The cotangent complex of a ring map

Let $A\to B$ be a homomorphism of commutative unital rings (Commutative ring) and let $ϵ\text{\hspace{0.17em}⁣}:P_{\bullet }\to B$ be its standard resolution (The standard simplicial resolution of a ring map, Simplicial objects, simplicial commutative rings and homotopy groups).

For each $n\ge 0$ the module of Kähler differentials ${\mathrm{\Omega }}_{P_n\mathrm{/}A}$ (Universal Kähler differential module, Derivation of an algebra) is a $P_n$-module, and the augmentation ${ϵ}_n\text{\hspace{0.17em}⁣}:P_n\to B$ makes $B$ a $P_n$-algebra, so $M_n:={\mathrm{\Omega }}_{P_n\mathrm{/}A}{\otimes }_{P_n}B$ is a $B$-module. The face maps $d_i\text{\hspace{0.17em}⁣}:P_n\to P_{n-1}$ induce $B$-linear maps $M_n\to M_{n-1}$ by $\omega \otimes b\mapsto d_i(\omega \otimes b)$, using the $P_n$-algebra structure on $M_{n-1}$ induced by $ϵ$; the alternating sum ${\mathrm{\partial }}_n={\sum }_{i=0}^n(-1{)}^i{d}_i\text{\hspace{0.17em}⁣}:M_n\to M_{n-1}(n\ge 1)$ is completed by ${\mathrm{\partial }}_0=0$ and satisfies ${\mathrm{\partial }}_{n-1}{\mathrm{\partial }}_n=0$ by the simplicial identities together with the Leibniz rule, so $(M_{\bullet },\mathrm{\partial })$ is a chain complex of $B$-modules concentrated in nonnegative degrees (Chain complex in an abelian category).

The cotangent complex $L_{B\mathrm{/}A}$ is this complex, indexed cohomologically by negating degrees: $L_{B\mathrm{/}A}^{-n}:=M_n$ with differential ${\mathrm{\partial }}_n\text{\hspace{0.17em}⁣}:L_{B\mathrm{/}A}^{-n}\to L_{B\mathrm{/}A}^{-(n-1)}$, the sign and reindexing conventions being those of the shift operation on complexes (The shift of a chain complex). Thus $L_{B\mathrm{/}A}$ is concentrated in degrees $\le 0$, so that $H^0(L_{B\mathrm{/}A})$ is the degree-zero cohomology of a map $M_1\to M_0$.

The complex is well defined without any choice: the standard resolution $P_{\bullet }$ is constructed explicitly by iterating the free polynomial algebra functor, and the tensor product, the differential and the reindexing are degreewise formulas. A simplicial resolution of $B$ over $A$ is any augmented simplicial $A$-algebra $Q_{\bullet }\to B$ with every $Q_n$ a polynomial $A$-algebra and the augmentation a weak equivalence (Simplicial objects, simplicial commutative rings and homotopy groups); its associated complex ${\mathrm{\Omega }}_{Q_{\bullet }\mathrm{/}A}{\otimes }_{Q_{\bullet }}B$ is formed by the same degreewise formula. The independence of $L_{B\mathrm{/}A}$ from the chosen resolution, up to canonical isomorphism in the derived category $D(B)$ (Quasi-isomorphism), is the comparison theorem proved separately; it is not part of the definition.