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Rajsekhar Bhattacharyya

Publications and source records attributed to Rajsekhar Bhattacharyya.

At least 19 recordsLinked to original sources
arXiv2026

On local cohomology modules over ramified regular local rings

In this paper, we show examples of local cohomology modules over ramified regular local ring, having finite set of associated primes. In doing so we consider our ramified regular local ring as Eisenstein extension of an unramified regular local ring sitting inside it. In ramified regular local ring for extended ideal (from the unramified one) set of associative primes of a local cohomology module is always finite. Using the Mayer-Vietoris spectral sequence, we construct examples of local cohomology of non extended ideal, whose set of associated primes is finite. In particular, we choose those non extended ideals whose minimal primes are extended ideals which are all cohomologically complete intersection. These examples of local cohomology modules include the cases whose associative primes also contain prime number $p$.

math.AC↗
arXiv2026

Asymptotic behaviour of bigraded components of local cohomology modules

Let $C$ be a commutative Noetherian ring containing a field $K$ of characteristic zero. Let $R=C[X_1, \ldots, X_n, Y_1, \ldots, Y_m]$ be a polynomial ring over $C$ with $\mathrm{bideg}~ c=(0,0)$ for all $c \in C$, $\mathrm{bideg}~ X_i=(1,0)$ and $\mathrm{bideg}~ Y_j=(0,1)$ for $i=1, \ldots, n$ and $j=1, \ldots, m$. Let $I$ be a bihomogeneous ideal in $R$. In this article, we study asymptotic behaviour of bigraded pieces of the local cohomology module $H^i_I(R)$. Moreover, under the extra assumption that $C$ is regular, we investigate the asymptotic stability of invariants associated to its bigraded components. Consequently, we obtain certain properties of components of the bigraded local cohomology module $H^i_I(R)$, where $C=K$ is a field and $I$ is a binomial edge ideal.

math.AC↗
arXiv2024

Eisenstein extension, connectedness and the second vanishing theorem

In this paper, at first, we show that for a ramified regular local ring $S$, which is an Eisenstein extension of an unramified regular local ring $R$, when an ideal $I$ of $S$ is extended from an ideal $J$ of $R$, the punctured spectrum of $R/J$ is connected if that of $S/JS$ is connected. Using this, we extend the result of SVT to complete ramified regular local ring only for the extended ideals. If the punctured spectrum of $S/JS$ is disconnected then that of $R/J$ is also disconnected when every minimal primes $\p$ of $J$, $R/\p$ is normal. Under this situation we prove that both of them have the same number of connected components. Finally, we show that for both unramified and ramified regular local rings (for extended ideal via Eisenstein extension), two top-most local cohomology modules satisfy the Conjecture 1 of \cite{L-Y}, although the conjecture is false in general.

math.AC↗
arXiv2016

Existence of almost Cohen-Macaulay algebras implies the existence of big Cohen-Macaulay algebras

In \cite{AB}, the dagger closure is extended over finitely generated modules over Noetherian local domain $(R,\fm)$ and it is proved to be a Dietz closure. In this short note we show that it also satisfies the `Algebra axiom' of \cite{R.G} and this leads to the following result of this paper: For a complete Noetherian local domain, if it is contained in an almost Cohen-Macaulay domain, then there exists a balanced big Cohen-Macaulay algebra over it.

math.AC↗
arXiv2015

Behaviour of Finiteness of the Set of Associated Primes under Ring Extensions

We study the behaviour of the finiteness of the set of associated primes of local cohomology modules, more generally of Lyubeznik functors, under various ring extensions. At first, we review the results for flat and faithfully flat extensions and we present new applications of them. Then, we focus how the finiteness property of the set of associated primes of local cohomology modules and Lyubeznik functors is transferred from extended ring to the base ring of pure and cyclically pure ring extensions. We show that finiteness property can be transferred from a ring to its pure local subring and this extends the result of Theorem 1.1 of \cite{Nu}. Further, we observed that under mild conditions on the rings, finiteness property comes down from cyclically pure ring extensions to its local base ring. In particular, we observe that the set of associated primes of Lyubeznik functors of a cyclically pure local subring (which turns out to be Cohen-Macaulay) of equicharacteristic or unramified regular local ring, is finite. There is an appendix on behaviour of the Bass numbers under pure and cyclically pure ring extensions.

math.AC↗
arXiv2015

Behaviour of the Support of Lyubeznik Functors under Ring Extensions

Let $R\rightarrow S$ be an arbitrary ring extension of Noetherian rings. In this article we study the behaviour of Zariski closedness of the support of Lyubeznik functors $\mathrm{T}$, when the ring extension $R\rightarrow S$ is namely `flat', `faithfully flat', `pure' and lastly `cyclically pure'. We show that the Zariski closedness of the support comes down from extended ring to the base ring for faithfully flat, pure and finally for cyclically pure ring extensions. Lastly, we focus on a special case of pure extension i.e. when $R$ is a direct summand of $S$ and we compare the sets $\mathrm{Supp}_S(\mathrm{T}(R)\otimes_R S)$ and $\mathrm{Supp}_S \mathrm{T}(S)$.

math.AC↗
arXiv2015

A Note on Associated Primes and Bockstein Homomorphisms of Local Cohomology Modules for Ramified Regular Local Rings

