##
**Almost ring theory.**
*(English)*
Zbl 1045.13002

Lecture Notes in Mathematics 1800. Berlin: Springer (ISBN 3-540-40594-1/pbk). 307 p. (2003).

Almost ring theory was developed by the reviewer for applications in higher-dimensional ramification theory, and only to the extent necessary for that purpose. The term “almost” refers to a non-discrete \(p\)-adic valuation and indicates that one ignores modules annihilated by any element of positive valuation. That is one works in a quotient of the category of modules. The one-dimensional case is due to J. Tate who observed that one highly ramified extension kills all other ramification.

In the present book the theory is presented in a much more systematic manner. Instead of non-discrete valuations one considers ideals equal to their square, with a thorough discussion about which additional assumptions should be made. After that much of classical commutative algebra gets revisited and reworked in a new perspective, as finiteness conditions, flat and projective modules, unramified, smooth and étale maps, henselian pairs, valuation rings. One difference to the classical treatments is the systematic use of the cotangent complex.

Finally, the last chapter investigates this complex for adic spaces, presumably as a preparation for an extension of the theory into a rigid context.

In the present book the theory is presented in a much more systematic manner. Instead of non-discrete valuations one considers ideals equal to their square, with a thorough discussion about which additional assumptions should be made. After that much of classical commutative algebra gets revisited and reworked in a new perspective, as finiteness conditions, flat and projective modules, unramified, smooth and étale maps, henselian pairs, valuation rings. One difference to the classical treatments is the systematic use of the cotangent complex.

Finally, the last chapter investigates this complex for adic spaces, presumably as a preparation for an extension of the theory into a rigid context.

Reviewer: Gerd Faltings (Bonn)

### MSC:

13B40 | Étale and flat extensions; Henselization; Artin approximation |

12J20 | General valuation theory for fields |

14G22 | Rigid analytic geometry |

13-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to commutative algebra |

13-02 | Research exposition (monographs, survey articles) pertaining to commutative algebra |

13A18 | Valuations and their generalizations for commutative rings |

14E22 | Ramification problems in algebraic geometry |