Krull ring

Let ${\displaystyle A}$ be an integral domain and let ${\displaystyle P}$ be the set of all prime ideals of ${\displaystyle A}$ of height one, that is, the set of all prime ideals properly containing no nonzero prime ideal. Then ${\displaystyle A}$ is a Krull ring if

1. ${\displaystyle A_{\mathfrak {p}}}$ is a discrete valuation ring for all ${\displaystyle {\mathfrak {p}}\in P}$,
2. ${\displaystyle A}$ is the intersection of these discrete valuation rings (considered as subrings of the quotient field of ${\displaystyle A}$).
3. Any nonzero element of ${\displaystyle A}$ is contained in only a finite number of height 1 prime ideals.

A Krull domain is a unique factorization domain if and only if every prime ideal of height one is principal.[1]

Let A be a Zariski ring (e.g., a local noetherian ring). If the completion ${\displaystyle {\widehat {A}}}$ is a Krull domain, then A is a Krull domain.[2]

1. Every integrally closed noetherian domain is a Krull ring. In particular, Dedekind domains are Krull rings. Conversely Krull rings are integrally closed, so a Noetherian domain is Krull if and only if it is integrally closed.
2. If ${\displaystyle A}$ is a Krull ring then so is the polynomial ring ${\displaystyle A[x]}$ and the formal power series ring ${\displaystyle A[[x]]}$.
3. The polynomial ring ${\displaystyle R[x_{1},x_{2},x_{3},\ldots ]}$ in infinitely many variables over a unique factorization domain ${\displaystyle R}$ is a Krull ring which is not noetherian. In general, any unique factorization domain is a Krull ring.
4. Let ${\displaystyle A}$ be a Noetherian domain with quotient field ${\displaystyle K}$, and ${\displaystyle L}$ be a finite algebraic extension of ${\displaystyle K}$. Then the integral closure of ${\displaystyle A}$ in ${\displaystyle L}$ is a Krull ring (Mori–Nagata theorem).[3]

A (Weil) divisor of a Krull ring A is a formal integral linear combination of the height 1 prime ideals, and these form a group D(A). A divisor of the form ${\displaystyle div(x)=\sum _{p\in P}v_{p}(x)\cdot p}$ for some non-zero x in K, the fractional field of ${\displaystyle A}$, is called a principal divisor, and the principal divisors form a subgroup of the group of divisors. The quotient of the group of divisors by the subgroup of principal divisors is called the divisor class group of A.

A Cartier divisor of a Krull ring is a locally principal (Weil) divisor. The Cartier divisors form a subgroup of the group of divisors containing the principal divisors. The quotient of the Cartier divisors by the principal divisors is a subgroup of the divisor class group, isomorphic to the Picard group of invertible sheaves on Spec(A).

Example: in the ring k[x,y,z]/(xy–z²) the divisor class group has order 2, generated by the divisor y=z, but the Picard subgroup is the trivial group.[4]

• N. Bourbaki. Commutative algebra.
• "Krull ring", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• Krull, Wolfgang (1931), "Allgemeine Bewertungstheorie", J. Reine Angew. Math., 167: 160–196
• Hideyuki Matsumura, Commutative Algebra. Second Edition. Mathematics Lecture Note Series, 56. Benjamin/Cummings Publishing Co., Inc., Reading, Mass., 1980. xv+313 pp. ISBN 0-8053-7026-9
• Hideyuki Matsumura, Commutative Ring Theory. Translated from the Japanese by M. Reid. Cambridge Studies in Advanced Mathematics, 8. Cambridge University Press, Cambridge, 1986. xiv+320 pp. ISBN 0-521-25916-9
• Samuel, Pierre (1964), Murthy, M. Pavman (ed.), Lectures on unique factorization domains, Tata Institute of Fundamental Research Lectures on Mathematics, 30, Bombay: Tata Institute of Fundamental Research, MR 0214579