A note on Bockstein homomorphisms in local cohomology (Q2906443)

The paper under review is an exposition of the paper [J. Reine Angew. Math. 655, 147--164 (2011; Zbl 1257.13004)].NEWLINENEWLINELet \(R\) be a commutative Noetherian ring and \(M\) an \(R\)-module. Let \(p\in R\) be a nonzero divisor on \(M\) and \(F^{\bullet}\) be a covariant \(\delta\)-functor on the category of \(R\)-modules and \(R\)-homomorphisms. The exact sequence NEWLINE\[CARRIAGE_RETURNNEWLINE0\rightarrow M\overset{p}{\rightarrow} M\overset{\pi}{\rightarrow} M/pM\rightarrow 0CARRIAGE_RETURNNEWLINE\]NEWLINE yields the following long exact sequence: NEWLINE\[CARRIAGE_RETURNNEWLINE\cdots \rightarrow F^k(M/pM)\overset{\delta_p^k}{\rightarrow} F^{k+1}(M)\overset{F^{k+1}(p)}{\rightarrow} F^{k+1}(M) \overset{F^{k+1}(\pi)}{\rightarrow} F^{k+1}(M/pM)\rightarrow \cdots .CARRIAGE_RETURNNEWLINE\]NEWLINE The composition NEWLINE\[CARRIAGE_RETURNNEWLINEF^{k+1}(\pi)\circ \delta_p^k:F^k(M/pM)\rightarrow F^{k+1}(M/pM)CARRIAGE_RETURNNEWLINE\]NEWLINE is called \textit{Bockstein homomorphism} and is denoted by \(\beta_p^k=\beta_p^k(F^{\bullet}(M))\).NEWLINENEWLINELet \(R\) be a regular ring, \(\mathfrak a=(f_1,\ldots, f_t)\) an ideal of \(R\) and let \(F^{\bullet}\) be the covariant \(\delta\)-functor \(H_{\mathfrak a}^{\bullet}(-)\). In 1993 Lyubeznik conjectured that each local cohomology module \(H_{\mathfrak a}^i(R)\) has finitely many associated prime ideals. One can easily check that this conjecture implies the vanishing of all but finitely many Bockstein homomorphism \(\beta_p^k:H_{\mathfrak a}^k(R/pR)\rightarrow H_{\mathfrak a}^{k+1}(R/pR);\) when \(p\) runs through prime numbers.NEWLINENEWLINEAlthough, Lyubeznik's conjeture has been established in many situations, it is not known whether it is true for the regular ring \(\mathbb{Z}[x_1,\ldots, x_n]\). This paper supports Lyubeznik's conjecture for this special regular ring. Let \(R=\mathbb{Z}[x_1,\ldots, x_n]\), \(k\) a nonnegative integer and \(p\) a prime number. The main result of this paper asserts that if \(p\) is a nonzero divisor on the Koszul cohomology module \(H^{k+1}(f_1,\ldots, f_t;R)\), then \(\beta_p^k:H_{\mathfrak a}^k(R/pR) \rightarrow H_{\mathfrak a}^{k+1}(R/pR)\) is zero.NEWLINENEWLINEFor the entire collection see [Zbl 1242.14003].