Deformation theory is often an important tool in dealing with a variety of questions related to moduli spaces. We have discussed the basic idea before: we study families of objects over Artinian rings restricting to a given object over Spec k, and this tells us, for instance, the dimension of the tangent space of a moduli space, or whether it is smooth at a point. We begin by developing this ge...

We introduce and investigate the notion of GC -projective modules over (possibly non-noetherian) commutative rings, where C is a semidualizing module. This extends Holm and Jørgensen’s notion of C-Gorenstein projective modules to the non-noetherian setting and generalizes projective and Gorenstein projective modules within this setting. We then study the resulting modules of finite GC-projectiv...

Let R <-* S be an embedding of associative noetherian rings such that 5 is finitely generated as a right Ä-module. There is a correspondence from the prime spectrum of S to the prime spectrum of R obtained by associating to a given prime ideal P of S the prime ideals of R minimal over P n R . The prime and primitive ideal theories for several specific noncommutative noetherian rings, including ...

In this paper we define and explore the analytic spread ℓ ( I ) $\ell (\mathcal {I})$ of a filtration in local ring. We show that, especially for divisorial symbolic filtrations, some basic properties an ideal extend to even when is non-Noetherian. also illustrate significant differences between with examples. case $I$ , have classical bounds ht ⩽ dim R $\mbox{ht}(I)\leqslant \ell (I)\leqslant ...

We consider families of parameterizations reduced curve singularities over a Noetherian base scheme and prove that the delta invariant is semicontinuous. In our setting, each singularity in family image parameterization not fiber morphism. The problem came up connection with right-left classification defined field positive characteristic. bound for determinacy terms delta, semicontinuity theore...

Let R be an associative ring. A map σ : R → R is called a ring endomorphism if σ(x+y) = σ(x)+σ(y) and σ(xy) = σ(x)σ(y) for all elements a,b ∈ R. A ring R is said to be rigid if it has only the trivial ring endomorphisms, that is, identity idR and zero 0R . Rigid left Artinian rings were described by Maxson [9] and McLean [11]. Friger [4, 6] has constructed an example of a noncommutative rigid r...

For any separated map f : X → Y of quasi-compact quasiseparated schemes, Rf∗ : D + qc(X) → D (Y ) has a right adjoint f . If f is proper and pseudo-coherent (e.g., finitely-presented and flat) then Duality and tor-independent Base Change hold for f . Preface This is a research summary written early in 1991, concerning results obtained by the author during a stay at MSRI in Berkeley during 1989–...