Efficient generation of ideals in discrete Hodge algebras and the behavior of Euler cycles under subintegral extensions

Jebasingh, R (2025) Efficient generation of ideals in discrete Hodge algebras and the behavior of Euler cycles under subintegral extensions. PhD thesis, Indian Institute of Science Education and Research Kolkata.

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Abstract

In 1975, Pavaman Murthy posed the question of whether every ideal I in k[x₁, · · · , xn], where k is a field, with trivial conormal bundle must necessarily complete intersection ideal. This statement, known as Murthy’s complete intersection conjecture, has seen limited progress, with the only significant result being due to Mohan Kumar (1978), who provided an affirmative answer when ht(I) = μ(I/I²) ≥ dim(k[X₁, · · · , Xd]/I) + 2. A discrete Hodge algebra over a Noetherian ring R is the coordinate ring of unions of intersections of coordinate hyperplanes. Ton vorst (1983) and AndreasWeimers (1992) demonstrated that large rank projective modules over such algebras behave similarly to those over polynomial rings. Motivated by the works of Mohan Kumar, Vorst, and Weimers, we investigate complete intersections in discrete Hodge algebras and establish an analogue of Mohan Kumar’s result. As applications, we derive important results on set-theoretic generation of ideals in discrete Hodge algebras. Let R → S be a ring extension such that the reduced structure of the conductor locus of R coincides with that of S, that is (R/C)red = (S/C)red, where C is the conductor ideal of R in S. An example of such an extension is finite subintegral extension of reduced rings. Let k be a field. We recall an integral extension of rings R → S is called subintegral if every k-valued point of R can be lifted uniquely to a k-valued point of S. For an extension R ,→ S of above kind, with dimension d and an ideal I ⊂ R of height n satisfying 2n ≥ d + 3, we prove any set of n generators of I/I² can be lifted to n generators of I if and only if the corresponding generators of IS/I²S can be lifted to n generators of IS. As an application, we analyze the behavior of Euler class groups under subintegral extensions. Finally, we address the splitting problem of algebraic vector bundles. Let R be a ring of dimension atleast 3 containing Q with height of the Jacobson radical J (R) atleast 2. Let P be a projective R[T, T⁻¹]-module of rank dim(R) with determinant L. We explore the Euler class group E(R[T, T⁻¹], L) of R[T, T⁻¹] with coefficients in the line bundle L, which serves as an obstruction group for detecting whether P splits off a free summand of rank one. The results are based on the following articles [R-Z 1], [R-Z 2], and [R-Z 3].

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