Amnon Yekutieli
Let A be a noetherian commutative ring, and let I be an ideal in A. We study questions of flatness and I-adic completeness for infinitely generated A-modules. This is done using the notions of decaying function and I-adically free A-module.
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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 whic…
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.
We prove a $p$-adic analog of Kunz's theorem: a $p$-adically complete noetherian ring is regular exactly when it admits a faithfully flat map to a perfectoid ring. This result is deduced from a more precise statement on detecting finiteness of projective dimension of finitely generated modules over noetherian rings via maps to perfectoid rings. We also establish a version of the $p$-adic Kunz's theorem where the flatness hypothesis is relaxed to almost flatness.
The Miracle of Flatness in Algebraic Geometry
研究代数几何中平坦性，在半环背景下获离散性结果等，构造不可下降的忠实平坦环映射并展示下降指数与基数关系，证明特定态射极大平展轨迹的仿射性，还有相关引理变体、\(D -\)模刻画及Tor - 独立性结果。
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.
Among reduced Noetherian prime characteristic commutative rings, we prove that a regular ring is precisely one where finite intersection of ideals commutes with taking bracket powers. However, reducedness is essential for this equivalence. Connections are made with Ohm-Rush content theory, intersection-flatness of the Frobenius map, and various flatness criteria.
Generic flatness of the cohomology of thickenings
本文研究了射影方案厚化方案的余弦学通用平坦性，通过构造非普遍自由的局部余弦学模块，证明其具有无限多个关联素理想，并探讨了相关问题的未解决状态。
The Proj of the Rees algebra of a graded family of ideals
研究Rees代数的渐进代数的Proj在Noetherian局部环中为Noetherian的条件，探讨高维情形下该性质不成立的例子及纤维锥的极小素理想有限性。
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 (…
We give, in a more general case than the noetherian case, an answer to the question posed by Shaul on the flatness of the completed tensor product.
Broadening existing results in the literature to much wider classes of rings, we prove among other things: 1. Reduced quotients of excellent regular rings of characteristic $p$ admit big test elements, 2. The set of F-jumping numbers of a principal ideal in a locally excellent regular ring is a discrete subset of $\mathbb Q$, and 3. If $R$ is a quotient of a locally excellent regular ring of prime characteristic, then there is a uniform upper bound on the Hartshorne-Speiser-Lyubeznik numbers of the injective hulls of the residue fields of $R$. To do so, we develop the parallel theories of Ohm-…
On torsion-free modules and semi-hereditary rings
本文研究了半遗传环的结构理论，重点探讨其与无挠模平坦性的关系，并考虑了Shimomoto关于Frobenius映射平坦性的问题。
Power Series Rings over Zero-Dimensional Rings
本文研究零维基环上的幂级数环，提出保持平坦性且维数有限的版本，并将Krull高度定理和纯混合性推广至非诺特环情形。
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 $R$-…
There are several theorems describing the intricate relationship between flatness and associated primes over commutative Noetherian rings. However, associated primes are known to act badly over non-Noetherian rings, so one needs a suitable replacement. In this paper, we show that the behavior of strong Krull primes most closely resembles that of associated primes over a Noetherian ring. We prove an analogue of a theorem of Epstein and Yao characterizing flat modules in terms of associated primes by replacing them with strong Krull primes. Also, we partly generalize a classical equational theor…
Local cohomology modules with nonclosed support
该研究构造了具有非闭支集的局部上同调模对应的诺特环，回答了Huneke-Lyubeznik提出的问题
We consider properties of extensions of Krull domains such as flatness that involve behavior of extensions and contractions of prime ideals. Let (R,m) be an excellent normal local domain with field of fractions K, let y be a nonzero element in m, and let R* denote the (y)-adic completion of R. For a finite set w of elements of yR* that are algebraically independent over R, we construct two Krull domains: an intersection domain A that is the intersection of R* with the field of fractions of K[w], and an approximation domain B to A. If R is countable with dim R at least 2, we prove that there ex…
We study the category of Sp-equivariant modules over the infinite variable polynomial ring, where Sp denotes the infinite symplectic group. We establish a number of results about this category: for instance, we show that every finitely generated module M fits into an exact triangle $T \to M \to F \to$ where T is a finite length complex of torsion modules and F is a finite length complex of "free" modules; we determine the Grothendieck group; and we (partially) determine the structure of injective modules. We apply these results to show that the twisted commutative algebras ${\rm Sym}({\bf C}^{…
We prove a Cohen-Macaulay version of a result by Avramov-Golod and Frankild-Jørgensen about Gorenstein rings, showing that if a noetherian ring $A$ is Cohen-Macaulay, and $a_1,\dots,a_n$ is any sequence of elements in $A$, then the Koszul complex $K(A;a_1,\dots,a_n)$ is a Cohen-Macaulay DG-ring. We further generalize this result, showing that it also holds for commutative DG-rings. In the process of proving this, we develop a new technique to study the dimension theory of a noetherian ring $A$, by finding a Cohen-Macaulay DG-ring $B$ such that $\mathrm{H}^0(B) = A$, and using the Cohen-Macaula…
