Can a football referee show a direct red card to a player because of an event happened 10 minutes ago?
Assume that the game was played in those minutes.
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They can, but they should not.
Law 12, section 3 Disciplinary Action begins
The referee has the authority to take disciplinary action from entering the field of play for the pre-match inspection until leaving the field of play after the match ends (including kicks from the penalty mark).
It later states in part Delaying the restart of play to show a card,
Once the referee has decided to caution or send off a player, play must not be restarted until the sanction has been administered, unless the non-offending team takes a quick free kick, has a clear goal-scoring opportunity and the referee has not started the disciplinary sanction procedure. The sanction is administered at the next stoppage; [two specific situations excepted]
Thus, if play has not been stopped, or was restarted under particular circumstances, the referee may delay the sending-off (red card) until the next stoppage in play.
But the following part Advantage clarifies,
Advantage should not be applied in situations involving serious foul play, violent conduct or a second cautionable offence unless there is a clear opportunity to score a goal. The referee must send off the player when the ball is next out of play, but if the player plays the ball or challenges/interferes with an opponent, the referee will stop play, send off the player and restart with an indirect free kick, unless the player committed a more serious offence.
Thus, IFAB clearly expects a player who should be sent off, to be sent off immediately, except when doing so would prevent a goalscoring opportunity for the opponent. This is more limiting than the previous statement, but allows for discretion to allow another sequence of play if it is more advantageous to the opponent.
It is not recommended that a referee decide or, through indecision, allow such a situation to occur, but, if it does occur due to a delay in making the decision to send off, that play can continue and the disciplinary sanction applied afterwards.
Ten minutes is a very long sequence of play without a stoppage, and, unless the opponent spent the entirety of that time in the midst of a goalscoring opportunity, should have been stopped earlier.