We give an intrinsic characterization of Cox sheaves on Krull schemes in terms of their valuative algebraic properties. We also provide a geometric characterization of their graded relative spectra in terms of good quotients of graded schemes, extending the existing theory on relative spectra of Cox sheaves on normal varieties. Moreover, we obtain an irredundant characterization of Cox rings which, in turn, produces a normality criterion for certain graded rings.

Let $R = \bigoplus _{\alpha \in \Gamma } R_{\alpha }$ be an integral domain graded by an arbitrary torsionless grading monoid $\Gamma $, $M$ a homogeneous maximal ideal of $R$ and $S(H) = R \setminus \bigcup _{P \in \text {h-}Spec (R)}P$. We show that $R$ is a graded Noetherian domain with $\text {h-}\dim (R) = 1$ if and only if $R_{S(H)}$ is a one-dimensional Noetherian domain. We then use this result to prove a graded Noetherian domain analogue of the Krull-Akizuki theorem. We prove that, if $R$ is a gr-valuation ring, then $R_M$ is a valuation domain, $\dim (R_M) = \text {h-}\dim (R)$ and $R_M$ is a discrete valuation ring if and only if $R$ is discrete as a gr-valuation ring. We also prove that, if $\{P_i\}$ is a chain of homogeneous prime ideals of a graded Noetherian domain $R$, then there exists a discrete valuation overring of $R$ which has a chain of prime ideals lying over $\{P_i\}$.

Let $R$ be a Gorenstein local ring with maximal ideal $\mathfrak{m} $ satisfying $\mathfrak{m} ^3=0\ne \mathfrak{m} ^2$. Set $\mathfrak{k} =R/\mathfrak{m} $ and $e=rank _{\mathfrak{k} }(\mathfrak{m} /\mathfrak{m} ^2)$. If $e>2$ and $M$, $N$ are finitely generated $R$-modules, we show that the formal power series \[ \sum _{i=0}^\infty rank _{\mathfrak{k} }\left (Ext ^i_R(M,N)\otimes _R\mathfrak{k} \right )t^i \] and \[ \sum _{i=0}^\infty rank _{\mathfrak{k} }\left (Tor _i^R(M,N)\otimes _R \mathfrak{k} \right )t^i \] are rational, with denominator $1-et+t^2$.

The notion of $F$-signature was defined by Huneke and Leuschke and this numerical invariant characterizes some singularities. This notion is extended to finitely generated modules by Sannai and is called dual $F$-signature. In this paper, we determine the dual $F$-signature of a certain class of Cohen-Macaulay modules (so-called ``special") over cyclic quotient surface singularities. Also, we compare the dual $F$-signature of a special Cohen-Macaulay module with that of its Auslander-Reiten translation. This gives a new characterization of the Gorensteinness.

We prove the existence of tilting objects on some global quotient stacks. As a consequence, we provide further evidence for a conjecture on the Rouquier dimension of derived categories formulated by Orlov.

Let $R$ be a commutative, Noetherian, local ring and $M$ a finitely generated $R$-module. Consider the module of homomorphisms $Hom _R(R/\mathfrak{a} ,M/\mathfrak{b} M)$ where $\mathfrak{b} \subseteq \mathfrak{a} $ are parameter ideals of $M$. When $M=R$ and $R$ is Cohen-Macaulay, Rees showed that this module of homomorphisms is isomorphic to $R/\mathfrak{a} $, and in particular, a free module over $R/\mathfrak{a} $ of rank one. In this work, we study the structure of such modules of homomorphisms for a not necessarily Cohen-Macaulay $R$-module $M$.