Indian Institute of Technology Palakkad&nbsp;(IIT Palakkad or IIT PKD) is a public autonomous&nbsp;engineering&nbsp;and&nbsp;research&nbsp;institute located in&nbsp;Palakkad,&nbsp;Kerala. It was&nbsp;one of the five new&nbsp;IITs&nbsp;proposed in the&nbsp;2014 Union budget of India. The campus was inaugurated on 3 August 2015. The institute has 94 faculty, 978 students, and 63 non-teaching staffs.

IT Palakkad currently offers four-year Bachelor of Technology (B.Tech) and two years MS programmes in 8 different engineering disciplines including&nbsp;Computer Science, Electrical, Mechanical, Civil among others.

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IIT Palakkad

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Department of Mathematics

Research in the Mathematical Sciences

Multiplicities and mixed multiplicities of arbitrary filtrations

In this paper we define and explore the analytic spread $\ell(\mathcal I)$ of a filtration in a local ring. We show that, especially for divisorial and symbolic filtrations, some basic properties of the analytic spread of an ideal extend to filtrations, even when the filtration is non Noetherian. We also illustrate some significant differences between the analytic spread of a filtration and the analytic spread of an ideal with examples. In the case of an ideal $I$, we have the classical bounds $\mbox{ht}(I)\le\ell(I)\le \dim R$. The upper bound $\ell(\mathcal I)\le \dim R$ is true for filtrations $\mathcal I$, but the lower bound is not true for all filtrations. We show that for the filtration $\mathcal I$ of symbolic powers of a height two prime ideal $\mathfrak p$ in a regular local ring of dimension three (a space curve singularity), so that $\mbox{ht}(\mathcal I) =2$ and $\dim R=3$, we have that $0\le \ell(\mathcal I)\le 2$ and all values of 0,1 and 2 can occur. In the cases of analytic spread 0 and 1 the symbolic algebra is necessarily non-Noetherian. The symbolic algebra is non-Noetherian if and only if $\ell(\mathfrak p^{(n)})=3$ for all symbolic powers of $\mathfrak p$ and if and only if $\ell(\mathcal I_a)=3$ for all truncations $\mathcal I_a$ of $\mathcal I$.

arXiv preprint arXiv:2104.

Analytic Spread of Filtrations and Symbolic Algebras

Let (R, m) be a local (Noetherian) ring of dimension d and let M be a ﬁnite length R-module with free resolution G•. De Stefani, Huneke, and Núñez-Betancourt explored two questions about the properties of resolutions of M. First, in characteristic p &gt; 0, what vanishing conditions on the Frobenius Betti numbers, βiF (M, R) := lime→∞ λ(Hi(Fe(G•)))/ped, force pdR M &lt; ∞? Second, if pdR M = ∞, does this force d+ 2nd or higher syzygies of M to have inﬁnite length? For the ﬁrst question, they showed, under rather restrictive hypotheses, that d + 1 consecutive vanishing Frobenius Betti numbers force pdR M &lt; ∞, and when d = 1 and R is CM then one vanishing Frobenius Betti number suﬃces. Using properties of stably phantom homology, we show that these results hold in general, i.e., d+1 consecutive vanishing Frobenius Betti numbers force pdR M &lt; ∞, and, under the hypothesis that R is CM, d consecutive vanishing Frobenius Betti numbers suﬃce. For the second question, they obtain very interesting results when d = 1. In particular, no third syzygy of M can have ﬁnite length. Their main tool is, if d = 1, to show that if the syzygy has a ﬁnite length, then it is an alternating sum of lengths of Tors. We are able to prove this fact for rings of arbitrary dimension, which allows us to show that if d = 2, no third syzygy of M can be of ﬁnite length! We also are able to show that the question has a positive answer if the dimension of the socle of Hm0 (R) is large relative to the rest of the module, generalizing the case of Buchsbaum rings. © 2020 American Mathematical Society

Proceedings of the American Mathematical Society

Frobenius betti numbers and syzygies of finite length modules

Let J⊂I be ideals in a formally equidimensional local ring with λ(I/J)&lt;∞. Rees proved that λ(In/Jn) is a polynomial P(I/J)(X) in n of degree at most dim R and J is a reduction of I if and only if deg⁡P(I/J)(X)≤dimR−1. We extend this result for all Noetherian filtrations of ideals in a formally equidimensional local ring and for (not necessarily Noetherian) filtrations of ideals in analytically irreducible rings. We provide certain classes of ideals such that deg⁡P(I/J) achieves its maximal degree. On the other hand, for ideals J⊂I in a formally equidimensional local ring, we consider the multiplicity function e(In/Jn) which is a polynomial in n for all large n. We explicitly determine the deg⁡e(In/Jn) in some special cases. For an ideal J of analytic deviation one, we give characterization of reductions in terms of deg⁡e(In/Jn) under some additional conditions. © 2019 Elsevier B.V.

