Paul C. Roberts
We define a closure operation for rings of mixed characteristic and verify that the closure is a ring. We then show that this closure produces a ring with good properties with respect to its Fontaine ring and give an example to show that rings that are not closed in this sense do not satisfy these properties.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
We introduce two closure operations on ideals in commutative rings related to the ring operation of root closure. One closure is the result of iterating a root-like operation on ideals infinitely many times, and the other closure arises as a graded component of the root closure of the Rees algebra applied to an ideal. We study the properties of these closures and prove they are distinct operations.
We study the Cohen-Macaulay property of a particular class of radical extensions of an unramified regular local ring having mixed characteristic.
We study how solid closure in mixed characteristic behaves after taking ultraproducts. The ultraproduct will be chosen so that we land in equal characteristic, and therefore can make a comparison with tight closure. As a corollary we get an asymptotic version of the Hochster-Roberts invariant theorem in dimension three: if $R$ is a mixed characteristic (cyclically) pure 3-dimensional local subring of a regular local ring $S$, then $R$ is Cohen-Macaulay, provided the ramification of $S$ is large with respect to its dimension and residual characteristic, and with respect to the multiplicity of $…
Extended plus (epf) closure and rank 1 (r1f) closure are two closure operations introduced by Raymond C. Heitmann for rings of mixed characteristic. Recently, he and Linquan Ma proved that epf closure satisfies the usual colon-capturing property under mild conditions. In this paper, we extend their result and prove that epf closure satisfies what we call the $p$-colon-capturing property. Based on that, we define a new closure notion called "weak epf closure", and prove that it satisfies the generalized colon-capturing property and some other colon-capturing properties. This gives a new proof o…
Representation and Embedding of Pseudo MV-algebras with Square Roots II. Closures
本文作为带平方根伪MV代数研究的第二部分，引入严格平方根与平方根闭包概念并比较二者，证明MV代数均有平方根闭包，还找到了特殊带弱平方根伪MV代数的最大子代数。
The (full) extended plus closure was developed as a replacement for tight closure in mixed characteristic rings. Here it is shown by adapting André's perfectoid algebra techniques that, for complete local rings that have F-finite residue fields, this closure has the colon-capturing property. In fact, more generally, if $R$ is a (possibly ramified) complete regular local ring of mixed characteristic that have F-finite residue fields, $I$ and $J$ are ideals of $R$, and the local domain $S$ is a finite $R$-module, then $(IS:J)\subseteq (I:J)S^{epf}$. A consequence is that all ideals in regular lo…
We prove that the absolute integral closure $R^{+}$ of an equicharacteristic zero noetherian complete local domain $R$ is not coherent, provided $\dim(R)\geq 2$. As a corollary, we give an elementary proof of the mixed characteristic version of the result due to Asgharzadeh and extend it to dimension $3$. Furthermore, we apply the methods of Aberbach and Hochster used to prove the positive characteristic version of this result to study F-coherent rings and our work naturally suggests a mixed characteristic analogue of a result of Smith.
Counterexamples to Minkowski-type inequalities and the Rees converse for modules via mixed Buchsbaum-Rim Multiplicities
本文构造混合Buchsbaum-Rim重数，证明其系数乘积公式，给出Minkowski型不等式等失效的反例，明确一维情形下Rees逆命题不成立，仅在特定条件下成立，并提出高维逆命题问题。
We prove that, modulo any power of a prime $p$, the absolute integral closure of an excellent noetherian domain is Cohen-Macaulay. A graded analog is also established, yielding variants of Kodaira vanishing "up to finite covers" in mixed characteristic. Our main tools are (log) prismatic cohomology (which yields a Frobenius action in mixed characteristic) and the $p$-adic Riemann-Hilbert functor for constructible étale $\mathbf{F}_p$-sheaves on varieties over a $p$-adic field (which almost controls perfectified prismatic cohomology).
To generalize the notion of Galois closure for separable field extensions, we devise a notion of $G$-closure for algebras of commutative rings $R\to A$, where $A$ is locally free of rank $n$ as an $R$-module and $G$ is a subgroup of $\mathrm{S}_n$. A $G$-closure datum for $A$ over $R$ is an $R$-algebra homomorphism $φ: (A^{\otimes n})^{G}\to R$ satisfying certain properties, and we associate to a closure datum $φ$ a closure algebra $A^{\otimes n}\otimes_{(A^{\otimes n})^G} R$. This construction reproduces the normal closure of a finite separable field extension if $G$ is the corresponding Galo…
We give short, closure-theoretic proofs for uniform bounds on the growth of symbolic powers of ideals in regular rings. The author recently proved these bounds in mixed characteristic using various versions of perfectoid/big Cohen-Macaulay test ideals, with special cases obtained earlier by Ma and Schwede. In mixed characteristic, we instead use Heitmann's full extended plus (epf) closure, Jiang's weak epf (wepf) closure, and R.G.'s results on closure operations that induce big Cohen-Macaulay algebras. Our strategy also applies to any Dietz closure satisfying R.G.'s algebra axiom and a Brianço…
A calculation of the perfectoidization of semiperfectoid rings
本文证明在半完美oid等情形下可通过$p$-根闭包计算perfectoidization，并借此建立了经典闭包运算与棱镜上同调理论中perfectoidization概念的联系。
In an earlier preprint (math.AG/9810142) we gave an explicit description of the algebraic closure of the field of power series over a field of characteristic p, in terms of "generalized power series". In this paper, we give an analogous description of the algebraic closure of the fraction field of the Witt ring of a field of characteristic p, using an analogue of generalized power series in mixed characteristic due to Poonen.
