Abstract A local ring R is regular if and only every finitely generated -module has finite projective dimension. Moreover, the residue field k a test module: This characterization can be extended to bounded derived category $\mathsf {D}^{\mathsf f}(R)$ , which contains small objects regular. Recent results of Pollitz, completing work initiated by Dwyer–Greenlees–Iyengar, yield an analogous for ...

We define an action of the 0-Hecke algebra of type A on the Stanley-Reisner ring of the Boolean algebra. By studying this action we obtain a family of multivariate noncommutative symmetric functions, which specialize to the noncommutative Hall-Littlewood symmetric functions and their (q, t)-analogues introduced by Bergeron and Zabrocki. We also obtain multivariate quasisymmetric function identi...

Several representations of the symmetric group, arising from different combinatorial, algebraic and geometric constructions, have lead to the same character, up to multiplication by the sign character: the homology of partition lattice (cf. [5, 7, 13]), the top component of a special quotient of the Stanley-Reisner ring of this same lattice [4], the top component of the cohomology algebra of th...

This paper addresses the problem of characterizing the possible h-vectors of a matroid complex ∆. We define the h-vector in Section 1, both in terms of the f -vector, and as the dimension vector of a graded ring S(∆) which is given as the Stanley-Reisner ring of ∆ modulo a linear system of parameters. Section 1 ends with a brief survey of the literature on h-vectors of matroid complexes. In Sec...

Adviser: Carina Curto Neurons in the brain represent external stimuli via neural codes. These codes often arise from stimulus-response maps, associating to each neuron a convex receptive field. An important problem confronted by the brain is to infer properties of a represented stimulus space without knowledge of the receptive fields, using only the intrinsic structure of the neural code. How d...

Unprojection theory analyzes and constructs complicated commutative rings in terms of simpler ones. Our main result is that, on the algebraic level of Stanley-Reisner rings, stellar subdivisions of Gorenstein* simplicial complexes correspond to unprojections of type Kustin-Miller. As an application of our methods we study the minimal resolution of Stanley– Reisner rings associated to stacked po...

for an $n$-gon with vertices at points $1,2,cdots,n$, the betti numbers of its suspension, the simplicial complex that involves two more vertices $n+1$ and $n+2$, is known. in this paper, with a constructive and simple proof, wegeneralize this result to find the minimal free resolution and betti numbers of the $s$-module $s/i$ where  $s=k[x_{1},cdots, x_{n}]$ and $i$ is the associated ideal to ...

We develop an algebraic theory of supports for \(R\)-linear codes fixed length, where \(R\) is a finite commutative unitary ring. A support naturally induces notion generalized weights and allows one to associate monomial ideal code. Our main result states that, under suitable assumptions, the code can be obtained from graded Betti numbers its associated ideal. In case \(\mathbb{F}_q\)-linear e...