1,577 research outputs found
Universal classes for algebraic groups
We exhibit cocycles representing certain classes in the rational cohomology
of of the general linear group with coefficients in the divided powers of a
Frobenius twist of the adjoint representation. These classes' existence was
anticipated by van der Kallen, and they intervene in the proof that reductive
linear algebraic groups have finitely generated cohomology algebras.Comment: 24 pages. v2, slight modifications : corollary 0.2 added, additional
comments in section 2, one reference added. To appear in Duke Math.
Ringel duality and derivatives of non-additive functors
We prove that Ringel duality in the category of strict polynomial functors
can be interpreted as derived functors of non-additive functors (in the sense
of Dold and Puppe). We give applications of this fact for both theories.Comment: Fourth version, 48 pages. Minor changes (typos corrected, comments
and references added). The article is self-contained (no prior knowledge of
Schur algebras, strict polynomial functors or derived functors of
non-additive functors is required
Troesch complexes and extensions of strict polynomial functors
We develop a new approach of extension calculus in the category of strict
polynomial functors, based on Troesch complexes. We obtain new short elementary
proofs of numerous classical Ext-computations as well as new results.
In particular, we get a cohomological version of the `fundamental theorems'
from classical invariant invariant theory for GL_n for n big enough (and we
give a conjecture for smaller values of n).
We also study the `twisting spectral sequence' E^{s,t}(F,G,r) converging to
the extension groups Ext^*(F^{(r)}, G^{(r)}) between the twisted functors
F^{(r)} and G^{(r)}. Many classical Ext-computations simply amount to the
collapsing of this spectral sequence at the second page (for lacunary reasons),
and it is also a convenient tool to study the effect of the Frobenius twist on
Ext-groups. We prove many cases of collapsing, and we conjecture collapsing is
a general fact.Comment: Revised version, 46 pages. Mathematics unchanged. Typos corrected,
Appendix 9 on Troesch complexes improve
M-curves of degree 9 with deep nests
The first part of Hilbert's sixteenth problem deals with the classification
of the isotopy types realizable by real plane algebraic curves of given degree
. For , one restricts the study to the case of the -curves. For
, the classification is still wide open. We say that an -curve of
degree 9 has a deep nest if it has a nest of depth 3. In the present paper, we
prohibit 10 isotopy types with deep nests and no outer ovals.Comment: 16 pages, 11 figures v.4 minimal correction
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