A note concerning the vanishing of local cohomology for roots in mixed characteristic
本文探讨了混合特征局部环中积分闭包的Cohen-Macaulay性质与局部上同调模块的关系，指出其等价于特定上同调模块为零，并等同于对偶模满足Serre条件(S_3)。
Let $S$ be an unramified regular local ring of mixed characteristic $p\geq 3$ and $S^p$ the subring of $S$ obtained by lifting to $S$ the image of the Frobenius map on $S/pS$. Let $R$ be the integral closure of $S$ in a biradical extension of degree $p^2$ of its quotient field obtained by adjoining $p$-th roots of sufficiently general square free elements $f,g\in S^p$. We show that $R$ admits a birational maximal Cohen-Macaulay module. It is noted that $R$ is not automatically Cohen-Macaulay.
For a local field F with finite residue field of characteristic p, we describe completely the structure of the filtered F_p[G]-module K^*/K^*p in characteristic 0 and $K^+/\wp(K^+) in characteristic p, where K=F(\root{p-1}\of F^*) and G=\Gal(K|F). As an application, we give an elementary proof of Serre's mass formula in degree p. We also determine the compositum C of all degree p separable extensions with solvable galoisian closure over an arbitrary base field, and show that C is K(\root p\of K^*) or K(\wp^{-1}(K)) respectively, in the case of the local field F. Our method allows us to compute…
Integral perfectoid big Cohen-Macaulay algebras via André's theorem
本文通过安德烈定理证明混合特征下Noetherian局部域可映射到积分完美体大Cohen-Macaulay代数，并探讨了其绝对积分闭包的相关性质。
On Abelian extensions in mixed characteristic and ramification in codimension one
针对Paul Roberts关于正则局部环整闭包性质的定理在模情形下不成立的问题，通过修改余维一p上分歧的相关表述，将该定理推广至混合特征下非分歧正则局部环的模情形（伽罗瓦群p挠元被p零化）。
Let $S$ be an unramified regular local ring of mixed characteristic two and $R$ the integral closure of $S$ in a biquadratic extension of its quotient field obtained by adjoining roots of sufficiently general square free elements $f,g\in S$. Let $S^2$ denote the subring of $S$ obtained by lifting to $S$ the image of the Frobenius map on $S/2S$. When at least one of $f,g\in S^2$, we characterize the Cohen-Macaulayness of $R$ and show that $R$ admits a birational small Cohen-Macaulay module. It is noted that $R$ is not automatically Cohen-Macaulay in case $f,g\in S^2$ or if $f,g\notin S^2$.
Almost Cohen-Macaulay algebras in mixed characteristic via Fontaine rings
本文通过Fontaine环和Witt向量方法，在混合特征下构造了完备局部域的弱几乎Cohen-Macaulay代数，并探讨了其与单项猜想的关系。
Tight closure test ideals have been central to the classification of singularities in rings of characteristic $p>0$, and via reduction to characteristic $p$, in equal characteristic zero as well. A summary of their properties and applications can be found in "A survey of test ideals" by Karl Schwede and Kevin Tucker. In this paper, we extend the notion of a test ideal to arbitrary closure operations, particularly those coming from big Cohen-Macaulay modules and algebras, and prove that it shares key properties of tight closure test ideals. Our main results show how these test ideals can be use…
Let $D$ be an integrally closed domain with quotient field $K$. Let $A$ be a torsion-free $D$-algebra that is finitely generated as a $D$-module. For every $a$ in $A$ we consider its minimal polynomial $μ_a(X)\in D[X]$, i.e. the monic polynomial of least degree such that $μ_a(a)=0$. The ring ${\rm Int}_K(A)$ consists of polynomials in $K[X]$ that send elements of $A$ back to $A$ under evaluation. If $D$ has finite residue rings, we show that the integral closure of ${\rm Int}_K(A)$ is the ring of polynomials in $K[X]$ which map the roots in an algebraic closure of $K$ of all the $μ_a(X)$, $a\i…
The log canonical threshold of products of ideals and mixed Łojasiewicz exponents
本文证明了环 $\mathcal O_n$ 中两理想之积 $IJ$ 的对数典范阈值关于其混合 Łojasiewicz 指数序列的精确下界，并给出了极大理想情形下等式成立的充要条件。
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。
Let $R$ be a commutative integral domain with unit, $f$ be a nonconstant monic polynomial in $R[t]$, and $I_f \subset R[t]$ be the ideal generated by $f$. In this paper we study the group of $R$-algebra automorphisms of the $R$-algebra without unit $I_f$. We show that, if $f$ has only one root (possibly with multiplicity), then $Aut (I_f) \cong R^\times$. We also show that, under certain mild hypothesis, if $f$ has at least two different roots in the algebraic closure of the quotient field of $R$, then $Aut(I_f)$ is a cyclic group and its order can be completely determined by analyzing the roo…
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。