Second symmetric powers of chain complexes

A Model for Random Chain Complexes

We introduce a model for random chain complexes over a finite field. The randomness in our complex comes from choosing the entries in the matrices that represent the boundary maps uniformly over $\mathbb{F}_q$, conditioned on ensuring that the composition of consecutive boundary maps is the zero map. We then investigate the combinatorial and homological properties of this random chain complex.

Bass Numbers and Semidualizing Complexes

Let R be a commutative local noetherian ring. We prove that the existence of a chain of semidualizing R-complexes of length (d+1) yields a degree-d polynomial lower bound for the Bass numbers of R. We also show how information about certain Bass numbers of R provide restrictions on the lengths of chains of semidualizing R-complexes. To make this article somewhat self-contained, we also include a survey of some of the basic properties of semidualizing modules, semidualizing complexes and derived categories.

实向量空间的随机链复形

Random chain complexes of real vector spaces

研究实向量空间随机链复形的同调性质，发现贝蒂数之和几乎必然达到最小值，表明非平凡同调罕见。

商多项式环上的组合自由链复形

Combinatorial free chain complexes over quotient polynomial rings

提出一种在商多项式环上组合构造自由模链复形的程序，证明其在特定环中为正合自由分解并导出内射维数无限，最后猜想该程序普适。

Contracting endomorphisms and dualizing complexes

We investigate how one can detect the dualizing property for a chain complex over a commutative local noetherian ring R. Our focus is on homological properties of contracting endomorphisms of R, e.g., the Frobenius endomorphism when R contains a field of positive characteristic.

Symmetric Auslander and Bass categories

We define the symmetric Auslander category A^s(R) to consist of complexes of projective modules whose left- and right-tails are equal to the left- and right tails of totally acyclic complexes of projective modules. The symmetric Auslander category contains A(R), the ordinary Auslander category. It is well known that A(R) is intimately related to Gorenstein projective modules, and our main result is that A^s(R) is similarly related to what can reasonably be called Gorenstein projective homomorphisms. Namely, there is an equivalence of triangulated categories: \underline{GMor}(R) --> A^s(R) / K^…

Jump loci for the rank of matrices and Betti numbers of chain complexes over Laurent polynomial rings

Let $K$ be a non-empty set of ideals of the commutative ring $R$, closed under taking smaller ideals. A subset $X$ of the group ring $R[\mathbb{Z}^s]$ is called a $K$-set if the ideal generated by the coefficients of the elements of $X$ is in $K$. For $X$ not a $K$-set we investigate the set of those homomorphisms $p \colon \mathbb{Z}^s \to \mathbb{Z}^t$ such that $p_*(X)$ is a $K$-set. We also consider corresponding notions of rank of matrices and Betti numbers of chain complexes; this includes an analysis of the case of McCoy rank. Our setup also recovers results on jump loci obtained by Koh…

具有支撑的链复形的导出等价

Derived equivalences for chain complexes with support

本文研究了具有有限A-维度支撑的链复形的导出等价，并探讨了其在K理论中的同伦纤维描述。

Multiplicativity and nonrealizable equivariant chain complexes

Let $G$ be a finite $p$-group and $\mathbb{F}$ a field of characteristic $p$. We filter the cochain complex of a free $G$-space with coefficients in $\mathbb{F}$ by powers of the augmentation ideal of $\mathbb{F} G$. We show that the cup product induces a multiplicative structure on the arising spectral sequence and compute the $E_1$-page as a bigraded algebra. As an application, we prove that recent counterexamples of Iyengar and Walker to an algebraic version of Carlsson's conjecture can not be realized topologically.

On growth in totally acyclic minimal complexes

Given a commutative Noetherian local ring, we provide a criterion under which a totally acyclic minimal complex of free modules has symmetric growth.

Pure resolutions of unbounded complexes of modules

Let $R$ be a commutative ring. We show that pure injective resolutions and pure projective resolutions can be constructed for unbounded complexes of $R$-modules. We use these to obtain a closed symmetric monoidal structure on the unbounded pure derived category.

Dynamical systems on chain complexes and canonical minimal resolutions

We introduce notions of vector field and its (discrete time) flow on a chain complex. The resulting dynamical systems theory provides a set of tools with a broad range of applicability that allow, among others, to replace in a canonical way a chain complex with a "smaller" one of the same homotopy type. As applications we construct in an explicit, canonical, and symmetry-preserving fashion a minimal free resolution for every toric ring and every monomial ideal. Our constructions work in all characteristics and over any base field. A key subtle new point is that in certain finitely many positiv…

Free $(\mathbb{Z}/p)^n$-complexes and $p$-DG modules

We reformulate the problem of bounding the total rank of the homology of perfect chain complexes over the group ring $\mathbb{F}_p[G]$ of an elementary abelian $p$-group $G$ in terms of commutative algebra. This extends results of Carlsson for $p=2$ to all primes. As an intermediate step, we construct an embedding of the derived category of perfect chain complexes over $\mathbb{F}_p[G]$ into the derived category of $p$-DG modules over a polynomial ring.

Alexander duals of symmetric simplicial complexes and Stanley-Reisner Ideals

Given an ascending chain $(I_n)_{n\in\mathbb{N}}$ of $\Sym$-invariant squarefree monomial ideals, we study the corresponding chain of Alexander duals $(I_n^\vee)_{n\in\mathbb{N}}$. Using a novel combinatorial tool, which we call \emph{avoidance up to symmetry}, we provide an explicit description of the minimal generating set up to symmetry in terms of the original generators. Combining this result with methods from discrete geometry, this enables us to show that the number of orbit generators of $I_n^\vee$ is given by a polynomial in $n$ for sufficiently large $n$. The same is true for the num…

Stanley-Reisner rings for symmetric simplicial complexes, G-semimatroids and Abelian arrangements

We extend the notion of face rings of simplicial complexes and simplicial posets to the case of finite-length (possibly infinite) simplicial posets with a group action. The action on the complex induces an action on the face ring, and we prove that the ring of invariants is isomorphic to the face ring of the quotient simplicial poset under a mild condition on the group action. We also identify a class of actions on simplicial complexes that preserve the homotopical Cohen-Macaulay property under quotients. When the acted-upon poset is the independence complex of a semimatroid, the $h$-polynomia…