##
**Cycles on arithmetic schemes and Euler characteristics of curves.**
*(English)*
Zbl 0654.14004

Algebraic geometry, Proc. Summer Res. Inst., Brunswick/Maine 1985, part 2, Proc. Symp. Pure Math. 46, No. 2, 421-450 (1987).

[For the entire collection see Zbl 0626.00011.]

This paper develops a theory of algebraic cycles and intersection numbers on regular arithmetic schemes and uses it to establish a formula describing the behaviour of the Euler characteristic in a degenerating family of curves in mixed characteristic case. To explain this formula, let \(f: X\to S=Spec(A)\) be a flat and proper morphism, with A a complete discrete valuation ring, X a regular scheme and the generic fibre \(X_ g\) smooth. If \(X_ s\) is the closed fibre of f, then one defines the Euler number of \(X_ z\) (with \(z=s\) or \(z=\) geometric generic point \(\bar g\)) \(\chi(X_ z)\) using the étale cohomology. The formula one is interested in asks to compute the difference \(\chi(X_ s)-\chi(X_{\bar g})\). In characteristic zero such a formula is well known. For example, if \(X_ s\) has only isolated singularities, then \((- 1)^{\dim(X)}(\chi(X_ s)-\chi(X_{\bar g}))\) is just the sum of the Milnor numbers of the singularities of X. If S is of pure characteristic \(p>0\) such a formula is in general wrong, but it can be “corrected” using the socalled Swan conductor. In the arithmetic case the author is able to define the intersection number \((\Delta_ X\cdot \Delta_ X)_ s=(-1)^{\dim (X)}c_{\dim (X)}(\Omega\) \(1_{X/S})\) as a 0-cycle of \(X_ s\) and then he conjectures that \((\Delta_ X\cdot \Delta_ X)=- sw(X/S)+\chi(X_ s)-\chi (X_{\bar g})\), where \(\Delta_ X\) is the diagonal of X and \(sw(X/S)\) is the Swan correction. Finally, he shows that this formula is true if the fibre dimension of X/S is one.

[See also the author’s paper in Algebraic geometry, Proc. Symp., Sendai/Jap. 1985, Adv. Stud. Pure Math. 10, 85-90 (1987; see review 14016)].

This paper develops a theory of algebraic cycles and intersection numbers on regular arithmetic schemes and uses it to establish a formula describing the behaviour of the Euler characteristic in a degenerating family of curves in mixed characteristic case. To explain this formula, let \(f: X\to S=Spec(A)\) be a flat and proper morphism, with A a complete discrete valuation ring, X a regular scheme and the generic fibre \(X_ g\) smooth. If \(X_ s\) is the closed fibre of f, then one defines the Euler number of \(X_ z\) (with \(z=s\) or \(z=\) geometric generic point \(\bar g\)) \(\chi(X_ z)\) using the étale cohomology. The formula one is interested in asks to compute the difference \(\chi(X_ s)-\chi(X_{\bar g})\). In characteristic zero such a formula is well known. For example, if \(X_ s\) has only isolated singularities, then \((- 1)^{\dim(X)}(\chi(X_ s)-\chi(X_{\bar g}))\) is just the sum of the Milnor numbers of the singularities of X. If S is of pure characteristic \(p>0\) such a formula is in general wrong, but it can be “corrected” using the socalled Swan conductor. In the arithmetic case the author is able to define the intersection number \((\Delta_ X\cdot \Delta_ X)_ s=(-1)^{\dim (X)}c_{\dim (X)}(\Omega\) \(1_{X/S})\) as a 0-cycle of \(X_ s\) and then he conjectures that \((\Delta_ X\cdot \Delta_ X)=- sw(X/S)+\chi(X_ s)-\chi (X_{\bar g})\), where \(\Delta_ X\) is the diagonal of X and \(sw(X/S)\) is the Swan correction. Finally, he shows that this formula is true if the fibre dimension of X/S is one.

[See also the author’s paper in Algebraic geometry, Proc. Symp., Sendai/Jap. 1985, Adv. Stud. Pure Math. 10, 85-90 (1987; see review 14016)].

Reviewer: L.Bădescu

### MSC:

14C17 | Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry |

14H10 | Families, moduli of curves (algebraic) |

14A15 | Schemes and morphisms |

14C99 | Cycles and subschemes |

57R20 | Characteristic classes and numbers in differential topology |

14F45 | Topological properties in algebraic geometry |