For a Noetherian regular ring $S$ and for a fixed ideal $J\subset S$, assume that the associated primes of local cohomology module $H^i_J(S)$ does not contain $p$ for some $i\geq 0$, and we call this as a property $\textit{\textbf{P}}^{i,p}_{J,S}$ or $\textit{\textbf{P}}$ for brevity. Recently, in Theorem 1.2 of \cite{Nu1}, it is proved that in a Noetherian regular local ring $S$ and for a fixed ideal $J\subset S$, associated primes of local cohomology module $H^i_J(S)$ for $i\geq 0$ is finite, if it does not contain $p$. In this paper, we study how the property $\textit{\textbf{P}}$ (as mentioned above) can come down from unramified regular ring to ramified regular local ring. In \cite{SW}, Bockstein homomorphism is studied in the context to the finiteness of associated primes of local cohomology modules for the ring of integers. There it is shown that if $p$ is nonzero divisor of Koszul homology then Bockstein homomorphism is a zero map (see, Theorem 3.1 of \cite{SW}). Here, in this paper, as a consequence of property $\textit{\textbf{P}}$, we extend the result of Theorem 3.1 of \cite{SW} to the ramified regular local ring.

math.AC↗
arXiv2015

Examples of local Cohomology Modules for Ramified Regular Local Rings having Finite Set of Associated Primes

Lyubeznik's conjecture, (\cite{Ly1}, Remark 3.7) asserts the finiteness of the set ssociated primes of local cohomology modules for regular rings. But, in the case of ramified regular local ring, it is open. Recently, in Theorem 1.2 of \cite{Nu1}, it is proved that in any Noetherian regular local ring $S$ and for a fixed ideal $J\subset S$, associated primes of local cohomology $H^i_J(S)$ for $i\geq 0$ is finite, if it does not contain $p$. In this paper, we use this result to construct examples of local cohomology modules for ramified regular local ring so that they have finitely many associated primes.

math.AC↗
arXiv2015

Smooth Algebra and Finiteness of the Set of Associated Primes of Local Cohomology Modules

In this article, we study the behaviour of smooth algebra $R$ over local Noetherian local ring $A$. At first, we observe that for every $f\in R$, $R_f$ has finite length in the category of $D(R,A)$-module if dimension of $A$ is zero. This extends the result of Theorem 2 of \cite{Ly3}. We use this fact to generalize the result of Theorem 4.1 of \cite{BBLSZ}, from the finiteness of the set of associated primes of local cohomology module to that of Lyubeznik functor. Finally, we introduce the definition of $Σ$-finite $D$-modulue for smooth algebra and we extend the result of Theorem 1.3 of \cite{Nu3} from polynomial and power series algebra to smooth algebra. Theorem 1.3 of \cite{Nu3} comes out as a partial answer to a question raised by Melvin Hochster. Thus, we extend the partial answer to the above question from polynomial and power series algebra to smooth algebra over an arbitrary Noetherian local ring.

math.AC↗
arXiv2008

Exact Multi-Restricted Schur Polynomial Correlators

We derive a product rule satisfied by restricted Schur polynomials. We focus mostly on the case that the restricted Schur polynomial is built using two matrices, although our analysis easily extends to more than two matrices. This product rule allows us to compute exact multi-point correlation functions of restricted Schur polynomials, in the free field theory limit. As an example of the use of our formulas, we compute two point functions of certain single trace operators built using two matrices and three point functions of certain restricted Schur polynomials, exactly, in the free field theory limit. Our results suggest that gravitons become strongly coupled at sufficiently high energy, while the restricted Schur polynomials for totally antisymmetric representations remain weakly interacting at these energies. This is in perfect accord with the half-BPS (single matrix) results of hep-th/0512312. Finally, by studying the interaction of two restricted Schur polynomials we suggest a physical interpretation for the labels of the restricted Schur polynomial: the composite operator $χ_{R,(r_n,r_m)}(Z,X)$ is constructed from the half BPS ``partons'' $χ_{r_n}(Z)$ and $χ_{r_m}(X)$.

hep-th↗
arXiv2008

Exact Multi-Matrix Correlators

We argue that restricted Schur polynomials provide a useful parameterization of the complete set of gauge invariant variables of multi-matrix models. The two point functions of restricted Schur polynomials are evaluated exactly in the free field theory limit. They have diagonal two point functions.

hep-th↗
arXiv2005

Brane Intersections in the Presence of a Worldvolume Electric Field

The study of brane intersections has provided important insights into a possible non-commutative structure of spacetime geometry. In this paper we focus on the D1$\bot$D3 system. We compare the D1 and D3 descriptions of the interesection and search for non-static solutions of the D3$\bot$D1 funnel equations in the presence of a worldvolume electric field. We find that the D1 and D3 descriptions do not agree. We find time dependent solutions that are a natural generalization of those found without the electric field.

hep-th↗
arXiv2005

Fluctuating Fuzzy Funnels

It is well known that a D-string ending on a D3, D5 or D7 brane is described in terms of a non-commutative fuzzy funnel geometry. In this article, we give a numerical study of the fluctuations about this leading geometry. This allows us to investigate issues related to the stability and moduli space of these solutions. We comment on the comparison to the linearized fluctuations in supergravity.

hep-th↗
arXiv2005

A Short Note on Multi-bion Solutions

Here we try to construct a form of multi-bion solution in the dual description of $D3 \bot D1$-system which connects the two separated bions each made up of 2 $D1-branes$ at large distance with a single $D3$-brane of four unit of magnetic charge at origin. Further we interested in the soluions which can interpolate between arbitrarily separated bions and single $D3$-brane with arbitrary amount of magnetic charges and we find that it is probably not possible to have the solution in each case.

hep-th↗