In this paper, we shall study finite generation of symbolic Rees rings of the defining ideal ${\frak p}$ of the space monomial curve $(t^a, t^b, t^c)$ for pairwise coprime integers $a$, $b$, $c$. Suppose that the base field is of characteristic $0$ and the above ideal ${\frak p}$ is minimally generated by three polynomials. Under the assumption that the homogeneous element $ξ$ of the minimal degree in ${\frak p}$ is the negative curve, we determine the minimal degree of an element $η$ such that the pair $\{ ξ, η\}$ satisfies Huneke's criterion in the case where the symbolic Rees ring is Noethe…
In this article, we generalize the well-known result that ideals of Noetherian polynomial rings have only finitely many initial ideals to the situation of ascending ideal chains in non-Noetherian polynomial rings. More precisely, we study ideal chains in the polynomial ring $R=K[x_{i,j}\,|\,1\leq i\leq c,j\in N]$ that are invariant under the action of the monoid $Inc(N)$ of strictly increasing functions on $N$, which acts on $R$ by shifting the second variable index. We show that for every such ideal chain, the number of initial ideal chains with respect to term orders on $R$ that are compatib…
On the infinitely generated locus of Frobenius algebras of rings of prime characteristic
本文研究素特征交换 Noether 环谱中 Frobenius 代数有限生成点集何时为 Zariski 开集，并证明对 Stanley–Reisner 环成立，还给出闭补集的显式算法计算。
In this note, it is proved that over a commutative noetherian henselian non-Gorenstein local ring there are infinitely many isomorphism classes of indecomposable totally reflexive modules, if there is a nonfree cyclic totally reflexive module.
A Counterexample to Problem 19 on Integer-valued Polynomial Rings
针对Cahen等人关于整值多项式环的平坦性与自由性的问题19，通过构造一个二元域上的二维诺特局部整环D，并证明D上的整值多项式环作为D-模不是平坦的，从而给出否定回答。
Noncatenary Unique Factorization Domains
本文构造了一类在无穷多个位置非悬链的局部诺特唯一分解整环，并证明其存在无穷多个高度为一的素理想使商环非悬链，还构造了所有高度为一素理想商环均非悬链的可数局部UFD。
Big pure projective modules over commutative noetherian rings: comparison with the completion
该研究针对交换诺特环上的纯投射模分类问题，利用有限生成数据刻画无限生成直和项，拓展经典理论至无限生成情形，还研究了相关幺半群结构并得到子幺半群结论与实现结果。
A mixed characteristic analogue of the perfection of rings and its almost Cohen-Macaulay property
本文在混合特征完备局部整环上构造了一个类似正特征完美闭包的 perfectoid 环，并利用 André 平坦性引理和 Riemann 延拓定理证明了其几乎 Cohen-Macaulay 性质。
Conditions on the Koszul complex of a noetherian local ring $R$ guarantee that $\mathrm{Tor}^{R}_{i}(M,N)$ is non-zero for infinitely many $i$, when $M$ and $N$ are finitely generated $R$-modules of infinite projective dimension. These conditions are obtained from results concerning Tor of differential graded modules over certain trivial extensions of commutative differential graded algebras.
This is a survey on the relation between homological properties of the Frobenius endomorphism and finiteness of various homological dimensions of the ring or of modules over it, such as global dimension and projective dimension. We begin with Kunz's surprising result in 1969 that the regularity of a Noetherian local ring is equivalent to the flatness of its Frobenius endomorphism, as well as the subsequent generalizations to the module setting by Peskine and Szpiro and continue up through the recent flurry of results in the last five years. An attempt is made to include proofs whenever feasibl…
Let $R$ be a commutative Noetherian local ring. Assume that $R$ has a pair $\{x,y\}$ of exact zerodivisors such that $\dim R/(x,y)\ge2$ and all totally reflexive $R/(x)$-modules are free. We show that the first and second Brauer--Thrall type theorems hold for the category of totally reflexive $R$-modules. More precisely, we prove that, for infinitely many integers $n$, there exists an indecomposable totally reflexive $R$-module of multiplicity $n$. Moreover, if the residue field of $R$ is infinite, we prove that there exist infinitely many isomorphism classes of indecomposable totally reflexiv…
Using recent work by Erman-Sam-Snowden, we show that finitely generated ideals in the ring of bounded-degree formal power series in infinitely many variables have finitely generated Gröbner bases relative to the graded reverse lexicographic order. We then combine this result with the first author's work on topological Noetherianity of polynomial functors to give an algorithmic proof of the following statement: ideals in polynomial rings generated by a fixed number of homogeneous polynomials of fixed degrees only have a finite number of possible generic initial ideals, independently of the numb…
Equivariant Free Resolutions of Sequences of Symmetric Module
本文研究一系列对称群表示模 $M_n$ 的等变自由分解问题，在 FI-模框架下提出了计算合系和 FI-等变微分的算法，并应用于计算无限多变量多项式环中理想的等变自由分解截断。
Let $R$ be a commutative Noetherian local ring of prime characteristic $p$ and $f:R\to R$ the Frobenius ring homomorphism. For $e\ge 1$ let $R^{(e)}$ denote the ring $R$ viewed as an $R$-module via $f^e$. Results of Peskine, Szpiro, and Herzog state that for finitely generated modules $M$, $M$ has finite projective dimension if and only if $\operatorname{Tor}_i^R(R^{(e)},M)=0$ for all $i>0$ and all (equivalently, infinitely many) $e\ge 1$. We prove this statement holds for arbitrary modules using the theory of flat covers and minimal flat resolutions.
