[Paper Review] Cohen-Macaulay fiber cones

Clare D’Cruz, K. N. Raghavan+1 more|ArXiv.org|Dec 9, 2002

Commutative Algebra and Its ApplicationsMathematics16 references21 citations

TL;DR

This paper establishes a necessary and sufficient condition for the fiber cone $ F(I) $ of an ideal $ I $ in a local ring to be Cohen-Macaulay, using the Hilbert series of $ F(I) $. The key result is that $ F(I) $ is Cohen-Macaulay if and only if its Hilbert series takes a specific rational form involving the lengths of certain quotient modules, with the reduction number $ r(I) $ determining the degree of the numerator. This criterion enables the construction of new examples of Cohen-Macaulay fiber cones and confirms Shah's conjecture on the converse of his necessary condition.

ABSTRACT

Cohen Macaulay property of fiber cones of ideals is characterized in terms of its Hilbert series. Hilbert series of fiber cones of ideals with minimal mixed multiplicity is calculated. It is proved that the fiber cone of an m-primary ideal I with minimal mixed multiplicity is Cohen-Macaulay if and only if its reduction number is atmost one. Hilbert series of fiber cones of ideals generated by quadratic sequences in standard graded rings is computed by deforming it to a face ring of a simplicial complex. Applications are given to defining ideals of monomial projective space curves lying on the quadric xw-yz=0, straightening-closed ideals in graded algebras with straightening law and Huckaba-Huneke ideals of analytic spread 1 and 2.

Motivation & Objective

• To establish a necessary and sufficient condition for the fiber cone $ F(I) $ of an ideal $ I $ in a local ring to be Cohen-Macaulay.
• To confirm K. Shah's conjecture that the necessary condition for $ F(I) $ to be Cohen-Macaulay—based on its Hilbert function—also suffices.
• To characterize Cohen-Macaulay fiber cones via the Hilbert series, particularly in terms of the reduction number $ r(I) $ and the structure of quotients $ I^n / (JI^{n-1} + mI^n) $.
• To apply the criterion to construct examples of fiber cones that are or are not Cohen-Macaulay, including in regular local rings.
• To analyze fiber cones of ideals with minimal mixed multiplicity and of quadratic sequences, showing that $ F(I) $ is CM if and only if $ r(I) eq 1 $.

Proposed method

• Use of the Hilbert series $ H(F(I), t) $ as the central analytical tool to characterize the Cohen-Macaulay property of $ F(I) $.
• Derivation of the Hilbert series formula $ H(F(I), t) = \frac{1}{(1-t)^a} \sum_{i=0}^r \ell\left(\frac{I^i}{JI^{i-1} + mI^i}\right) t^i $, where $ a = \dim F(I) $, $ r = r_J(I) $, and $ J $ is a minimal reduction of $ I $.
• Application of the fact that $ JF(I) $ is generated by a regular sequence when $ F(I) $ is Cohen-Macaulay, allowing the use of the standard formula $ H(F(I), t) = \frac{1}{(1-t)^a} H(F(I)/JF(I), t) $.
• Computation of the Hilbert series of $ F(I)/JF(I) $ via the direct sum decomposition involving the quotients $ I^n / (JI^{n-1} + mI^n) $.
• Use of the equivalence between $ F(I) $ being Cohen-Macaulay and the equality $ e(F(I)) = \sum_{n=0}^r \ell\left(\frac{I^n}{JI^{n-1} + mI^n}\right) $, linking multiplicity to the structure of the fiber cone.
• Verification of the criterion through explicit computations in examples, including ideals in regular local rings and ideals generated by quadratic sequences.

Experimental results

Research questions

• RQ1 Under what conditions on the Hilbert series is the fiber cone $ F(I) $ of an ideal $ I $ in a local ring Cohen-Macaulay?
• RQ2 Is Shah's necessary condition for $ F(I) $ to be Cohen-Macaulay also sufficient?
• RQ3 What is the relationship between the reduction number $ r(I) $ and the Cohen-Macaulay property of $ F(I) $?
• RQ4 Can the Hilbert series criterion be used to construct new examples of Cohen-Macaulay fiber cones?
• RQ5 For ideals of minimal mixed multiplicity in a Cohen-Macaulay local ring, when is $ F(I) $ Cohen-Macaulay?

Key findings

• The fiber cone $ F(I) $ is Cohen-Macaulay if and only if its Hilbert series has the form $ H(F(I), t) = \frac{1}{(1-t)^a} \sum_{i=0}^r \ell\left(\frac{I^i}{JI^{i-1} + mI^i}\right) t^i $, where $ a = \dim F(I) $, $ r = r_J(I) $, and $ J $ is a minimal reduction of $ I $.
• The multiplicity of $ F(I) $ satisfies $ e(F(I)) = \sum_{n=0}^r \ell\left(\frac{I^n}{JI^{n-1} + mI^n}\right) $ if and only if $ F(I) $ is Cohen-Macaulay.
• For ideals of minimal mixed multiplicity in a Cohen-Macaulay local ring, $ F(I) $ is Cohen-Macaulay if and only if $ r(I) \leq 1 $.
• An example is constructed of a zero-dimensional ideal in a two-dimensional regular local ring whose fiber cone is not Cohen-Macaulay.
• In the case of ideals generated by quadratic sequences, such as $ I = (x_i y_j) $ in $ k[x_1,x_2,y_1,\dots,y_s] $, the fiber cone $ F(I) $ is Cohen-Macaulay with Hilbert series $ \frac{1 + (s-1)t}{(1-t)^{s+1}} $, and $ e(F(I)) = s $, showing minimal multiplicity.
• For the ideal $ I = (x^2, y^2, z^3) $ in $ k[x,y,z] $, $ F(I) $ is Cohen-Macaulay, as verified by showing $ r(I) = 2 $, $ \ell(I^2 / JI + mI^2) = 1 $, and the Hilbert series matches the criterion.