Swan conductors for characters of degree one in the imperfect residue field case.

*(English)*Zbl 0716.12006
Algebraic \(K\)-theory and algebraic number theory, Proc. Semin., Honolulu/Hawaii 1987, Contemp. Math. 83, 101-131 (1989).

[For the entire collection see Zbl 0655.00010.]

Let K be a complete discrete valuation field with the residue field F. Let L be a finite Galois extension of K with Galois group Gal(L/K). In this paper, Swan conductors (and the refined Swan conductors) are defined for characters of degree one of Gal(L/K) without separability assumption on F, and some of their geometrical properties are discussed. The paper consists of two parts: Part I deals with the definitions of Swan (and the refined Swan) conductors, and Part II with their geometrical properties.

Let \(H^ q(K):=H^ q(K,({\mathbb{Q}},{\mathbb{Z}})(q-1))\), \(q>0\) be a Galois cohomology group. For \(n\geq 1\), let \(fil_ nH^ q(K)\) be the subgroup of \(H^ q(K)\) consisting of elements of \(H^ q(K)\) such that the cup product with \(\chi\) annihilates, in some universal sense, the \((n+1)\)-th unit group. The Swan conductor, sw(\(\chi\)), is defined to be the smallest integer \(n\geq 0\) such that \(\chi \in fil_ nH^ q(K)\). This definition, however, is valid only for characters \(\chi\) of degree one.

One of the main results of the paper is formulated in the following Theorem. Let \(\Omega_ F^{\bullet}\) denote the exterior algebra of the absolute differential module \(\Omega^ 1_ F=\Omega^ 1_{F/{\mathbb{Z}}}\). Then there is an injective homomorphism \(fil_ n/fil_{n-1}\to \Omega^ q_ F\oplus \Omega_ F^{q-1}.\) This homomorphism is defined canonically once a prime element of K is fixed.

To remedy the dependence of the above homomorphism on a prime element of K, the refined Swan conductors are defined. Discussions on characteristic properties of Swan conductors and the refined Swan conductors as well as explicit description of the image of the above homomorphism when K has characteristic \(p>0\) take up the rest of Part I.

Geometric properties of Swan conductors are discussed in Part II. Integral property of Swan conductors, and behaviours of Swan conductors under blow-ups, specializations and restrictions are studied. Swan conductors are shown to get larger after specialization. A result on Swan conductors of the “restrictions to curves” is formulated as follows. Let A be a regular local ring of dimension \(\geq 2\) with the field of quotients M and the maximal ideal \(m_ A\). Let \({\mathfrak p}\) be a prime ideal of A of height one such that A/\({\mathfrak p}\) is regular. Let \(A_{{\mathfrak p}}\) denote the local ring of A at \({\mathfrak p}\), which is a discrete valuation ring.

Theorem. Let \(q\geq 0\) and let \(\chi \in H^ q(M,({\mathbb{Q}}/{\mathbb{Z}})(q-1))\) be a character which is “unramified” on Spec(A)-\(\{\) \({\mathfrak p}\}\). Let \(n=sw(\chi)\) be the Swan conductor of \(\chi\) with respect to \(A_{{\mathfrak p}}\). Let \({\mathfrak p},{\mathfrak p}'\in Spec(A)\). Assume that dim(A/\({\mathfrak p}')=\dim (A/{\mathfrak p}'')=1\), \({\mathfrak p}+{\mathfrak p}'={\mathfrak p}+{\mathfrak p}''={\mathfrak m}_ A\). Then, if \(length_ A(A/({\mathfrak p}'+{\mathfrak p}''))\geq n\), the “restriction” of \(\chi\) to Spec \(\kappa\) (\({\mathfrak p}')\) and Spec \(\kappa\) (\({\mathfrak p}'')\) have the same Swan conductor sw(\(\chi|\) \({\mathfrak p}')=sw(\chi |\) \({\mathfrak p}'')\).

