Dieudonne

derived category D(A) of an abelian category A

The axiomatics required an innovation, the concept of triangulated category

The [length] of a finite resolution is the index n such that _________. If M admits a finite projective resolution, the minimal length among all finite projective resolutions of M is called its [projective dimension] and denoted pd(M). 

If M does not admit a finite projective resolution, then by convention the projective dimension is said to be _____.

As an example, consider a module M such that pd(M) = 0. In this situation, the exactness of the sequence 0 → P0 → M → 0 indicates that the arrow in the center is an isomorphism, and hence M itself is_____.

#3.1

Given a module, M, a [projective resolution] of M is an infinite exact sequence of modules

___________________________

with all the Pi s projective.

#3.2

Every module possesses a projective resolution. 

In fact a free resolution (resolution by free modules) exists.

#2.19

Qp/Zp has discrete topology, not trivial topology, Zp is not dense in Qp, actually take any z in Qp\Zp, z+Zp gives an open nbhd which has trivial intersection with Zp.

#2.18

change basis ——— moving the variable

#2.17

Flat implies torsionfree

#2.16

flat module over DVR

#2.15

Jacobson ring

#2.14

Change of coefficients for flatness

#2.13

Def of noetherian module: sub modules be fg