The Stacks project

90.2 Advice for the reader

In writing this chapter we have tried to minimize the use of simplicial techniques. We view the choice of a resolution $P_\bullet $ of a ring $B$ over a ring $A$ as a tool to calculating the homology of abelian sheaves on the category $\mathcal{C}_{B/A}$, see Remark 90.5.5. This is similar to the role played by a “good cover” to compute cohomology using the Čech complex. To read a bit on homology on categories, please visit Cohomology on Sites, Section 21.38. The derived lower shriek functor $L\pi _!$ is to homology what $R\Gamma (\mathcal{C}_{B/A}, -)$ is to cohomology. The category $\mathcal{C}_{B/A}$, studied in Section 90.4, is the opposite of the category of factorizations $A \to P \to B$ where $P$ is a polynomial algebra over $A$. This category comes with maps of sheaves of rings

\[ \underline{A} \longrightarrow \mathcal{O} \longrightarrow \underline{B} \]

where over the object $U = (P \to B)$ we have $\mathcal{O}(U) = P$. It turns out that we obtain the cotangent complex of $B$ over $A$ as

\[ L_{B/A} = L\pi _!(\Omega _{\mathcal{O}/\underline{A}} \otimes _\mathcal {O} \underline{B}) \]

see Lemma 90.4.3. We have consistently tried to use this point of view to prove the basic properties of cotangent complexes of ring maps. In particular, all of the results can be proven without relying on the existence of standard resolutions, although we have not done so. The theory is quite satisfactory, except that perhaps the proof of the fundamental triangle (Proposition 90.7.4) uses just a little bit more theory on derived lower shriek functors. To provide the reader with an alternative, we give a rather complete sketch of an approach to this result based on simple properties of standard resolutions in Remarks 90.7.5 and 90.7.6.

Our approach to the cotangent complex for morphisms of ringed topoi, morphisms of schemes, morphisms of algebraic spaces, etc is to deduce as much as possible from the case of “plain ring maps” discussed above.


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