(also nonabelian homological algebra)
homotopy theory, (∞,1)-category theory, homotopy type theory
flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed…
models: topological, simplicial, localic, …
see also algebraic topology
Introductions
Definitions
Paths and cylinders
Homotopy groups
Basic facts
Theorems
The homological resolution of a chain complex by a complex of free modules.
Assuming the axiom of choice, then every abelian group $A$ admits a free resolution of just length 2, hence with trivial syzygies, hence there exists a short exact sequence of abelian groups of the form
where both $F_1$ and $F_2$ are free abelian groups.
Let $F_1 = \mathbb{Z}[A]$ be the free abelian group on the underlying set of $A$, and let $F_1 \to A$ be the canonical morphism that sends a generator to itself (the adjunction counit of the free-forgetful adjunction).
Then let $F_2 \to F_1$ be the kernel of that map, hence $F_2$ a subgroup of $F_1$. Assuming the axiom of choice then every subgroup of a free abelian group is itself free abelian, hence $F_2$ is free abelian (prop.).
Prop. 2 implies that all Ext-groups between abelian groups are concentrated in degrees 0 and 1.
(By the discussion at derived functor in homological algebra.)
(Koszul complex is free resolution of quotient ring)
For $R$ a commutative ring and $(x_1, \cdots, x_d)$ a regular sequence of elements, then the Koszul complex $K(x_1, \cdots, x_d)$ is a free resolutions of the quotient ring $R/(x_1, \cdots, x_d)$ by free $R$-modules.
projective object, projective presentation, projective cover, projective resolution
injective object, injective presentation, injective envelope, injective resolution
free object, free resolution
flat object, flat resolution
Lecture notes include for instance
around page 5 of