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Eisenstein extension, connectedness and the second vanishing theorem

Rajsekhar Bhattacharyya

Abstract

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.

Eisenstein extension, connectedness and the second vanishing theorem

Abstract

In this paper, at first, we show that for a ramified regular local ring , which is an Eisenstein extension of an unramified regular local ring , when an ideal of is extended from an ideal of , the punctured spectrum of is connected if that of 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 is disconnected then that of is also disconnected when every minimal primes of , 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.

Paper Structure

This paper contains 3 sections, 4 theorems, 3 equations.

Table of Contents

  1. introduction
  2. the main result
  3. when punctured spectrum is not connected

Key Result

Lemma 2.1

Let $(R, \mathfrak{m})$ be a $d$-dimensional complete unramified regular local ring of mixed characteristic. Let $(S, \mathfrak{n})$ be a ramified complete regular local ring obtained via Eisenstein extension of $R$ where $f(X)\in R[X]$ is the Eisenstein polynomial. (1) If the residue field of $S$ i

Theorems & Definitions (10)

  • Lemma 2.1
  • proof
  • Theorem 2.2
  • proof
  • Theorem 3.1
  • proof
  • Remark 1
  • Corollary 1
  • proof
  • Remark 2