Fulltext

Journal of Pure and Applied Algebra

Rees' theorem for filtrations, multiplicity function and reduction criteria

In this paper we define and explore properties of mixed multiplicities of (not necessarily Noetherian) filtrations of mR-primary ideals in a Noetherian local ring R, generalizing the classical theory for mR-primary ideals. We construct a real polynomial whose coefficients give the mixed multiplicities. This polynomial exists if and only if the dimension of the nilradical of the completion of R is less than the dimension of R, which holds, for instance, if R is excellent and reduced. We show that many of the classical theorems for mixed multiplicities of mR-primary ideals hold for filtrations, including the famous Minkowski inequalities of Teissier, and of Rees and Sharp. © 2019 American Mathematical Society

Transactions of the American Mathematical Society

Mixed multiplicities of filtrations

We relate the set of complete reduction vectors of a Zs-graded admissible filtration of ideals F with the set of multigraded regularities of G(F). We prove reg(G(F))=reg(R(F)). We establish a relation between the sets of complete reduction vectors of F and postulation vectors of F under some cohomological conditions. © 2017 Elsevier Inc.

Journal of Algebra

Multigraded regularity, reduction vectors and postulation vectors

We find conditions on the local cohomology modules of multi-Rees algebras of admissible filtrations which enable us to predict joint reduction numbers. As a consequence, we are able to prove a generalization of a result of Reid, Roberts and Vitulli in the setting of analytically unramified local rings for completeness of power products of complete ideals. © 2016 by The Editorial Board of the Nagoya Mathematical Journal Â.

Nagoya Mathematical Journal

Local cohomology of multi-rees algebras, joint reduction numbers and product of complete ideals

We study the relationship between postulation and reduction vectors of admissible multigraded filtrations F = [[[[F(n)]]]]n∈Zs of ideals in Cohen-Macaulay local rings of dimension at most two. This is enabled by a suitable generalization of the Kirby-Mehran complex. An analysis of its homology leads to an analogue of Huneke's fundamental lemma which plays a crucial role in our investigations. We also clarify the relationship between the Cohen-Macaulay property of the multigraded Rees algebra of F and reduction vectors with respect to complete reductions of F. © 2017 Rocky Mountain Mathematics Consortium.

Journal of Commutative Algebra

Postulation and reduction vectors of multigraded filtrations of ideals

In this expository paper, we present proofs of Grothendieck-Serre formula for multi-graded algebras and Rees algebras for admissible multi-graded filtrations. As applications, we derive formulas of Sally for postulation number of admissible filtrations and Hilbert coefficients. We also discuss a partial solution of Itoh’s conjecture by Kummini and Masuti. We present an alternate proof of Huneke-Ooishi Theorem and a generalisation for multi-graded filtrations. © Springer Science+Business Media Singapore 2016.

Springer Proceedings in Mathematics and Statistics

Variations on the Grothendieck-Serre formula for Hilbert functions and their applications

Hilbert functions and Hilbert polynomials of Zs-graded admissible filtrations of ideals {F(n)}n∈Zs such that λ(R/F(n)) is finite for all n∈Zs are studied. Conditions are provided for the Hilbert function HF(n):=λ(R/F(n)) and the corresponding Hilbert polynomial PF(n) to be equal for all n∈Ns. A formula for the difference HF(n)-PF(n) in terms of local cohomology of the extended Rees algebra of F is proved which is used to obtain sufficient linear relations analogous to the ones given by Huneke and Ooishi among coefficients of PF(n) so that HF(n)=PF(n) for all n∈Ns. A theorem of Rees about joint reductions of the filtration {IrJs}r,s∈Z is generalised for admissible filtrations of ideals in two-dimensional Cohen-Macaulay local rings. Necessary and sufficient conditions are provided for the multi-Rees algebra of an admissible Z2-graded filtration F to be Cohen-Macaulay. © 2015 Elsevier Inc.