A simplicial commutative rings is a simplicial object in the category of commutative rings (the prototypical example is , where is a simplicial set). Observe that the singular complex of a topological commutative ring is a simplicial commutative ring, and the geometric realization of a simplicial commutative ring is a topological commutative ring. The motivation comes from Dold-Kan correspondence which says when is not an abelian cateogory, we can still do ‘homological algebra’ by working with simplicial objects instead.
Recall given a ring map , we can form its Kahler differential which is well-behaved if is smooth over . In general we want to resolve by smooth -algebras. What should be ‘free’ simplicial -algebras if is a simplicial commutative ring? See definition A.6.4.
Existence of resolution: If is a map of classical commutative rings, then there exists a free simplicial -algebras and a map s.t. the composite is . Intuitively the resolution is where we either apply or just multiply within . This is weakly equivalent to by the extra degeneracy argument (same argument why the bar complex is a resolution). More generally, we can consider a monad and the algebra over the monad. A mondad often arises from unit/counit associated to an adjunction (in this case it is the functor sending to ), and actually every monad appears as in this way though the adjunction is not unique. One can also think of monad as a cateogorification of idempotents, and Beck’s monadicity theorem is related to the effectiveness of descent in the sense that a morphism is an effective descent morphism iff the base-change functor it induces is (co)monadic (monadicity/comonadicity of an adjunction between and is expressing whether we can view as the category of algebras over or vice versa, and applying this to the push-pull adjunction between and it is exactly the effectivity of descent data).
We may now define the cotangent complex of a morphism , as invented by Quillen and developed by Illusie. Let be a simplicial resolution of as a free simplicial -algebra. Then let be the simplicial -module obtained by forming Kahler differentials level-wise: . Finally we define the cotangent complex to be .
The relative cotangent sequence is not left exact in general, the reason being is not true in general (think of and a curve). This prompt us that there could be a cohomology theory explaining the lack of left exactness.
The relative conormal sequence also suggests we can further continue the relative cotangent sequence to the left. For a eady version of the idea of simplicial resolution first look at the naive cotangent complex.
Reference: https://math.uchicago.edu/~amathew/SCR.pdf