## 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

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

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.

## Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like `$\pi$`

). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

## Comments (0)