Related papers: Almost Buchsbaumness of some rings arising from co…

We show that, under certain constraints, the Stanley-Reisner ring of an infinite simplicial complex is Cohen-Macaulay in the sense of ideals and weak Bourbaki unmixed. We apply this result to prove the wanted claim -- that initial complexes…

We establish ring isomorphisms between quantum Grothendieck rings of certain remarkable monoidal categories of finite-dimensional representations of quantum affine algebras of types $A_{2n-1}^{(1)}$ and $B_n^{(1)}$. Our proof relies in part…

In an attempt to get some information on the multiplicative structure of the Green ring we study algebraic modules for simple groups, and associated groups such as quasisimple and almost-simple groups. We prove that, for almost all groups…

Projective monomial curves are associated with rings generated by monomials of equal degree in two variables. In this paper, we give an infinite class of non-smooth, non Cohen-Macaulay $k$-Buchsbaum projective monomial curves for any $k\geq…

Holomorphic principal G-bundles over a complex manifold M can be studied using non-abelian cohomology groups H^1(M,G). On the other hand, if M=\Sigma is a closed Riemann surface, there is a correspondence between holomorphic principal…

Recently, I defined a squarefree module over a polynomial ring $S = k[x_1, >..., x_n]$ generalizing the Stanley-Reisner ring $k[\Delta] = S/I_\Delta$ of a simplicial complex $\Delta \subset 2^{1, ..., n}$. In this paper, from a squarefree…

In this work we introduce a new set of invariants associated to the linear strands of a minimal free resolution of a $\mathbb{Z}$-graded ideal $I\subseteq R=\Bbbk[x_1, \ldots, x_n]$. We also prove that these invariants satisfy some…

Let $K$ be a field and $I$ a monomial ideal of the polynomial ring $S=K[x_1,\ldots, x_n]$. We show that if either: 1) $I$ is almost complete intersection, 2) $I$ can be generated by less than four monomials; or 3) $I$ is the Stanley-Reisner…

Levelness and nearly Gorensteinness are well-studied properties of graded rings as a generalized notion of Gorensteinness. In this paper, we compare the strength of these properties. For any Cohen-Macaulay homogeneous affine semigroup ring…

The Golodness of a simplicial complex is defined algebraically in terms of the Stanley-Reisner ring, and it has been a long-standing problem to find its combinatorial characterization. The tightness of a simplicial complex is a…

The coefficient forms \( {}_{a} \ell_{k} \) and the para-Eisenstein series \(\alpha_{k}\) are simplicial Drinfeld modular forms. We study the attached simplicial complexes \(\mathcal{BT}^{r}( {}_{a} \ell_{k})\) and…

For a simplicial complex X on {1,2, ..., n} we define enriched homology and cohomology modules. They are graded modules over k[x_1, ..., x_n] whose ranks are equal to the dimensions of the reduced homology and cohomology groups. We…

Isotopes of C-loops with unique non-identity squares are shown to be both C-loops and A-loops. The relationship between C-loops and Steiner loops is further studied. Central loops with the weak and cross inverse properties are also…

We study the relationship between the reduction number of a primary ideal of a local ring relative to one of its minimal reductions and the multiplicity of the corresponding Sally module. This paper is focused on three goals: (i) To develop…

In a k-linear triangulated category (where k is a field) we show that the existence of Auslander-Reiten triangles implies that objects are determined, up to shift, by knowing dimensions of homomorphisms between them. In most cases the…

We give a bound for the Betti numbers of the Stanley-Reisner ring of a stellar subdivision of a Gorenstein* simplicial complex by applying unprojection theory. From this we derive a bound for the Betti numbers of iterated stellar…

Generalizing the strong Lefschetz property for an $\mathbb{N}$-graded algebra, we introduce the multigraded strong Lefschetz property for an $\mathbb{N}^m$-graded algebra. We show that, for $\mathbf{a} \in \mathbb{N}^m_+$, the generic…

We study modules over stacks of deformation quantization algebroids on complex Poisson manifolds. We prove finiteness and duality theorems in the relative case and construct the Hochschild class of coherent modules. We prove that this class…

We study the general linear groups of infinite fields (or more generally connected semi-local rings with infinite residue fields) from the perspective of $E_\infty$-algebras. We prove that there is a vanishing line of slope 2 for their…

Let $G$ be an adjoint quasi-simple group defined and split over a non-archimedean local field $K$. We prove that the dual of the Steinberg representation of $G$ is isomorphic to a certain space of harmonic cochains on the Bruhat-Tits…