Characteristic-free test ideals

Tight closure test ideals have been central to the classification of singularities in rings of characteristic p > 0 p>0 , and via reduction to characteristic p > 0 p>0 , in equal characteristic 0 as well. Their properties and applications have been described by Schwede and Tucker [Progress in commutative algebra 2, Walter de Gruyter, Berlin, 2012]. In this paper, we extend the notion of a test ideal to arbitrary closure operations, particularly those coming from big Cohen-Macaulay modules and algebras, and prove that it shares key properties of tight closure test ideals. Our main results show how these test ideals can be used to give a characteristic-free classification of singularities, including a few specific results on the mixed characteristic case. We also compute examples of these test ideals.

Mechanical colon obstruction due to the alimentary limb after Roux-en-Y gastric bypass: a case report

Background Small bowel obstruction is a known and potentially lethal complication after gastric bypass surgery, in both the early and the late postoperative course. Colon or large bowel obstruction, on the other hand, seems to be rare after gastric bypass surgery and thus is not routinely considered. Case presentation We present the case of a 21-year old morbidly obese caucasian patient who underwent laparoscopic Roux-en-Y gastric bypass surgery and developed an early severe transverse colon obstruction due to compression of the transverse colon by the antecolic alimentary limb. Emergency revisional surgery showed a short and tense alimentary limb mesentery and possibly tight closure of Petersen’s space contributing to the compression. Through opening of Petersen’s space and mobilization of alimentary limb mesentery, decompression was achieved, and the patient fully recovered. Conclusions This is a rare case of colon obstruction caused by direct compression of the transverse colon by the antecolic alimentary limb. We propose that a combination of short tense alimentary limb mesentery and perhaps tight closure of Petersen’s space was responsible for the obstruction in this case. Surgeons and treating physicians need to be aware of such rare causes of early postoperative bowel obstruction and take these into consideration when evaluating patients.

Changes in the structural organization of bone after amputation

Introduction: Despite a considerable amount of work on limb amputations, questions of the mechanism and conditions for the occurrence of pathological rearrangement in the bone stump and ways to prevent it remain unexplored. Aim: To study the nature of changes in the structural organization of bone stump after amputation. Material and methods: Nine series of experiments were conducted on 129 rabbits with amputation of the thigh and closure of the filing with fascia, muscles with varying degrees of tension and a bone plate. Duration of observation is 1, 3, and 6 months. The research method is histological with a filling of blood vessels with a mascara-gelatin mixture. Results and discussion: Tight closure of the bone marrow cavity and uniform muscle tension during plastic surgery in the I–III series of experiments make it possible to obtain cylindrical stumps with the formation of a bone closure plate and the completion of the reparative process. In the majority of observations of the IV–IX series, significant violations of the structural organization of the bone occurred in the form of conical, club-shaped, swollen stumps, creeping fractures. Conclusions: The rapid completion of the reparative process and the normalization of intraosseous circulation with a locking bone plate, while maintaining normal bone marrow tissue, is possible only with a dense closure of the filing. The lack of tight closure of the bone marrow canal and uneven muscle tension cause a violation of regeneration with changes in the structural organization of the bone. These changes are manifested by a thickening of the bone diameter, axis curvature, creeping fractures.

Tight closure of powers of ideals and tight hilbert polynomials

Let (R, ) be an analytically unramified local ring of positive prime characteristic p. For an ideal I, let I* denote its tight closure. We introduce the tight Hilbert function $$H_I^*\left( n \right) = \Im \left( {R/\left( {{I^n}} \right)*} \right)$$ and the corresponding tight Hilbert polynomial $$P_I^*\left( n \right)$$, where I is an m-primary ideal. It is proved that F-rationality can be detected by the vanishing of the first coefficient of $$P_I^*\left( n \right)$$. We find the tight Hilbert polynomial of certain parameter ideals in hypersurface rings and Stanley-Reisner rings of simplicial complexes.

Length of Local Cohomology in Positive Characteristic and Ordinarity

Let D be the ring of Grothendieck differential operators of the ring R of polynomials in d ≥ 3 variables with coefficients in a perfect field of characteristic p. We compute the D-module length of the 1st local cohomology module ${H^{1}_{f}}(R)$ with respect to a polynomial f with an isolated singularity, for p large enough. The expression we give is in terms of the Frobenius action on the top coherent cohomology of the exceptional fibre of a resolution of the singularity. Our proof rests on a tight closure computation of Hara. Since the above length is quite different from that of the corresponding local cohomology module in characteristic zero, we also consider a characteristic zero D-module whose length is expected to equal that above, for ordinary primes.