# Semistable filtered vector spaces, a Tannakian category.

Let $k$ be a field (char = 0, perhaps). Let $(V,F)$ be a pair, where $V$ is a finite-dimensional $k$-vector space, and $F$ is a filtration of $V$, indexed by rational numbers, satisfying:

1. $F^i V \supset F^j V$ when $i < j$.
2. $F^i V = V$ for $i << 0$. $F^i V = \{ 0 \}$ for $i >> 0$.
3. $F^i V = \bigcap_{j < i} F^j V$.

We define: $$F^{i+} V = \bigcup_{j > i} F^j V.$$

The slope of $(V,F)$ (when $V \neq \{ 0 \}$) is the rational number: $$M(V,F) = \frac{1}{dim(V)} \sum_{i \in Q} i \cdot dim(F^i V / F^{i+} V).$$

The pair $(V,F)$ is called semistable if $M(W, F_W) \leq M(V, F)$ for every subspace $W \subset V$, with the subspace filtration $F_W$.

A paper of Faltings and Wustholz constructs an additive category with tensor products, whose objects are semistable pairs $(V,F)$. A paper of Fujimori, "On Systems of Linear Inequalities", Bull. Soc. Math. France, seems to imply that the full subcategory of slope-zero objects (together with the zero object) is Tannakian (the abelian category axioms require semistability), with fibre functor to the category of $k$-vector spaces (though Fujimori considers quite a bit more).

Does anyone know another good reference for the properties of this Tannakian category? Can you describe the associated affine group scheme over $k$? I'm particularly interested, when $k$ is a finite field or a local field.

Update: I think the slope-zero requirement is too strong (though it is assumed in Fujimori). It seems to exclude almost all the semistable pairs $(V,F)$, if my linear algebra is correct. Anyone want to explain this to me too?

## 1 Answer

I am not sure if you are still interested in this, one year later.

Check out the papers by Yves Andre (see this and this) as well as the articles of Burt Totaro and Laurent Fargues listed in the second.

• Not too late at all. Thank you very much! I'll check out the articles you mention. – Marty Apr 14 '11 at 0:40
• Theorem 0.7 in the first paper linked above (of Andre) seems to be what you are looking for. – SGP Apr 16 '11 at 11:46