Mixed characteristic homological theorems in low degrees

群扩张的Gorenstein同调维数

Gorenstein homological dimension of group extensions

该研究针对交换环k上FP_∞型群，证明其三类自然维数一致，建立Fel'dman定理的Gorenstein同调类似结果等，拓展了群同调理论相关结论。

A note on homological dimension of a family of coherent sheaves

We prove a theorem on how a conclusion on homological dimension of a family of coherent sheaves over a scheme can be done from homological dimension of the restriction of this family to the reduction of the base.

Homological Algebra for Persistence Modules

We develop some aspects of the homological algebra of persistence modules, in both the one-parameter and multi-parameter settings, considered as either sheaves or graded modules. The two theories are different. We consider the graded module and sheaf tensor product and Hom bifunctors as well as their derived functors, Tor and Ext, and give explicit computations for interval modules. We give a classification of injective, projective, and flat interval modules. We state Kunneth theorems and universal coefficient theorems for the homology and cohomology of chain complexes of persistence modules i…

Singularities in mixed characteristic. The perfectoid approach

The homological conjectures, which date back to Peskine, Szpiro and Hochster in the late sixties, make fundamental predictions about syzygies and intersection problems in commutative algebra. They were settled long ago in the presence of a base field and led to tight closure theory, a powerful tool to investigate singularities in characteristic p. Recently, perfectoid techniques coming from p-adic Hodge theory have allowed us to get rid of any base field; this solves the direct summand conjecture and establishes the existence and weak functoriality of big Cohen-Macaulay algebras, which solve i…

Dedekind 整环上一般线性群的同调稳定性

Homological stability for general linear groups over Dedekind domains

该研究针对 Dedekind 整环上有限生成投射模的自同构群，提出了考虑所有稳定化映射的新型同调稳定性定理，并验证了约化 Borel--Serre 空间也具备该稳定性。

A Homological Approach to Factorization

Mott noted a one-to-one correspondence between saturated multiplicatively closed subsets of a domain D and directed convex subgroups of the group of divisibility D. With this, we construct a functor between inclusions into saturated localizations of D and projections onto partially ordered quotient groups of G(D). We use this functor to construct many cochain complexes of o-homomorphisms of po-groups. These complexes naturally lead to some fundamental structure theorems and some natural homology theory that provide insight into the factorization behavior of D.

A short proof of Reisner's Theorem on Cohen-Macaulay simplicial complexes

We present a short proof of Reisner's Theorem, characterizing which simplicial complexes have a Cohen-Macaulay face ring. In some cases, we can also express some homological invariants of the face ring in terms of the reduced homology of the complex.

Generalized local cohomology modules and homological Gorenstein dimensions

Let \fa be an ideal of a commutative Noetherian ring R and M and N two finitely generated R-modules. Let \cd_{\fa}(M,N) denote the supremum of the i's such that H^i_{\fa}(M,N)\neq 0. First, by using the theory of Gorenstein homological dimensions, we obtain several upper bounds for \cd_{\fa}(M,N). Next, over a Cohen-Macaulay local ring (R,\fm), we show that \cd_{\fm}(M,N)=\dim R-\grade(\Ann_RN,M), provided that either projective dimension of M or injective dimension of N is finite. Finally, over such rings, we establish an analogue of the Hartshorne-Lichtenbaum Vanishing Theorem in the context…

Homological invariants of modules over contracting endomorphisms

It is proved that when R is a local ring of positive characteristic, $ϕ$ is its Frobenius endomorphism, and some non-zero finite R-module has finite flat dimension or finite injective dimension for the R-module structure induced through $ϕ$, then R is regular. This broad generalization of Kunz's characterization of regularity in positive characteristic is deduced from a theorem concerning a local ring R with residue field of k of arbitrary characteristic: If $ϕ$ is a contracting endomorphism of R, then the Betti numbers and the Bass numbers over $ϕ$ of any non-zero finitely generated R-module …

混合特征局部余幂级数模块上的Frobenius作用

The Frobenius action on local cohomology modules in mixed characteristic

本文在混合特征局部环中利用标准化长度和Frobenius映射，证明了局部余幂级数的相关结果，并对splinters研究产生影响。

Homological algebra of modules over posets

Homological algebra of modules over posets is developed, as closely parallel as possible to that of finitely generated modules over noetherian commutative rings, in the direction of finite presentations and resolutions. Centrally at issue is how to define finiteness to replace the noetherian hypothesis which fails. The tameness condition introduced for this purpose captures finiteness for variation in families of vector spaces indexed by posets in a way that is characterized equivalently by distinct topological, algebraic, combinatorial, and homological manifestations. Tameness serves both the…