Abstract:
Inspired by recent works on rings satisfying Auslander's conjecture, we study invariants, which we call Auslander bounds, and prove that they have strong relations to some homological conjectures.

Abstract:
In this paper, various Homological Conjectures are studied for local rings which are locally finitely generated over a discrete valuation ring $V$ of mixed characteristic. Typically, we can only conclude that a particular Conjecture holds for such a ring provided the residual characteristic of $V$ is sufficiently large in terms of the complexity of the data, where the complexity is primarily given in terms of the degrees of the polynomials over $V$ that define the data, but possibly also by some additional invariants such as (homological) multiplicity. Thus asymptotic versions of the Improved New Intersection Theorem, the Monomial Conjecture, the Direct Summand Conjecture, the Hochster-Roberts Theorem and the Vanishing of Maps of Tors Conjecture are given. That the results only hold asymptotically, is due to the fact that non-standard arguments are used, relying on the Ax-Kochen-Ershov Principle, to infer their validity from their positive characteristic counterparts. A key role in this transfer is played by the Hochster-Huneke canonical construction of big Cohen-Macaulay algebras in positive characteristic via absolute integral closures.

Abstract:
We obtain various characterizations of commutative Noetherian local rings $(R, \fm)$ in terms of homological dimensions of certain finitely generated modules. For example, we establish that $R$ is Gorenstein if the Gorenstein injective dimension of the maximal ideal $\fm$ of $R$ is finite. Furthermore we prove that $R$ must be regular if a single $\Ext_{R}^{n}(I,J)$ vanishes for some integrally closed $\fm$-primary ideals $I$ and $J$ of $R$ and for some integer $n\geq \dim(R)$. Along the way we observe that local rings that admit maximal Cohen-Macaulay Tor-rigid modules are Cohen-Macaulay.

Abstract:
Homological mirror symmetry for crepant resolutions of Gorenstein toric singularities leads to a pair of conjectures on certain hypergeometric systems of PDEs. We explain these conjectures and verify them in some cases.

Abstract:
This paper builds on work of Hochster and Yao that provides nice embeddings for finitely generated modules of finite G-dimension, finite projective dimension, or locally finite injective dimension. We extend these results by providing similar embeddings in the relative setting, that is, for certain modules of finite G_C-dimension, finite P_C-projective dimension, locally finite GI_C-injective dimension, or locally finite I_C-injective dimension where C is a semidualizing module. Along the way, we extend some results for modules of finite homological dimension to modules of locally finite homological dimension in the relative setting.

Abstract:
We study homological properties of test modules that are, in principle, modules that detect finite homological dimensions. The main outcome of our results is a generalization of a classical theorem of Auslander and Bridger: we prove that, if a commutative Noetherian complete local ring R admits a test module of finite Gorenstein dimension, then R is Gorenstein.

Abstract:
Given a good $n$-tilting module $T$ over a ring $A$, let $B$ be the endomorphism ring of $T$, it is an open question whether the kernel of the left-derived functor $T\otimes^L_B-$ between the derived module categories of $B$ and $A$ could be realized as the derived module category of a ring $C$ via a ring epimorphism $B\rightarrow C$ for $n\ge 2$. In this paper, we first provide a uniform way to deal with the above question both for tilting and cotilting modules by considering a new class of modules called Ringel modules, and then give criterions for the kernel of $T\otimes^L_B-$ to be equivalent to the derived module category of a ring $C$ with a ring epimorphism $B\rightarrow C$. Using these characterizations, we display both a positive example of $n$-tilting modules from noncommutative algebra, and a counterexample of $n$-tilting modules from commutative algebra to show that, in general, the open question may have a negative answer. As another application of our methods, we consider the dual question for cotilting modules, and get corresponding criterions and counterexamples. The case of cotilting modules, however, is much more complicated than the case of tilting modules.

Abstract:
We investigate the notion of the C-projective dimension of a module, where C is a semidualizing module. When C=R, this recovers the standard projective dimension. We show that three natural definitions of finite C-projective dimension agree, and investigate the relationship between relative cohomology modules and absolute cohomology modules in this setting. Finally, we prove several results that demonstrate the deep connections between modules of finite projective dimension and modules of finite C-projective dimension. In parallel, we develop the dual theory for injective dimension and C-injective dimension.

Abstract:
Let $k$ be a commutative Noetherian ring. In this paper we consider filtered modules of the category FI firstly introduced by Nagpal. We show that a finitely generated FI-module $V$ is filtered if and only if its higher homologies all vanish, and if and only if a certain homology vanishes. Using this homological characterization, we characterize finitely generated FI-modules $V$ whose projective dimension is finite, and describe an upper bound for it. Furthermore, we give a new proof for the fact that $V$ induces a finite complex of filtered modules, and use it as well as a result of Church and Ellenberg to obtain another upper bound for homological degrees of $V$.

Abstract:
We introduce the concept of quasirational relation modules for discrete (pro-$p$) presentations of discrete (pro-$p$)groups. We have proved that aspherical presentations and their subpresentations are quasirational. In pro-$p$-case we have quasirationality of pro-$p$-presentations of pro-$p$-groups with a single defining relation. For every quasirational (pro-$p$-)relation module we construct a so called $p$-adic rationalization, which is the pro-fd-module $\overline{R}\widehat{\otimes}_{\mathbb{Z}_p}\mathbb{Q}_p= \varprojlim R/[R,R\mathcal{M}_n]\otimes_{\mathbb{Z}_p}\mathbb{Q}_p$. We have proved the isomorphism $\overline{R}\widehat{\otimes}_{\mathbb{Z}_p}\mathbb{Q}_p=\overline{R^{\wedge}_w}(\mathbb{Q}_p)$, where $\overline{R^{\wedge}_w}(\mathbb{Q}_p)$ stands for rational points of the abelianization of $p$-adic prounipotent completion of $R$. We show how $\overline{R^{\wedge}_{w}}$ embeds into a sequence of prounipotent groups. This sequence arises naturally (from certain prounipotent crossed module, the latter considered as concrete examples of proalgebraic homotopy types. The old-standing open problem of Serre, slightly corrected by Gildenhuys, in a modern form states that pro-$p$-groups with a single defining relation are aspherical. We give motivation behind a rationalized version of the conjectured Identity Theorem