More Like this

1. A partial converse ghost lemma for the derived category of a commutative Noetherian ring

   https://doi.org/10.1090/proc/16294  

   Liu, Jian; Pollitz, Josh (January 2023, Proceedings of the American Mathematical Society)

   In this article a condition is given to detect the containment among thick subcategories of the bounded derived category of a commutative noetherian ring. More precisely, for a commutative noetherian ring $R$and complexes of $R$-modules with finitely generated homology $M$and $N$, we show $N$is in the thick subcategory generated by $M$if and only if the ghost index of $N_{\mathfrak{p}}$with respect to $M_{\mathfrak{p}}$is finite for each prime $\mathfrak{p}$of $R$. To do so, we establish a “converse coghost lemma” for the bounded derived category of a non-negatively graded DG algebra with noetherian homology. 

   more » « less
2. Bounds on cohomological support varieties

   https://doi.org/10.1090/btran/182  

   Briggs, Benjamin; Grifo, Eloísa; Pollitz, Josh (March 2024, Transactions of the American Mathematical Society, Series B)

   Over a local ring $R$, the theory of cohomological support varieties attaches to any bounded complex $M$of finitely generated $R$-modules an algebraic variety ${\mathrm{V}}_R\left(M\right)$that encodes homological properties of $M$. We give lower bounds for the dimension of ${\mathrm{V}}_R\left(M\right)$in terms of classical invariants of $R$. In particular, when $R$is Cohen–Macaulay and not complete intersection we find that there are always varieties that cannot be realized as the cohomological support of any complex. When $M$has finite projective dimension, we also give an upper bound for $\dim <\#comment/>{\mathrm{V}}_R\left(M\right)$in terms of the dimension of the radical of the homotopy Lie algebra of $R$. This leads to an improvement of a bound due to Avramov, Buchweitz, Iyengar, and Miller on the Loewy lengths of finite free complexes, and it recovers a result of Avramov and Halperin on the homotopy Lie algebra of $R$. Finally, we completely classify the varieties that can occur as the cohomological support of a complex over a Golod ring. 

   more » « less
3. 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. 

   more » « less
4. On the Boundary Behavior of Mass-Minimizing Integral Currents

   https://doi.org/10.1090/memo/1446  

   De_Lellis, Camillo; De_Philippis, Guido; Hirsch, Jonas; Massaccesi, Annalisa (November 2023, Memoirs of the American Mathematical Society)

   Let $\mathrm{\Sigma <\#comment/>}$be a smooth Riemannian manifold, $\mathrm{\Gamma <\#comment/>}\subset <\#comment/>\mathrm{\Sigma <\#comment/>}$a smooth closed oriented submanifold of codimension higher than $2$and $T$an integral area-minimizing current in $\mathrm{\Sigma <\#comment/>}$which bounds $\mathrm{\Gamma <\#comment/>}$. We prove that the set of regular points of $T$at the boundary is dense in $\mathrm{\Gamma <\#comment/>}$. Prior to our theorem the existence of any regular point was not known, except for some special choice of $\mathrm{\Sigma <\#comment/>}$and $\mathrm{\Gamma <\#comment/>}$. As a corollary of our theorem we answer to a question in Almgren’sAlmgren’s big regularity paperfrom 2000 showing that, if $\mathrm{\Gamma <\#comment/>}$is connected, then $T$has at least one point $p$of multiplicity $\frac{1}{2}$, namely there is a neighborhood of the point $p$where $T$is a classical submanifold with boundary $\mathrm{\Gamma <\#comment/>}$; we generalize Almgren’s connectivity theorem showing that the support of $T$is always connected if $\mathrm{\Gamma <\#comment/>}$is connected; we conclude a structural result on $T$when $\mathrm{\Gamma <\#comment/>}$consists of more than one connected component, generalizing a previous theorem proved by Hardt and Simon in 1979 when $\mathrm{\Sigma <\#comment/>}={\mathbb{R}}^{m+1}$and $T$is $m$-dimensional. 

   more » « less
5. 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 $\mathrm{\partial <\#comment/>}D$. 

   more » « less