We describe a nonsingular hermitian form of rank 4 over the group ring Z[Z] which is not extended from the integers. Moreover, we show that under certain indefiniteness asumptions, every nonsingular hermitian form on a free Z[Z]module is extended from the integers. As a corollary, there exists a closed oriented 4-dimensional manifold with fundamental group Z which is not the connected sum of S ...

It is shown that the face ring of a pure simplicial complex modulo m generic linear forms is a ring with finite local cohomology if and only if the link of every face of dimension m or more is nonsingular. 2000 Mathematics Subject Classification: 13F55.

A fundamental result of Springer says that a quadratic form over field characteristic ≠2 is isotropic if it so after an odd degree extension. In this paper we generalize Springer’s theorem as follows. Let R be arbitrary semilocal ring, let S finite R-algebra constant degree, which étale or generated by one element, and q nonsingular R-quadratic whose base ring extension qS isotropic. We show th...

We defined the Grothendieck groups KX and K0X. They are vector bundles, respectively coherent sheaves, modulo the relation [E] = [E ] + [E ]. We have a pullback on K: f : KX → KY. KX is a ring: [E] · [F] = [E ⊗ F]. We have a pushforward on K0: f∗[F ] = ∑ i≥0(−1) [Rf∗F ]. We obviously have a homomorphism KX → K0X. K0X is aK X-module: KX⊗K0X → X is given by [E] · [F ] = [E⊗ F ]. Unproved fact: If...

In this section we will define the sheaf of relative differential forms of one scheme over another. In the case of a nonsingular variety over C, which is like a complex manifold, the sheaf of differential forms is essentially the same as the dual of the tangent bundle in differential geometry. However, in abstract algebraic geometry, we will define the sheaf of differentials first, by a purely ...

A module $M$ is called a $C_{21}$-module if, whenever $A$ and $B$ are submodules of with $A \cong B$, nonsingular direct summand $M$, then $M$. Various examples $C_{21}$-modules presented. Some basic properties these modules investigated. It shown that the class rings $R$ over which every $C_2$-module exactly right SI-rings. Also, we prove for ring $R$, $R$-module has $(C_{21})$ if only t-semis...

It is well-known, that the ring C[X1, . . . , Xn]n of polynomial invariants of the alternating group An has no finite SAGBI basis with respect to the lexicographical order for any number of variables n ≥ 3. This note proves the existence of a nonsingular matrix δn ∈ GL(n,C) such that the ring of polynomial invariants C[X1, . . . ,Xn] δn n , where An n denotes the conjugate of An with respect to...

We prove a Cohen-Macaulay version of result by Avramov-Golod and Frankild-J{\o}rgensen about Gorenstein rings, showing that if noetherian ring $A$ is Cohen-Macaulay, $a_1,\dots,a_n$ any sequence elements in $A$, then the Koszul complex $K(A;a_1,\dots,a_n)$ DG-ring. further generalize this result, it also holds for commutative DG-rings. In process proving this, we develop new technique to study ...

This paper deals with the following question:whether a ring of matrices or classes over an adequate elementary divisor inherits property adequacy?
 The to being in matrix rings and commutative is studied.Let us denote by $\mathfrak{A}$ $\mathfrak{E}$ domains, respectively. Also $\mathfrak{A}_2$ $\mathfrak{E}_2$ $2 \times 2$ them. We prove that full nonsingular from are singular set $\mathf...