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The notion of Frobenius Betti numbers generalizes the Hilbert-Kunz multiplicity theory and serves as an invariant that measures singularity. However, the explicit computation of the Frobenius Betti numbers of rings has been limited to very few specific cases. This article focuses on the explicit computation of Frobenius Betti numbers of Cohen-Macaulay graded rings of finite Cohen-Macaulay type.

In this note, we give a description of the parameter test submodule of Rees algebras. This, in turn, describes the non-$F$-rational locus., Comment: 5 pages

In this paper, we show the equality of the (local) $\mathrm{v}$-number and Castelnuovo-Mumford regularity of certain classes of Gorenstein algebras, including the class of Gorenstein monomial algebras. Also, for the same classes of algebras with the assumption of level, we show that the (local) $\mathrm{v}$-number serves as an upper bound for the regularity. Moreover, we investigate the $\mathrm{v}$-number of Frobenius powers of graded ideals in prime characteristic setup. In this study, we demonstrate that the $\mathrm{v}$-numbers of Frobenius powers of graded ideals have an asymptotically linear behaviour. In the case of unmixed monomial ideals, we provide a method for computing the $\mathrm{v}$-number without prior knowledge of the associated primes., Comment: 15 pages, comments are welcome

In this article, we characterize Cohen-Macaulay permutation graphs. In particular, we show that a permutation graph is Cohen-Macaulay if and only if it is well-covered and there exists a unique way of partitioning its vertex set into $r$ disjoint maximal cliques, where $r$ is the cardinality of a maximal independent set of the graph. We also provide some sufficient conditions for a comparability graph to be a uniquely partially orderable (UPO) graph., Comment: 9 pages, 4 figures; comments are welcome

In this paper, we prove some sufficient conditions for Cohen-Macaulay normal Rees algebras to be $F$-rational. Let $(R,\mathfrak{m})$ be a Gorenstein normal local domain of dimension $d\geq 2$ and of characteristic $p > 0$. Let $I$ be a $\mathfrak{m}$-primary ideal. Our first set of results give conditions on the test ideals $\tau(I^n)$, $n \geq 1$ which would imply that the normalization of the Rees algebra $R[It]$ is $F$-rational. Another sufficient condition is that the socle of $\mathrm{H}_{\overline{G}_+}^d(\overline{G})$ (where $\overline{G}$ is the associated graded ring for the integral closure filtration) is entirely in degree $-1$, if $R$ is $F$-rational (but not necessarily Gorenstein). Then we show that if $R$ is a hypersurface of degree $2$ or is three-dimensional and $F$-rational, and $\mathrm{Proj} (R[\mathfrak{m} t ])$ is $F$-rational, then $R[\mathfrak{m} t ]$ is $F$-rational.