Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userFlatness and Completion Revisited
117
0
0.0
(
0
)
Authors
Amnon Yekutieli
Ask ChatGPT about the research
No Arabic abstract
We continue investigating the interaction between flatness and $mathfrak{a}$-adic completion for infinitely generated modules over a commutative ring $A$. We introduce the concept of $mathfrak{a}$-adic flatness, which is weaker than flatness. We prove that $mathfrak{a}$-adic flatness is preserved under completion when the ideal $mathfrak{a}$ is weakly proregular. We also prove that when $A$ is noetherian, $mathfrak{a}$-adic flatness coincides with flatness (for complete modules). An example is worked out of a non-noetherian ring $A$, with a weakly proregular ideal $mathfrak{a}$, for which the completion $hat{A}$ is not flat. We also study $mathfrak{a}$-adic systems, and prove that if the ideal $mathfrak{a}$ is finitely generated, then the limit of any $mathfrak{a}$-adic system is a complete module.
rate research
Read More
This paper has two parts. In the first part we recall the important role that weak proregularity of an ideal in a commutative ring has in derived completion and in adic flatness. We also introduce the new concepts of idealistic and sequential derived completion, and prove a few results about them, including the fact that these two concepts agree iff the ideal is weakly proregular. In the second part we study the local nature of weak proregularity, and its behavior w.r.t. ring quotients. These results allow us to prove that weak proregularity occurs in the context of bounded prisms, in the sense of Bhatt and Scholze. We anticipate that the concept of weak proregularity will help simplify and improve some of the more technical aspects of the groundbreaking theory of perfectoid rings and prisms (that has transformed arithmetic geometry in recent years).
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 $R$-module $M$ such that $M = PM$ for every minimal prime $P$ of $R$. We study the notion of intersection flatness and use it to prove that in certain graded cases it suffices to examine just one closed fiber to prove the stable prime extension property.
In a paper in 1962, Golod proved that the Betti sequence of the residue field of a local ring attains an upper bound given by Serre if and only if the homology algebra of the Koszul complex of the ring has trivial multiplications and trivial Massey operations. This is the origin of the notion of Golod ring. Using the Koszul complex components he also constructed a minimal free resolution of the residue field. In this article, we extend this construction up to degree five for any local ring. We describe how the multiplicative structure and the triple Massey products of the homology of the Koszul algebra are involved in this construction. As a consequence, we provide explicit formulas for the first six terms of a sequence that measures how far the ring is from being Golod.
We study the homological algebra of an R = Q/I module M using A-infinity structures on Q-projective resolutions of R and M. We use these higher homotopies to construct an R-projective bar resolution of M, Q-projective resolutions for all R-syzygies of M, and describe the differentials in the Avramov spectral sequence for M. These techniques apply particularly well to Golod modules over local rings. We characterize R-modules that are Golod over Q as those with minimal A-infinity structures. This gives a construction of the minimal resolution of every module over a Golod ring, and it also follows that if the inequality traditionally used to define Golod modules is an equality in the first dim Q+1 degrees, then the module is Golod, where no bound was previously known. We also relate A-infinity structures on resolutions to Avramovs obstructions to the existence of a dg-module structure. Along the way we give new, shorter, proofs of several classical results about Golod modules.
We introduce the notions of Koszul $N$-complex, $check{mathrm{C}}$ech $N$-complex and telescope $N$-complex, explicit derived torsion and derived completion functors in the derived category $mathbf{D}_N(R)$ of $N$-complexes using the $check{mathrm{C}}$ech $N$-complex and the telescope $N$-complex. Moreover, we give an equivalence between the category of cohomologically $mathfrak{a}$-torsion $N$-complexes and the category of cohomologically $mathfrak{a}$-adic complete $N$-complexes, and prove that over a commutative noetherian ring, via Koszul cohomology, via RHom cohomology (resp. $otimes$ cohomology) and via local cohomology (resp. derived completion), all yield the same invariant.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
