Derived algebraic geometry

... ∞-categories is a hairy business, the ∞-groupoids are well-understood: they are essentially the same thing as spaces (say, CW-complexes), as studied in homotopy theory. If X is any space, then it gives rise to an ∞-groupoid as follows: the objects are the points of X, the morphisms are the paths bet ...

... ∞-categories is a hairy business, the ∞-groupoids are well-understood: they are essentially the same thing as spaces (say, CW-complexes), as studied in homotopy theory. If X is any space, then it gives rise to an ∞-groupoid as follows: the objects are the points of X, the morphisms are the paths bet ...

Locally normal subgroups of totally disconnected groups. Part II

... said to be h.j.i. if every non-trivial closed locally normal subgroup is open. • atomic type: |LN (G)| > 2 but LC(G) = {0, ∞}, there is a unique least element of LN (G) r {0}, the action of G on LN (G) is trivial and G is not abstractly simple. • non-principal filter type (abbreviated by NPF type): ...

... said to be h.j.i. if every non-trivial closed locally normal subgroup is open. • atomic type: |LN (G)| > 2 but LC(G) = {0, ∞}, there is a unique least element of LN (G) r {0}, the action of G on LN (G) is trivial and G is not abstractly simple. • non-principal filter type (abbreviated by NPF type): ...

Sheaves of Modules

... and only if ϕ ◦ α = 0. Moreover, on the level of stalks we have Ker(ϕ)x = Ker(ϕx ). On the other hand, we define Coker(ϕ) as the sheaf of OX -modules associated to the presheaf of OX -modules defined by the rule U 7−→ Coker(G(U ) → F(U )) = F(U )/ϕ(G(U )). Since taking stalks commutes with taking sh ...

... and only if ϕ ◦ α = 0. Moreover, on the level of stalks we have Ker(ϕ)x = Ker(ϕx ). On the other hand, we define Coker(ϕ) as the sheaf of OX -modules associated to the presheaf of OX -modules defined by the rule U 7−→ Coker(G(U ) → F(U )) = F(U )/ϕ(G(U )). Since taking stalks commutes with taking sh ...