Let K be a complete discrete valuation field with the residue field F. Let L be a finite Galois extension of K with Galois group Gal(L/K). In this paper, Swan conductors (and the refined Swan conductors) are defined for characters of degree one of Gal(L/K) without separability assumption on F, and some of their geometrical properties are discussed. The paper consists of two parts: Part I deals with the definitions of Swan (and the refined Swan) conductors, and Part II with their geometrical properties.

Let \(H^ q(K):=H^ q(K,({\mathbb{Q}},{\mathbb{Z}})(q-1))\), \(q>0\) be a Galois cohomology group. For \(n\geq 1\), let \(fil_ nH^ q(K)\) be the subgroup of \(H^ q(K)\) consisting of elements of \(H^ q(K)\) such that the cup product with \(\chi\) annihilates, in some universal sense, the \((n+1)\)-th unit group. The Swan conductor, sw(\(\chi\)), is defined to be the smallest integer \(n\geq 0\) such that \(\chi \in fil_ nH^ q(K)\). This definition, however, is valid only for characters \(\chi\) of degree one.

One of the main results of the paper is formulated in the following Theorem. Let \(\Omega_ F^{\bullet}\) denote the exterior algebra of the absolute differential module \(\Omega^ 1_ F=\Omega^ 1_{F/{\mathbb{Z}}}\). Then there is an injective homomorphism \(fil_ n/fil_{n-1}\to \Omega^ q_ F\oplus \Omega_ F^{q-1}.\) This homomorphism is defined canonically once a prime element of K is fixed.

To remedy the dependence of the above homomorphism on a prime element of K, the refined Swan conductors are defined. Discussions on characteristic properties of Swan conductors and the refined Swan conductors as well as explicit description of the image of the above homomorphism when K has characteristic \(p>0\) take up the rest of Part I.

Geometric properties of Swan conductors are discussed in Part II. Integral property of Swan conductors, and behaviours of Swan conductors under blow-ups, specializations and restrictions are studied. Swan conductors are shown to get larger after specialization. A result on Swan conductors of the “restrictions to curves” is formulated as follows. Let A be a regular local ring of dimension \(\geq 2\) with the field of quotients M and the maximal ideal \(m_ A\). Let \({\mathfrak p}\) be a prime ideal of A of height one such that A/\({\mathfrak p}\) is regular. Let \(A_{{\mathfrak p}}\) denote the local ring of A at \({\mathfrak p}\), which is a discrete valuation ring.

Theorem. Let \(q\geq 0\) and let \(\chi \in H^ q(M,({\mathbb{Q}}/{\mathbb{Z}})(q-1))\) be a character which is “unramified” on Spec(A)-\(\{\) \({\mathfrak p}\}\). Let \(n=sw(\chi)\) be the Swan conductor of \(\chi\) with respect to \(A_{{\mathfrak p}}\). Let \({\mathfrak p},{\mathfrak p}'\in Spec(A)\). Assume that dim(A/\({\mathfrak p}')=\dim (A/{\mathfrak p}'')=1\), \({\mathfrak p}+{\mathfrak p}'={\mathfrak p}+{\mathfrak p}''={\mathfrak m}_ A\). Then, if \(length_ A(A/({\mathfrak p}'+{\mathfrak p}''))\geq n\), the “restriction” of \(\chi\) to Spec \(\kappa\) (\({\mathfrak p}')\) and Spec \(\kappa\) (\({\mathfrak p}'')\) have the same Swan conductor sw(\(\chi|\) \({\mathfrak p}')=sw(\chi |\) \({\mathfrak p}'')\).

Reviewer: N.Yui

##### MSC:

12J12 | Formally \(p\)-adic fields |

11S31 | Class field theory; \(p\)-adic formal groups |

19F05 | Generalized class field theory (\(K\)-theoretic aspects) |

12G05 | Galois cohomology |

11S70 | \(K\)-theory of local fields |

19D45 | Higher symbols, Milnor \(K\)-theory |

11S25 | Galois cohomology |

13H05 | Regular local rings |