Ascent and descent of the Golod property along algebra retracts

We study ascent and descent of the Golod property along an algebra retract. We characterise trivial extensions of modules, fibre products of rings to be Golod rings. We present a criterion for a graded module over a graded affine algebra of characteristic zero to be a Golod module.Comment: 16 page

Non-conservation of Density of States in Bi2_2Sr2_2CaCu2_2Oy_y: Coexistence of Pseudogap and Superconducting gap

The tunneling spectra obtained within the ab-plane of Bi$_2$Sr$_2$CaCu$_2$O$_y$ (Bi2212) for temperatures below and above the critical temperature (T$_c$) are analyzed. We find that the tunneling conductance spectra for the underdoped compound in the superconducting state do not follow the conservation of states rule. There is a consistent loss of states for the underdoped BI2212 implying an underlying depression in the density of states (DOS) and hence the pseudogap near the Fermi energy (E$_F$). Such an underlying depression can also explain the peak-dip-hump structure observed in the spectra. Furthermore, the conservation of states is recovered and the dip-hump structure disappears after normalizing the low temperature spectra with that of the normal state. We argue that this is a direct evidence for the coexistence of a pseudogap with the superconducting gap.Comment: 5 pages, 4 figure

On the existence of unimodular elements and cancellation of projective modules over noetherian and non-noetherian rings

Let $R$ be a commutative ring of dimension $d$, $S = R[X]$ or $R[X, 1/X]$ and $P$ a finitely generated projective $S$ module of rank $r$. Then $P$ is cancellative if $P$ has a unimodular element and $r \geq d + 1$. Moreover if $r \geq \dim (S)$ then $P$ has a unimodular element and therefore $P$ is cancellative. As an application we have proved that if $R$ is a ring of dimension $d$ of finite type over a Pr\"{u}fer domain and $P$ is a projective $R[X]$ or $R[X, 1/X]$ module of rank at least $d + 1$, then $P$ has a unimodular element and is cancellative.Comment: 21 page

Characterizations of regular local rings via syzygy modules of the residue field

Let $R$ be a commutative Noetherian local ring with residue field $k$. We show that if a finite direct sum of syzygy modules of $k$ surjects onto `a semidualizing module' or `a non-zero maximal Cohen-Macaulay module of finite injective dimension', then $R$ is regular. We also prove that $R$ is regular if and only if some syzygy module of $k$ has a non-zero direct summand of finite injective dimension.Comment: 7 page