Transferring algebra structures on complexes

Big Cohen-Macaulay modules, morphisms of perfect complexes, and intersection theorems in local algebra

There is a well known link from the first topic in the title to the third one. In this paper we thread that link through the second topic. The central result is a criterion for the tensor nilpotence of morphisms of perfect complexes over commutative noetherian rings, in terms of a numerical invariant of the complexes known as their level. Applications to local rings include a strengthening of the Improved New Intersection Theorem, short direct proofs of several results equivalent to it, and lower bounds on the ranks of the modules in every finite free complex that admits a structure of differe…

On the cohomology of highly connected covers of finite complexes

Relying on the computation of the Andre-Quillen homology groups for unstable Hopf algebras, we prove that the mod p cohomology of the n-connected cover of a finite H-space is always finitely generated as algebra over the Steenrod algebra.

Gröbner bases and Betti numbers of monoidal complexes

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 ring, and determine its graded Betti numbers. Our results generalize celebrated theorems of Hochster in combinatorial commutative algebra.

DG Structure on Length 3 Trimming Complexes and Applications to Tor Algebras

In this paper, we consider the iterated trimming complex associated to data yielding a complex of length $3$. We compute an explicit algebra structure in this complex in terms of the algebra structures of the associated input data. Moreover, it is shown that many of these products become trivial after descending to homology. We apply these results to the problem of realizability for Tor-algebras of grade $3$ perfect ideals, and show that under mild hypotheses, the process of "trimming" an ideal preserves Tor-algebra class. In particular, we construct new classes of ideals in arbitrary regular …

Rigid Complexes via DG Algebras

Let A be a commutative ring, B a commutative A-algebra and M a complex of B-modules. We begin by constructing the square Sq_{B/A} M, which is also a complex of B-modules. The squaring operation is a quadratic functor, and its construction requires differential graded (DG) algebras. If there exists an isomorphism ρ: M \iso Sq_{B/A} M then the pair (M,ρ) is called a rigid complex over B relative to A (there are some finiteness conditions). There is an obvious notion of rigid morphism between rigid complexes. We establish several properties of rigid complexes, including their uniqueness, existenc…

Linear resolutions over Koszul complexes and Koszul homology algebras

Let $R$ be a standard graded commutative algebra over a field $k$, let $K$ be its Koszul complex viewed as a differential graded $k$-algebra, and let $H$ be the homology algebra of $K$. This paper studies the interplay between homological properties of the three algebras $R$, $K$, and $H$. In particular, we introduce two definitions of Koszulness that extend the familiar property originally introduced by Priddy: one which applies to $K$ (and, more generally, to any connected differential graded $k$-algebra) and the other, called strand-Koszulness, which applies to $H$. The main theoretical res…

交换代数的半纤维积与复形的提升

Semi-fiber products of algebras and lifting of complexes

本文研究了交换代数的半纤维积结构，并探讨了复形提升的条件与方法。

Pontryagin algebras and the LS-category of moment-angle complexes in the flag case

For any flag simplicial complex $K$, we describe the multigraded Poincare series, the minimal number of relations and the degrees of these relations in the Pontryagin algebra of the corresponding moment-angle complex $Z_K$. We compute the LS-category of $Z_K$ for flag complexes and give a lower bound in the general case. The key observation is that the Milnor-Moore spectral sequence collapses at the second sheet for flag $K$. We also show that the results of Panov and Ray about the Pontryagin algebras of Davis-Januszkiewicz spaces are valid for arbitrary coefficient rings, and introduce the $(…

从点到复形：单纯复形的一种意外性概念

From points to complexes: a concept of unexpectedness for simplicial complexes

本文受意外超曲面理论启发，为无平方单项式理想提出了意外参数系概念，证明其存在会导致特定代数不满足弱Lefschetz性质，并揭示其与平衡复形理论的内在联系及Rees代数的应用价值。

Examples of finite free complexes of small rank and small homology

This work concerns finite free complexes over commutative noetherian rings, in particular over group algebras of elementary abelian groups. The main contribution is the construction of complexes such that the total rank of their underlying free modules, or the total length of their homology, is less than predicted by various conjectures in the theory of transformation groups and in local algebra.

拟阵复形的探索

Explorations of Matroid Complexes

受Kontsevich图复形启发，本文系统研究拟阵复形，通过构造删除-收缩双复形并建立Hopf代数结构，证明广泛acyclicity结果，并通过计算小规模复形检测非平凡同调，提出与奇轮拟阵相关的猜想。

Torus actions, combinatorial topology and homological algebra

The paper surveys some new results and open problems connected with such fundamental combinatorial concepts as polytopes, simplicial complexes, cubical complexes, and subspace arrangements. Particular attention is paid to the case of simplicial and cubical subdivisions of manifolds and, especially, spheres. We describe important constructions which allow to study all these combinatorial objects by means of methods of commutative and homological algebra. The proposed approach to combinatorial problems relies on the theory of moment-angle complexes, currently being developed by the authors. The …