On flatness and coherence with respect to modules of flat dimension at most one

Extensions of primes, flatness, and intersection flatness

We study when $R \to S$ has the property that prime ideals of $R$ extend to prime ideals or the unit ideal of $S$, and the situation where this property continues to hold after adjoining the same indeterminates to both rings. We prove that if $R$ is reduced, every maximal ideal of $R$ contains only finitely many minimal primes of $R$, and prime ideals of $R[X_1,\dots,X_n]$ extend to prime ideals of $S[X_1,\dots,X_n]$ for all $n$, then $S$ is flat over $R$. We give a counterexample to flatness over a reduced quasilocal ring $R$ with infinitely many minimal primes by constructing a non-flat $R$-…

A new proof of the local criterion of flatness

Let (A,m_A) -> (B,m_B) be a local morphism of local noetherian rings and M a finitely generated B-module. Then it follows from Tor^A_1(M,A/m_A) = 0 that M is a flat A-module. This is usually called the "local criterion of flatness". We give a proof that proceeds along different lines than the usual textbook proofs, using completions and only elementary properties of flat modules and the Tor-functor.

Descent of Flatness and the Direct Summand Conjecture

In the central theorem of this article we prove the following: if $R$ is a complete regular local ring and $B$ is the integral closure of $R$ in the algebraic closure of the fraction field of $R$, then $\Hom_R(B, R) \neq 0$. Our proof of this theorem does not involve almost mathematics and it works for all characteristics. As consequences we derive the validity of a) descent of flatness for integral extensions of Noetherian rings and b) direct summand conjecture due to M. Hochster.

Criteria for flatness and injectivity

Let $R$ be a commutative Noetherian ring. We give criteria for flatness of $R$-modules in terms of associated primes and torsion-freeness of certain tensor products. This allows us to develop a criterion for regularity if $R$ has characteristic $p$, or more generally if it has a locally contracting endomorphism. Dualizing, we give criteria for injectivity of $R$-modules in terms of coassociated primes and (h-)divisibility of certain $\Hom$-modules. Along the way, we develop tools to achieve such a dual result. These include a careful analysis of the notions of divisibility and h-divisibility (…

On $τ_q$-flatness and $τ_q$-coherence

In this paper, we introduce and study the notions of $τ_q$-flat modules and $τ_q$-coheret rings. First, by investigating the Nagata rings of $τ_q$-torsion theory, we show that the small finitistic dimensions of T$(R[x])$ are all equal to $0$ for any ring $R$. Then, we introduce the notion of $τ_q$-VN regular rings (i.e. over which all modules are $τ_q$-flat), and show that a ring $R$ is a $τ_q$-VN regular ring if and only if T$(R[x])$ is a von Neumann regular ring. Finally, we obtain the Chase theorem for $τ_q$-coheret rings: a ring $R$ is $τ_q$-coherent if and only if any direct product of $R…

平坦仿射群概型的平坦性纤维判别法与同态

Fiber criteria for flatness and homomorphisms of flat affine group schemes

本文提出了避免“有限性”假设的模平坦性纤维判别法，并基于 Tannakian 观点推导了其在仿射与平坦群概型理论中的应用。