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Kohli gained two points for his half-century to get to 888 while Smith (911) lost 10 points in the table after scoring 1 and 1 in his two innings.
Marnus Labuschagne’s innings of 47 and 6 in the low-scoring match pushed him to a career-best 839 points in the latest ICC rankings for Test batsmen.
Australia skipper Tim Paine’s player-of-the-match innings of 73 not out has lifted him to a career-best 33rd with 592 points.
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Spinner R Ashwin has now become India’s top-ranked bowler after taking four wickets in Australia’s first innings. Ashwin, now ranked ninth, overtook pacer Jasprit Bumrah.
Ravindra Jadeja, who did not play the Adelaide Test, was unchanged at number three in the all-rounders’ list.
Australian fast bowler Pat Cummins gained six points for his seven-wicket haul as he moved from 904 to 910 for a comfortable lead over Stuart Broad in second place.
Cummins’ pace colleague, Josh Hazelwood’s figures of 5/8 in India’s second innings helped him gain four spots and move back into the top five with 805 points, the first time since March 2018.