Related papers: $\mathbb{Z}^r$-graded rings and their canonical mo…

It is well-known that the category of Kleisli algebras for a monoidal monad carries a canonical monoidal structure. We define the notion of a commutative graded monad and present a strictly two-categorical proof that Kleisli algebras for…

We construct Quillen equivalences between the model categories of monoids (rings), modules and algebras over two Quillen equivalent model categories under certain conditions. This is a continuation of our earlier work where we established…

We compute the canonical class of certain Rees algebras. Our formula generalizes that of Herzog and Vasconcelos. Its proof relies on the fact that the formation of the canonical module commutes with subintersections in important cases. As…

Let H be a non semi-simple Ariki-Koike algebra. According to [18] and [14], there is a generalisation of Lusztig's a-function which induces a natural order (parametrised by a tuple m) on Specht modules. In some cases, Geck and Jacon have…

We study the properties of F-rationality and F-regularity in multigraded rings and their diagonal subalgebras. The main focus is on diagonal subalgebras of bigraded rings: these constitute an interesting class of rings since they arise…

We introduce the notion of E-depth of graded modules over polynomial rings to measure the depth of certain Ext modules. First, we characterize graded modules over polynomial rings with (sufficiently) large E-depth as those modules whose…

In this short note, we give a characterization of domains satisfying Serre's condition $(\mathrm{R}_1)$ in terms of their canonical modules. In the special case of toric rings, this generalizes a result of the second author (K. Yanagawa,…

Over $d$-dimensional Cohen-Macaulay rings with a canonical module, $d$-cotilting classes containing the maximal and balanced big Cohen-Macaulay modules are classified. Particular emphasis is paid to the direct limit closure of the balanced…

We define a general notion of centrally $\Gamma$-graded sets and groups and of their graded products, and prove some basic results about the corresponding categories: most importantly, they form braided monoidal categories. Here, $\Gamma$…

Let R be a ring and G a group. An R-module A is said to be artinian-by-(finite rank) if TorR(A) is artinian and A/TorR(A) has finite R-rank. The authors study ZG-modules A such that A/CA(H) is artinian-by-(finite rank) (as a Z-module) for…

Let $K$ be a field and $S=K[x_1,\ldots,x_m, y_1,\ldots,y_n]$ be the standard bigraded polynomial ring over $K$. In this paper, we explicitly describe the structure of finitely generated bigraded "sequentially Cohen--Macaulay" $S$-modules…

This work builds on earlier work of the first three authors where a notion of congruence modules in higher codimension is introduced. The main new results are a criterion for detecting regularity of local rings in terms of congruence…

In this paper we study the local cohomology of all finitely generated bigraded modules over a standard bigraded polynomial ring which have only one nonvanishing local cohomology with respect to one of the irrelevant bigraded ideals.

We shall describe the divisor class group and the graded canonical module of the multi-section ring for a normal projective variety X and Weil divisors D_1,..., D_s on X under a mild condition. In the proof, we use the theory of Krull…

We describe "quasi canonical modules" for modular invariant rings $R$ of finite group actions on factorial Gorenstein domains. From this we derive a general "quasi Gorenstein criterion" in terms of certain 1-cocycles. This generalizes a…

In this paper we lay the basis of the theory of rational modules of corings extending results on rational modules for coalgebras to the case of arbitrary ground rings. We apply these results mainly to categories of entwined modules (e.g.…

The trace of the canonical module (the canonical trace) determines the non-Gorenstein locus of a local Cohen--Macaulay ring. We call a local Cohen--Macaulay ring nearly Gorenstein, if its canonical trace contains the maximal ideal. Similar…

The main result of this article is a fantastic generalization of a classical result in graded ring theory. In fact, our result states that if $S$ is a multiplicative set of homogeneous elements of an $M$-graded commutative ring…

Let $A$ be a noetherian ring, $\fa$ an ideal of $A$, and $M$ an $A$--module. Some uniform theorems on the artinianness of certain local cohomology modules are proven in a general situation. They generalize and imply previous results about…

Various classification theorems of thick subcategories of a triangulated category have been obtained in many areas of mathematics. In this paper, as a higher-dimensional version of the classification theorem of thick subcategories of the…