Related papers: Dualizing Complex of a Toric Face Ring II: Non-nor…
A "toric face ring", which generalizes both Stanley-Reisner rings and affine semigroup rings, is studied by Bruns, Roemer and their coauthors recently. In this paper, under the "normality" assumption, we describe a dualizing complex of a…
We characterize the seminormality of an affine semigroup ring in terms of the dualizing complex, and the normality of a Cohen-Macaulay semigroup ring by the "shape" of the canonical module. We also characterize the seminormality of a toric…
We characterize the toric face rings that are normal (respectively seminormal). Extending results about local cohomology of Brun, Bruns, Ichim, Li and R\"omer of seminormal monoid rings and Stanley toric face rings, we prove the vanishing…
We introduce the class of weakly log canonical singularities, a natural generalization of semi-log canonical singularities. Toric varieties (associated to toric face rings, possibly non-normal or reducible) which have weakly (semi-) log…
In this note we consider monoidal complexes and their associated algebras, called toric face rings. These rings generalize Stanley-Reisner rings and affine monoid algebras. We compute initial ideals of the presentation ideal of a toric face…
Following a construction of Stanley we consider toric face rings associated to rational pointed fans. This class of rings is a common generalization of the concepts of Stanley--Reisner and affine monoid algebras. The main goal of this…
A finite poset $P$ is called "simplicial", if it has the smallest element $\hat{0}$, and every interval $[\hat{0}, x]$ is a boolean algebra. The face poset of a simplicial complex is a typical example. Generalizing the Stanley-Reisner ring…
Toric face rings is a generalization of the concepts of affine monoid rings and Stanley-Reisner rings. We consider several properties which imply Koszulness for toric face rings over a field $k$. Generalizing works of Laudal, Sletsj\o{}e…
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,…
We introduce toric complexes as polyhedral complexes consisting of rational cones together with a set of integral generators for each cone, and we define their associated face rings. Abstract simplicial complexes and rational fans can be…
A ring with an Auslander dualizing complex is a generalization of an Auslander-Gorenstein ring. We show that many results which hold for Auslander-Gorenstein rings also hold in the more general setting. On the other hand we give criteria…
We construct an explicit Deligne - Du Bois complex for algebraic varieties which are locally analytically isomorphic to the spectrum of a toric face ring.
We study dualizing complexes on algebraic stacks. In particular, we show their existence for (tame) Deligne--Mumford stacks of equicharacteristic in great generality.
Crystalline graded rings are generalizations of certain classes of rings like generalized twisted group rings, generalized Weyl algebras, and generalized skew crossed products. When the base ring is a commutative Dedekind domain, two…
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…
In this survey article we present several new developments of `toric topology' concerning the cohomology of face rings (also known as Stanley-Reisner algebras). We prove that the integral cohomology algebra of the moment-angle complex Z_K…
Let A be a noetherian local commutative ring and let M be a suitable complex of A-modules. This paper proves that M is a dualizing complex for A if and only if the trivial extension A \ltimes M is a Gorenstein Differential Graded Algebra.…
Generalizing the concepts of Stanley-Reisner and affine monoid algebras, one can associate to a rational pointed fan the toric face ring. Assuming that this ring is Cohen-Macaulay, the main result of this paper is to characterize the…
We give explicit descriptions of rings of differential operators of toric face rings in characteristic $0$. For quotients of normal affine semigroup rings by radical monomial ideals, we also identify which of their differential operators…
A semidualizing module is a generalization of Grothendieck's dualizing module. For a local Cohen-Macaulay ring $R$, the ring itself and its canonical module are always realized as (trivial) semidualizing modules. Reasonably, one might…