Semistable filtered vector spaces, a Tannakian category. - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T22:57:12Z http://mathoverflow.net/feeds/question/18538 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/18538/semistable-filtered-vector-spaces-a-tannakian-category Semistable filtered vector spaces, a Tannakian category. Marty 2010-03-18T02:06:48Z 2011-04-14T00:17:26Z <p>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: </p> <ol> <li>$F^i V \supset F^j V$ when $i &lt; j$.</li> <li>$F^i V = V$ for $i &lt;&lt; 0$. $F^i V = { 0 }$ for $i >> 0$.</li> <li>$F^i V = \bigcap_{j &lt; i} F^j V$.</li> </ol> <p>We define: $$F^{i+} V = \bigcup_{j > i} F^j V.$$</p> <p>The slope of $(V,F)$ (when $V \neq { 0 }$) is the rational number: $$M(V,F) = \frac{1}{dim(V)} \sum_{i \in Q} i \cdot dim(F^i V / F^{i+} V).$$</p> <p>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 $F_W$. </p> <p>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).</p> <p>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.</p> <p>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?</p> http://mathoverflow.net/questions/18538/semistable-filtered-vector-spaces-a-tannakian-category/61612#61612 Answer by SGP for Semistable filtered vector spaces, a Tannakian category. SGP 2011-04-14T00:17:26Z 2011-04-14T00:17:26Z <p>I am not sure if you are still interested in this, one year later.</p> <p>Check out the papers by Yves Andre (see <a href="http://arxiv.org/abs/1008.1553" rel="nofollow">this</a> and <a href="http://arxiv.org/abs/0812.3921" rel="nofollow">this</a>) as well as the articles of Burt Totaro and Laurent Fargues listed in the second. </p>