# On a conjecture by Novakovic

In his recent preprint No full exceptional collections on non-split Brauer–Severi varieties of dimension $\leq 3$ Sasa Novakovic conjectured the following:

**Conjecture** *Let $X\neq\mathbb{P}_k^n$ be a Brauer–Severi variety. Then $X$ does not admit a full (strong) exceptional collection consisting of arbitrary objects.*

In his paper he proves this (as the preprint's title already suggests) in dimension less than 3, by using the transitivity of the braid group action on the set of exceptional collections. Bondal and Polishchuk conjectured this for arbitrary triangulated categories, but it is only known in very special cases, mainly for the projective line, plane and three-space (over an algebraically closed field). From this he can prove the conjecture in the special cases where the degree of the central simple algebra is at most 4.

Recently Theo Raedschelders showed how one can use the results of the preprint Noncommutative motives of separable algebras by Tabuada–Van den Bergh to prove the conjecture in complete generality. He has written a note Non-split Severi–Brauer varieties do not admit full exceptional collections about it, available on his website. It is a very cool application of some heavy and abstract machinery (namely noncommutative motives)!

### Semi-orthogonal decompositions

Recall that there is a well-known semi-orthogonal decomposition for the Brauer–Severi variety of a central simple algebra. In A semiorthogonal decomposition for Brauer–Severi schemes Bernardara gave a semi-orthogonal decomposition generalising Orlov's projective bundle formula (or Beilinson's decomposition for projective space if you like), categoryfing Quillen's result for algebraic K-theory. For a central simple algebra $A$ of degree $n$ it says that $\mathbf{D}^{\mathrm{b}}(A^{\otimes i})$ is an admissible subcategory, and that there exists a semi-orthogonal decomposition \begin{equation} \mathbf{D}^{\mathrm{b}}(\mathrm{BS}(A)=\langle\mathbf{D}^{\mathrm{b}}(k),\mathbf{D}^{\mathrm{b}}(A),\ldots,\mathbf{D}^{\mathrm{b}}(A^{\otimes n})\rangle \end{equation}

Observe that $\mathbf{D}^{\mathrm{b}}(A^{\otimes i})$ in general is *not* equivalent to the subcategory generated by an exceptional object, sometimes one says that it is given by a semi-exceptional object. So Novakovic's conjecture tells you that (non-trivial) Brauer–Severi varieties cannot have a better choice of objects giving rise to a full exceptional collection. For the matrix algebra the Brauer–Severi variety is just $\mathbb{P}_k^{n-1}$, for which Beilinson's collection is a full collection.

### On the proof

Observe that it is even possible to immediately appeal to a result which is already in the Tabuada–Van den Bergh preprint. For this it suffices to realise that having a full exceptional collection in the derived category of the Brauer–Severi variety indeed means that its noncommutative motive is isomorphic to the noncommutative motive of the Brauer–Severi variety of the matrix algebra (of the same degree). But by theorem 3.12 (whose proof goes along the lines of the argument in Theo's note) this means that the central simple algebra must generate the same subgroup of the Brauer group as the matrix algebra, hence is necessarily trivial itself.

To get a feel for how one proves such a result regarding noncommutative motives, I advise you to take a look at Tabuada's Additive invariants of toric and twisted projective homogeneous varieties via noncommutative motives, where all the details are collected in section 9 and the proof of Novakovic's conjecture follows immediately from proposition 9.2 (bypassing some of the steps in the more general Tabuada–Van den Bergh paper). Given the framework of noncommutative motives, it boils down to the fact that everything in the derived category of a central simple algebra is formal, that the Grothendieck group of a central simple algebra is always just the integers, and some very explicit linear algebra over the integers.