c6 9. Ba4 O-O 10. Bg5 h6 11. Bh4 Bc7 12. Bc2 b5 13. d4 d6 14. Bd3 g5 15.
Bg3 Qe7 16. Re1 Bb6 17. Nc2 Bd7 18. h3 Rfd8 19. Qf3 Ne8 20. Rad1 Ng7 21. Bf1
Bc7 22. c4 a6 23. Ne3 Re8 24. b4 Rad8 25. d5 Rf8 26. dxc6 Bxc6 27. cxb5 axb5
28. Bxb5 Ba8 29. Bc4 Bb6 30. Ng4 Kh7 31. Nf6+ Kg6 32. Nd5 Bxd5 33. Rxd5 Ne6 34.
Qf5+ Kg7 35. h4 Nd4 36. Qg4 Rc8 37. Bd3 Rc3 38. Rd1 Ra8 39. Bb1 Rc4 40. Kh2
Rxb4 41. f4 exf4 42. Bxf4 Qe6 43. Qxe6 fxe6 44. Bxd6 exd5 45. Bxb4 dxe4 46. Bc3
Ra4 47. Bxe4 gxh4 48. Rb1 Bc7+ 49. Kh3 Be5 50. Bd5 Kf6 51. Rb4 Ra3 52. Bb3 Nxb3
53. Bxe5+ Kxe5 54. axb3 Ra2 55. Rxh4 Rb2 56. Rb4 Kf5 57. g4+ Kg6 58. Kg3 Rb1
59. Kf4 Rf1+ 60. Ke5 Rb1 61. Kf4 Rf1+ 62. Kg3 Rg1+ 63. Kf3 Rf1+ 64. Kg2 Rd1 65.
Rb5 Rd3 66. b4 Rb3 67. Kf2 Kf6 68. Rf5+ Kg6 69. b5 Kg7 70. Rc5 Kg6 71. Rd5 Kf6
72. Rc5 Kg6 73. Ke2 Kf6 74. Rf5+ Kg6 75. Kd2 Rb4 76. Kc3 Rxg4 77. Rc5 Rg1 78.
Kb4 Rb1+ 79. Ka5 Ra1+ 80. Kb6 h5 81. Kc7 h4 82. b6 h3 83. b7 h2 84. b8=Q h1=Q
85. Qg8+ Kh6 86. Rc6+ Kh5 87. Qg6+ Kh4 88. Rc4+ Kh3 89. Qg4+ 1-0


With 56. Rh6 , white could transfer in a won endgame. And black slips in move
81 (instead of 81... h4? the move 81 ... Ra5 would have drawn). Could white
win the rook-endgame before move 77 without the slip 55 ...Rb2 ?
Well, according to 7-man tablebase your analysis is correct:



Rxh6 would win, reaching a Lucena position.

81...h4? was a mistake, ...Ra5 was the only drawing move.
Neither player made a mistake in between those two points.


Also, Black could have made it more difficult with 86...Qxc6+, reaching QvR which can be hard to win. White didn't have to allow that possibility.

Of course, there's a big difference between checking tablebase, and giving useful advice to use during a game.  I guess we could say that White should really have seen the chance to go into Lucena; and Black must have miscalculated something when playing ...h4 -- or possibly overlooked the Ra5 plan (it is unusual).