An example of an infinite set of associated primes of a local cohomology module

Let $(R,m)$ be a local Noetherian ring, let $I\subset R$ be any ideal and let $M$ be a finitely generated $R$-module. In 1990 Craig Huneke conjectured that the local cohomology modules $H^i_I(M)$ have finitely many associated primes for all $i$. In this paper I settle this conjecture by constructing a local cohomology module of a local $k$-algebra with an infinite set of associated primes, and I do this for any field $k$

Bahrain: Reform, Security, and U.S. Policy

[Excerpt] An uprising that began in Bahrain on February 14, 2011, following the revolt that overthrew Egypt’s President Hosni Mubarak three days earlier, began a political crisis that defies resolution. Bahrain’s unrest demonstrates that Shiite grievances over the distribution of power and economic opportunities were not satisfied by the efforts during 1999-2010 to increase the role of the Shiite majority in governance; most Bahraini Shiites now say they seek a constitutional monarchy in which governments are established by an elected parliament. Reflecting increasing polarization, many Sunnis in Bahrain believe the Shiite majority will settle for nothing less than outright rule. As protests escalated in March 2011, Bahrain’s government bucked U.S. advice by inviting direct security assistance from other Gulf Cooperation Council countries, declaring a state of emergency, forcefully suppressing demonstrations, and arresting dissident leaders and pro-opposition health care workers. Although the state of emergency ended on June 1, 2011, the continued imprisonment of dissidents contributed to the resulting failure of a “national dialogue,” held in July 2011, to reach on more than just a few political reform recommendations. Hopes for resolution were raised by a pivotal report by a government-appointed “Independent Commission of Inquiry” (BICI) on the unrest, released November 23, 2011, which was critical of the government’s actions against the unrest as well as the opposition’s responses to government proposals early in the crisis. The government, through an appointed national commission, has begun to implement most of the BICI recommendations, but the stalemate on major political reforms has contributed to the resumption of some renewed violent demonstrations and dashed hopes that a complete solution is in sight. The Obama Administration has not called for a change of the Al Khalifa regime, but it has opposed the regime’s use of force against protesters and urged further and faster political reform. The U.S. position on Bahrain has been criticized by those who believe the United States is downplaying regime abuses because the U.S. security relationship with the Al Khalifa regime is critical to U.S. efforts to contain Iran and preserve security in the Persian Gulf more broadly. In exchange for a tacit security guarantee against Iran or other aggressors, Bahrain has provided key support for U.S. interests by hosting U.S. naval headquarters for the Gulf for over 60 years and by providing facilities and small numbers of personnel for U.S. war efforts in Iraq and Afghanistan. U.S. officials are concerned that the instability in Bahrain could render U.S. use of the naval headquarters facilities untenable, but there are no evident moves to relocate it. Beyond the naval facility, the United States signed a formal defense pact with Bahrain in 1991 and has designated Bahrain as a “major non-NATO ally,” entitling it to sales of sophisticated U.S. weapons systems. Bahrain also receives small amounts of U.S. security assistance. New U.S. sales and aid are coming under criticism from human rights and other groups and, in response, the Administration put on hold a significant proposed sale of armored vehicles and anti-tank weapons while approving smaller sales of military spare parts. Factoring into the U.S. position is a perception that Iran might seek to take advantage of Shiite unrest in Bahrain to reduce U.S. influence and the U.S. military presence in the Persian Gulf. Consumed by its own crisis, Bahrain has joined with but deferred to other GCC powers in initiatives to resolve uprisings in Libya, Syria, and Yemen. Fueling Shiite unrest is the fact that Bahrain, having largely run out of crude oil reserves, is poorer than most of the other Persian Gulf monarchies. The country has tried to compensate through diversification, particularly with banking and some manufacturing. In September 2004, the United States and Bahrain signed a free trade agreement (FTA); legislation implementing itwas signed January 11, 2006 (P.L. 109-169). The unrest in 2011 has further strained Bahrain’s economy

The Hilbert series of algebras of Veronese type

This paper gives a fairly explicit formula for the Hilbert series of algebras of Veronese type

The support of top graded local cohomology modules

Let $R_0$ be any domain, let $R=R_0[U_1, ..., U_s]/I$, where $U_1, ..., U_s$ are indeterminates of some positive degrees, and $I\subset R_0[U_1, ..., U_s]$ is a homogeneous ideal. The main theorem in this paper is states that all the associated primes of $H:=H^s_{R_+}(R)$ contain a certain non-zero ideal $c(I)$ of $R_0$ called the ``content'' of $I$. It follows that the support of $H$ is simply V(\content(I)R + R_+) (Corollary 1.8) and, in particular, $H$ vanishes if and only if $c(I)$ is the unit ideal. These results raise the question of whether local cohomology modules have finitely many minimal associated primes-- this paper provides further evidence in favour of such a result. Finally, we give a very short proof of a weak version of the monomial conjecture based on these results

F-stable submodules of top local cohomology modules of Gorenstein rings

This paper applies G. Lyubeznik's notion of F-finite modules to describe in a very down-to-earth manner certain annihilator submodules of some top local cohomology modules over Gorenstein rings. As a consequence we obtain an explicit description of the test ideal of Gorenstein rings in terms of ideals in a regular ring

Parameter test ideals of Cohen Macaulay rings

We describe an algorithm for computing parameter-test-ideals in certain local Cohen-Macaulay rings. The algorithm is based on the study of a Frobenius map on the injective hull of the residue field of the ring and on the application of Rodney Sharp's notion of ``special ideals''. Our techniques also provide an algorithm for computing indices of nilpotency of Frobenius actions on top local cohomology modules of the ring and on the injective hull of its residue field. The study of nilpotent elements on injective hulls of residue fields also yields a great simplification of the proof of the fact that for a power series ring $R$ of prime characteristic, for all nonzero $f\in R$, $1/f$ generates $R_f$ as a $D_R$-module.Comment: 16 pages To appear in Compositio Mathematic

On ideals of minors of matrices with indeterminate entries

This paper has two aims. The first is to study ideals of minors of matrices whose entries are among the variables of a polynomial ring. Specifically, we describe matrices whose ideals of minors of a given size are prime. The main result in the first part of this paper is a theorem which gives sufficient conditions for the ideal of minors of a matrix to be prime. This theorem is general enough to include interesting examples, such as the ideal of maximal minors of catalecticant matrices and their generalisations discussed in the second part of the paper. The second aim of this paper is to settle a specific problem raised by David Eisenbud and Frank-Olaf Schreyer on the primary decomposition of an ideal of maximal minors. We solve this problem by applying the theorem above together with some ad-hoc techniques