In this paper, we investigate the relationship between Milnor's K-group and Galois cohomology for the quotient eld of a 2-dimensional complete regular local ring with a nite residue eld. The results given in this paper are considered as a partial answer to the Bloch-Kato conjecture for such a eld.

In [1], Anscombe and Koenigsmann give an existential ∅-definition of the ring of formal power series F [[t]] in its quotient field in the case where F is finite. We extend their method in several directions to give general definability results for henselian valued fields with finite or pseudo-algebraically closed residue fields. §

Let S be an unramified regular local ring having mixed characteristic p > 0 and R the integral closure of S in a pth root extension of its quotient field. We show that R admits a finite, birational module M such that depth(M) = dim(R). In other words, R admits a maximal Cohen-Macaulay module.

The theory of groups, rings and modules is developed to a great depth. Group theory results include Zassenhaus’s theorem and the Jordan-Hoelder theorem. The ring theory development includes ideals, quotient rings and the Chinese remainder theorem. The module development includes the Nakayama lemma, exact sequences and Tensor products.

Abstract This article is about Lehn–Lehn–Sorger–van Straten eightfolds $Z$ and their anti-symplectic involution $\iota$ . When birational to the Hilbert scheme of points on a K3 surface, we give an explicit formula for action Chow group $0$ -cycles The in agreement with Bloch–Beilinson conjectures has some non-trivial consequences ring quotient.

2 Rings and Polynomials 30 2.1 Rings, Integral Domains and Fields . . . . . . . . . . . . . . . 30 2.2 Ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 2.3 Quotient Rings and Homomorphisms . . . . . . . . . . . . . . 33 2.4 The Characteristic of a Ring . . . . . . . . . . . . . . . . . . . 35 2.5 Polynomial Rings . . . . . . . . . . . . . . . . . . . . . . . . . 35 2.6 Ga...

Let f1, . . . , fr ∈ K[x], K a field, be homogeneous polynomials and put F = ∑r i=1 yifi ∈ K[x, y]. The quotient J = K[x, y]/I, where I is the ideal generated by the ∂F/∂xi and ∂F/∂yj , is the Jacobian ring of F . We describe J by computing the cohomology of a certain complex whose top cohomology group is J .