Let A be an integral domain with field of fractions K. We investigate the structure of the overrings B of A (contained in K) that are well-centered on A in the sense that each principal ideal of B is generated by an element of A. We consider the relation of well-centeredness to the properties of flatness, localization and sublocalization for B over A. If B = A[b] is a simple extension of A, we prove that B is a localization of A if and only if B is flat and well-centered over A. If the integral closure of A is a Krull domain, in particular, if A is Noetherian, we prove that every finitely gene…
Infinitely generated symbolic Rees rings of positive characteristic
本文针对环面簇爆破后的Cox环，给出有限生成判据，并证明特定理想对应的扩展符号Rees环在特征为2或3时为Noether环。
We show that a ring $R$ is regular if $Tor_{i}^{R}(R^{+},k) = 0$ for some $i\geq 1$ assuming further that $R$ is a $\mathbb{N}$-graded ring of dimension $2$ finitely generated over an equi-characteristic zero field $k$. This answers a question of Bhatt, Iyengar, and Ma. We use almost mathematics over $R^{+}$ to deduce properties of the noetherian ring $R$ and rational surface singularities. Moreover we show that $R^{+}$ in equi-characteristic zero is $m$-adically ideal(wise) separated, a condition which appears in the proof of local criterion for flatness. In dimension $2$ it is Ohm-Rush and i…
Nonlocality and the central geometry of dimer algebras
本文证明不可消去 dimer 代数及其中心非 Noether、中心模无限生成，并刻画其约化中心的 Gorenstein 几何。
It is proved that a module $M$ over a Noetherian ring $R$ of positive characteristic $p$ has finite flat dimension if there exists an integer $t\ge 0$ such that $\operatorname{Tor}_i^R(M, {}^{f^{e}}\!R)=0$ for $t\le i\le t+\dim R$ and infinitely many $e$. This extends a result of Herzog, who proved it when $M$ is finitely generated, and strengthens a result of the third author and Webb in the case $M$ is arbitrary. It is also proved that when $R$ is a Cohen-Macaulay local ring, it suffices that the Tor vanishing holds for one $e\ge \log_{p}e(R)$, where $e(R)$ is the multiplicity of $R$.
A Mayer-Vietoris calculus for regular denominators
本文研究交换环上理想的正则分母，探讨其与理想交、和的关系，在诺特环上实现分母等价类分类，证明平坦基变换等性质，为具体情形的primary分解计算提供替代方案。
The reciprocal complement of a polynomial ring in several variables over a field
研究域上n≥2元多项式环的倒数补R(D)的性质，确定其为n维局部非诺特等特性的原子G-域，且除高度0和n外各高度有无穷素理想。
In this paper, we classify certain subcategories of modules over a ring R. A wide subcategory of R-modules is an Abelian subcategory of R-Mod that is closed under extensions. We give a complete classification of wide subcategories of finitely presented modules when R is a quotient of a regular commutative coherent ring by a finitely generated ideal. This includes all finitely presented algebras over a principal ideal domain, as well as polynomial rings on infinitely many variable over a PID. The classification is in terms of subsets of Spec R, and depends heavily on Thomason's classification o…
It is proved that a module $M$ over a Noetherian local ring $R$ of prime characteristic and positive dimension has finite flat dimension if Tor$_i^R({}^e R, M)=0$ for dim $R$ consecutive positive values of $i$ and infinitely many $e$. Here ${}^e R$ denotes the ring $R$ viewed as an $R$-module via the $e$th iteration of the Frobenius endomorphism. In the case $R$ is Cohen-Macualay, it suffices that the Tor vanishing above holds for a single $e\geq \log_p e(R)$, where $e(R)$ is the multiplicity of the ring. This improves a result of D. Dailey, S. Iyengar, and the second author, as well as genera…
Baer Splitting Beyond $τ_q$-Semisimplicity
本文证明 Baer 模均投射的环不一定是 $\tau_q$-半单环，通过非约化 Noether 环和约化凝聚 Bézout 环两类反例，推翻了原有猜想。
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