Against the Slow Slav, after 1. d4 d5 2. c4 c6 3. Nf3 Nf6 4. e3, is the 4... a6 Chebanenko still good for Black? Someone told me that:


Against the move order 3. Nf3 Nf6 4. Nc3 or 3. Nc3 Nf6 4. Nf3, the move 4...a6 is averagely effective.
Against the move order 3. Nc3 Nf6 4. e3, the move 4...a6 is extremely effective, since White has already committed to playing e3, so White has lost many of his options (White now only has two options: 5. Nf3 or 5. Qc2).
Against the move order 3. Nf3 Nf6 4. e3, the move 4...a6 is not very effective (it's still 'playable' but it doesn't make as much sense as in the usual Chebanenko), because White has not committed his b1 Knight to c3, so the ...b5 push is not as strong since White can manoeuvre his Knight to the c5 square more easily and ...b4 wouldn't gain a tempo on a White Knight.


Is this really true?

I've looked into many games databases (openings explorers), but they don't seem to help in answering this question. Against 1. d4 d5 2. c4 c6 3. Nf3 Nf6 4. e3 the move 4...a6 is still played very often, though Black scores very slightly worse than average against 5. Bd3, 5. b3, 5. Nbd2 and 5. Qc2.
 But the fact that it is played often doesn't necessarily reflect if it's good or not, it might simply be due to the fact that many Chebanenko players play 4...a6 against this move order anyways, hoping that White kindly transposes into a usual Chebanenko with 5. Nc3.
And the fact that it scores slightly worse than usual for Black might simply be a statistical fluctuation, or it might just be caused by the fact that Chebanenko players are maybe less prepared against these White options (5. Bd3, 5. b3, 5. Nbd2 and 5. Qc2) because they are rarer than the usual Chebanenko lines.

I would like to know if 4... a6 against the Slow Slav is objectively good or not.
I am by no means an openings expert, but I have played the Slow Slav as White for a long time so I will just make some comments.

In general, lines with an early b7-b5 in Chebanenko positions tend to work better with a Nc3 than a Nd2.  As you have already noted, a Nd2 makes it much more likely that White will be able to find some way to exploit the holes on Black's queenside (particularly c5, a5, and c6, often exploited by a well-timed a2-a4) created by an early b7-b5.  Positions with b7-b5 and Nb1-d2 are not necessarily very good for White, but they do seem to give a better chance of White gaining a small advantage than the main lines, and a small advantage is usually the best that White can hope for against the Chebanenko these days.  Positions with a Nc3 make it much more difficult for White to exploit the holes Black makes on the queenside with b7-b5, and give Black more dynamic compensation for such holes due to the possibility in many lines of a well-timed b5-b4, forcing the Nc3 to an awkward square.

Since an eventual b7-b5 is the general idea behind a7-a6, I always respond to 4...a6 by playing Nb1-d2 (either right away, or more commonly a little later after Bf1-d3). This is probably White's best try for an advantage in these lines. If Black goes ahead and plays b7-b5 anyways, I generally get the kind of position I want, which I would describe as lightly +=.

However, if Black refrains from playing a quick b7-b5 and plays something like an immediate Bc8-g4 (say, after 4...a6 5.Bd3), he still has a very solid Slav position in which the "wasted" move a7-a6 is compensated for by the fact that d2 is not an ideal square for the White N in lines where Black does not play b7-b5 (that is, because it removes some of the pressure from d5 and because in some variations the Nd2 can also get in the way of the Nf3).  Typically, in positions with a strong center like this (pawns at c4/d4/e3), White would prefer his Ns on the most active squares, c3 and f3, where they simultaneously pressure Black's center and support possible pawn advances by White.  So the move Nbd2 after a7-a6 may also be viewed as a kind concession.

I like to look at the exchange as a kind of exercise in prophylaxis: Black plays a7-a6 as prophylaxis against White's ideal set-up in the center with Nc3; White then plays Nb1-d2 (or just refrains for committing the N to c3 for a while) as prophylaxis against Black's idea of b7-b5.  Black, having now discouraged White from playing his ideal set-up with the N on c3 (I use the word "discouraged" very loosely here because positions with Nc3 and b7-b5 are fully playable for White), then plays Bc8-g4 with the idea that this set-up is even better after the exchange of a7-a6 and Nb1-d2, because a7-a6 does not hurt Black and the Nd2 is slightly misplaced.  White is not entirely convinced of this last point, because Nd2 is at least a developing move, while a7-a6 is not.  And so the discussion continues, with White trying to find lines in which his Nbd2 is actually useful.

As far as I know, there is no clear way to an advantage for White in lines with the Nbd2/Bd3 set-up against a7-a6, but the play is different and less forcing that the positions with a Nc3.  Having played many games with this set-up and a number of games with the main line Slow Slav positions (that is, with an immediate Bc8-f5 or Bc8-g4 instead of a7-a6), I would evaluate both set-ups as roughly the same: +=/=.  They are very slightly easier to play for White, but White has no significant advantage if Black knows what he/she is doing.

The upshot of all this is that 4...a7-a6 appears to be fine against the Slow Slav set-up as long as Black refrains from an early b7-b5 before White commits his N to c3.  In such positions, the move a7-a6 once again proves to be a high-class waiting move, with a dollop of added prophylaxis.