Tags » Cohomology

impossible objects

Recently in between studying for my finals I’ve been learning about a whole load of different things in algebraic geometry, a main one being cohomology… 733 more words


Forms and Galois cohomology

Yesterday¬†we gave a brief and abstract description of Galois descent, the punchline of which was that Galois descent could abstractly be described as a natural equivalence… 1,352 more words


Projective representations are homotopy fixed points

Yesterday we described how a (finite-dimensional) projective representation of a group functorially gives rise to a -linear action of on such that the Schur class classifies this action. 870 more words


Projective representations give categorical representations

Today we’ll resolve half the puzzle of why the cohomology group appears both when classifying projective representations of a group over a field and when classifying -linear actions of on the category of -vector spaces by describing a functor from the former to the latter. 898 more words


Projective representations

Three days ago we stated the following puzzle: we can compute that isomorphism classes of -linear actions of a group on the category of vector spaces over a field correspond to elements of the cohomology group… 662 more words


Fixed points of group actions on categories

Previously we described what it means for a group to act on a category (although we needed to slightly correct our initial definition). Today, as the next step in our attempt to understand… 1,318 more words


Group actions on categories

Yesterday we decided that it might be interesting to describe various categories as “fixed points” of Galois actions on various other categories, whatever that means: for example, perhaps real Lie algebras are the “fixed points” of a Galois action on complex Lie algebras. 752 more words