Possible noncommutative progress on the standard conjectures
Two days ago Matilde Marcolli and Goncalo Tabuada uploaded a survey article to the arXiv titled Noncommutative motives and their applications. It is a nice introduction to the notion of noncommutative motives, and taking dg categories as the main player in noncommutative geometry.
But one particular paragraph in this article intrigued me, paragraph 5.2 if you can't wait to read it. Recall that the theory of motives is tightly related to Grothendieck's standard conjectures on algebraic cycles. These conjectures are so far unproven, although lots of work have gone into them, mostly in trying to connect these conjectures to possibly easier to tackle problems. But there is hope to prove one of them using noncommutative algebraic geometry.
In their article Marcolli and Tabuada survey the "noncommutative standard conjectures". And they summarize what is known about them: the commutative conjecture implies the noncommutative conjecture, but not vice versa. But there is an equivalence in one case! Namely for the smash-nilpotence conjecture as proposed by Voevodsky.
Conjecture D of the standard conjectures (they all have a letter, there's also A, B and C) says that two types of equivalence relations on algebraic cycles, namely homological equivalence (depending on your choice of a Weil cohomology) and numerical equivalence, are the same. It is easy to prove (if I remember correctly) that numerical equivalence is weaker than homological equivalence (numerical equivalence being the weakest equivalence relation that satisfies some nice properties). Hence this conjecture says that it doesn't matter which Weil cohomology you choose, you will get the same equivalence relation.
So far, parts of this conjecture have been proven apparently, in cases like divisors with coefficients in a field of characteristic zero (Matsusaka's theorem, which even predates the standard conjectures!).
Another approach to these conjectures is proposed by Vladimir Voevodsky. He introduced the smash-nilpotence equivalence (again, see the Wikipedia page on equivalence relations), and proved that it is stronger than homological equivalence. Then he conjectured that it is actually equal to numerical equivalence too, so if one can prove this you get conjecture D for free.
And the coolest thing is that the noncommutative smash-nilpotence conjecture is equivalent to the commutative smash-nilpotence conjecture! The proof of this can be found in Unconditional motivic Galois groups and Voevodsky's nilpotence conjecture in the noncommutative world by Marcolli and Tabuada, in theorem 4.1. The proof seems almost trivial, but dependent on some of the constructions and I'm not familiar with these, so I can't say more on the result (for now). If I ever have more to say about it I will. Possibly when they have made the progress they are hoping for, as indicated in the article.