A generalization of the Shestakov-Umirbaev inequality

We recover some data of a module $M$ from the Ext-family $\{Ext^i_R(M,R)\}$. In this regard, we investigate homological subsets of $Spec(R)$, defined by the help of Ext-family. For example, we present a situation for which $\bigcup_{i=0}^\infty Ass(Ext^i_R(M,R))$ is infinite, thereby answering a question of Vasconcelos. We extend Grothendieck's calculation of $dim(Ext^g_R(M,R))$ and present some new situations to calculate $dim(Ext^\ast_R(M,R))$. Also, we compute support and the set of all associated prime ideals of the Ext-family in a serial of nontrivial cases.

math.AC↗

arXiv2026

On Stanley-Reisner Rings with Minimal Betti Numbers

We study simplicial complexes with a given number of vertices whose Stanley-Reisner ring has the minimal sum of Betti numbers. We prove that every full subcomplex of them is homotopy equivalent either to a point or to a sphere. Moreover, we prove that such kind of simplicial complexes coincide with those having minimal Taylor resolutions.

math.AC↗

arXiv2026

Local-Cohen-Macaulay DG-rings with unequal finitistic projective dimensions

For every pair of integers $d,n\geq1$, we construct a commutative noetherian local-Cohen-Macaulay DG-ring $A$ such that $\dim H^0(A)=d$, $\mathrm{amp}(A)=n$, $\mathrm{fpd}(A)=d-n$, and $\mathrm{FPD}(A)=d$. This gives a negative answer to the implication from local-Cohen-Macaulayness to equality of the two finitistic projective dimensions in a question of Bird, Shaul, Sridhar, and Williamson.

math.AC↗