In this paper we present a new approach to Grothendieck duality over commutative rings. Our approach is based on the idea of rigid dualizing complexes, which was introduced by Van den Bergh in the context of noncommutative algebraic geometry. The method of rigidity was modified to work over general commutative base rings in our paper [YZ5]. In the present paper we obtain many of the important l...

‎let $i$ be an ideal in a regular local ring $(r,n)$‎, ‎we will find‎ ‎bounds on the first and the last betti numbers of‎ ‎$(a,m)=(r/i,n/i)$‎. ‎if $a$ is an artinian ring of the embedding‎ ‎codimension $h$‎, ‎$i$ has the initial degree $t$ and $mu(m^t)=1$‎, ‎we call $a$ a {it $t-$extended stretched local ring}‎. ‎this class of‎ ‎local rings is a natural generalization of the class of stretched ...

We prove that in a ring Rwith an identity there exists an element a ∈ R and a nonzero derivation d ∈ Der R such that ad(a) 6= 0. A ring R is said to be a d-rigid ring for some derivation d ∈ Der R if d(a) = 0 or ad(a) 6= 0 for all a ∈ R. We study rings with rigid derivations and establish that a commutative Artinian ring R either has a non-rigid derivation or R = R1 ⊕ · · · ⊕ Rn is a ring direc...

Using the method of breaking of circular symmetry and of subsequent symmetry restoration via projection techiques, we present calculations for the ground-state energies and excitation spectra of N -electron parabolic quantum dots in strong magnetic fields in the medium-size range 10 ≤ N ≤ 30. The physical picture suggested by our calculations is that of finite rotating electron molecules (REMs)...

Archaeal proteasomes share many features with their eukaryotic counterparts and serve as important models for assembly. Proteasomes are also found in certain bacterial lineages yet their assembly mechanism is thought to be fundamentally different. Here we investigate α-ring formation using recombinant proteasomes from the archaeon Methanococcus maripaludis. Through an engineered disulfide cross...

Front Cover: New bat siloxanes fly to the moon: Tricyclic laddersiloxanes with larger side rings are prepared for first time from Janus rings. The obtained novel, 12-8-12-membered ring laddersiloxane has a rigid silsesquioxane body flexible oligosiloxane “wings”. Like real bat, they can move wings freely apply into host molecules or highfunctional materials. This work is found in article number...

We present a formal scheme based cycle model for the motivic cohomology of fat points defined by truncated polynomial rings $$k[t]/(t^m)$$ with $$m \ge 2$$ , in one variable over field k. compute their Milnor range class groups when has sufficiently many elements. With some aids from rigid analytic geometry and Gersten conjecture K-theory resolved M. Kerz, we prove that resulting are isomorphic...