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1. Compatible ideals in ℚ-Gorenstein rings

   https://doi.org/10.1090/proc/16331  

   Polstra, Thomas; Schwede, Karl (October 2023, Proceedings of the American Mathematical Society)

   Suppose $R$is a $F$-finite and $F$-pure $\mathbb{Q}$-Gorenstein local ring of prime characteristic $p>0$. We show that an ideal $I\subseteq <\#comment/>R$is uniformly compatible ideal (with all $p^{-<\#comment/>e}$-linear maps) if and only if exists a module finite ring map $R\to <\#comment/>S$such that the ideal $I$is the sum of images of all $R$-linear maps $S\to <\#comment/>R$. In other words, the set of uniformly compatible ideals is exactly the set of trace ideals of finite ring maps. 

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2. Regularity, singularities and ℎ-vector of graded algebras

   https://doi.org/10.1090/tran/9089  

   Dao, Hailong; Ma, Linquan; Varbaro, Matteo (January 2024, Transactions of the American Mathematical Society)

   Let $R$be a standard graded algebra over a field. We investigate how the singularities of $\mathrm{Spec}<\#comment/>R$or $\mathrm{Proj}<\#comment/>R$affect the $h$-vector of $R$, which is the coefficient of the numerator of its Hilbert series. The most concrete consequence of our work asserts that if $R$satisfies Serre’s condition $\left(S_r\right)$and has reasonable singularities (Du Bois on the punctured spectrum or $F$-pure), then $h_0$, …, $h_r\ge <\#comment/>0$. Furthermore the multiplicity of $R$is at least $h_0+h_1+\cdots <\#comment/>+h_{r-<\#comment/>1}$. We also prove that equality in many cases forces $R$to be Cohen-Macaulay. The main technical tools are sharp bounds on regularity of certain $\mathrm{Ext}$modules, which can be viewed as Kodaira-type vanishing statements for Du Bois and $F$-pure singularities. Many corollaries are deduced, for instance that nice singularities of small codimension must be Cohen-Macaulay. Our results build on and extend previous work by de Fernex-Ein, Eisenbud-Goto, Huneke-Smith, Murai-Terai and others. 

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3. On regularity of \overline∂-solutions on 𝑎_{𝑞} domains with 𝐶² boundary in complex manifolds

   https://doi.org/10.1090/tran/9315  

   Gong, Xianghong (March 2025, Transactions of the American Mathematical Society)

   We study regularity of solutions $u$to $\stackrel{¯<\#comment/>}{\mathrm{\partial <\#comment/>}}u=f$on a relatively compact $C^2$domain $D$in a complex manifold of dimension $n$, where $f$is a $(0,q)$form. Assume that there are either $(q+1)$negative or $(n-<\#comment/>q)$positive Levi eigenvalues at each point of boundary $\mathrm{\partial <\#comment/>}D$. Under the necessary condition that a locally $L^2$solution exists on the domain, we show the existence of the solutions on the closure of the domain that gain $1/2$derivative when $q=1$and $f$is in the Hölder–Zygmund space ${\mathrm{\Lambda <\#comment/>}}^r\left(D\right)$with $r>1$. For $q>1$, the same regularity for the solutions is achieved when $\mathrm{\partial <\#comment/>}D$is either sufficiently smooth or of $(n-<\#comment/>q)$positive Levi eigenvalues everywhere on $\mathrm{\partial <\#comment/>}D$. 

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4. Products of normal subsets

   https://doi.org/10.1090/tran/8960  

   Larsen, Michael; Shalev, Aner; Tiep, Pham (February 2024, Transactions of the American Mathematical Society)

   In this paper we consider which families of finite simple groups $G$have the property that for each $\mathrm{ϵ<\#comment/>}>0$there exists $N>0$such that, if $|G|\ge <\#comment/>N$and $S,T$are normal subsets of $G$with at least $\mathrm{ϵ<\#comment/>}|G|$elements each, then every non-trivial element of $G$is the product of an element of $S$and an element of $T$. We show that this holds in a strong and effective sense for finite simple groups of Lie type of bounded rank, while it does not hold for alternating groups or groups of the form ${\mathrm{P}\mathrm{S}\mathrm{L}}_n\left(q\right)$where $q$is fixed and $n\to <\#comment/>\mathrm{\infty <\#comment/>}$. However, in the case $S=T$and $G$alternating this holds with an explicit bound on $N$in terms of $\mathrm{ϵ<\#comment/>}$. Related problems and applications are also discussed. In particular we show that, if $w_1,w_2$are non-trivial words, $G$is a finite simple group of Lie type of bounded rank, and for $g\in <\#comment/>G$, $P_{w_1\left(G\right),w_2\left(G\right)}\left(g\right)$denotes the probability that $g_1{g}_2=g$where $g_i\in <\#comment/>w_i\left(G\right)$are chosen uniformly and independently, then, as $|G|\to <\#comment/>\mathrm{\infty <\#comment/>}$, the distribution $P_{w_1\left(G\right),w_2\left(G\right)}$tends to the uniform distribution on $G$with respect to the $L^{\mathrm{\infty <\#comment/>}}$norm. 

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5. Residual intersections and linear powers

   https://doi.org/10.1090/btran/127  

   Eisenbud, David; Huneke, Craig; Ulrich, Bernd (October 2023, Transactions of the American Mathematical Society, Series B)

   If $I$is an ideal in a Gorenstein ring $S$, and $S/I$is Cohen-Macaulay, then the same is true for any linked ideal $I^{\prime }$; but such statements hold for residual intersections of higher codimension only under restrictive hypotheses, not satisfied even by ideals as simple as the ideal $L_n$of minors of a generic $2\times <\#comment/>n$matrix when $n>3$. In this paper we initiate the study of a different sort of Cohen-Macaulay property that holds for certain general residual intersections of the maximal (interesting) codimension, one less than the analytic spread of $I$. For example, suppose that $K$is the residual intersection of $L_n$by $2n-<\#comment/>4$general quadratic forms in $L_n$. In this situation we analyze $S/K$and show that $I^{n-<\#comment/>3}\left(S/K\right)$is a self-dual maximal Cohen-Macaulay $S/K$-module with linear free resolution over $S$. The technical heart of the paper is a result about ideals of analytic spread 1 whose high powers are linearly presented. 

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