Research Projects

Studying the symmetric and Rees algebra of an ideal in a commutative ring is an important area of research in commutative algebra. Various notions of the sequences (generalizing regular sequences) have been investigated over the past five decades. They have become crucial to the theme of the study of symmetric and Rees algebra. A sequence here refers to a collection of elements in a commutative ring that satisfies specific properties. For a quick survey on a list of sequences under consideration in this proposal, see: Relative regular sequences (M. Fiorentini, J. Algebra, 18 (1971)); d-sequence (J. Algebra, 62 (1980)) to show isomorphism of the symmetric and Rees algebra; weak d-sequences (C. Huneke, J. Algebra, 68 (1981)); proper sequence (J. Herzog, A. Simis, W. V. Vasconcelos, J. Algebra, 74 (1982)) to study approximation complexes of blowing-up rings; quadratic sequences (K. N. Raghavan, Trans. Amer. Math. Soc., 343 (1994)); M-sequence and sequence of interval type (A. Conca, E. De Negri, J. Algebra, 211 (1999)); s-sequences, and symmetric algebras (J. Herzog, G. Restuccia, Z. Tang, Manuscripta Math., Manuscripta Math. 104 (2001)); c-sequences (H. Kulosman, Illinois J. Math., 52 (2008)). The vanishing of x-regularly helps us check whether the Castelnuovo-Mumford regularity of powers of an ideal has a linear resolution (T. Romer, Illinois J. Math, (2001), and W. Bruns, A. Conca, M. Varbaro, Commutative algebra, (2021)), and y-regularly can be used to detect the nature of the ideal being generated by the d-sequence (T. Romer, (2001)). The bigraded Betti numbers, in general, capture information regarding the corresponding ideals/modules for the Hilbert series, and in particular, x-regularly and y-regularity. These notions are very much related and share a great deal of information about the corresponding ideal under investigation. In 1994, M. Cipu and M. Fiorentini summarised results concerning relative regular sequences, d-sequence, and proper sequences that appeared in the conference proceedings in honor of Michael Artin. The subject has continued to grow; however, only a single reference source studied the relationship among these sequences. We have begun the literature survey, and initial observations seem exciting. The subject is interesting and relevant to the commutative algebra community; at the same time, there is enough scope for new results and progress to be made on this front. This particular reason has motivated us to take on this project. The goal of this project is to review the literature carefully, relate the implications among the ideals generated by sequences, study the corresponding symmetric and Rees algebra, and in particular, find new characterization for the ideal of linear type. Computational packages have evolved in the past two decades. One of the goals of this project is to develop packages by writing algorithms and code for computing homological invariants.