A surprising fact about D-modules in characteristic p>0

The vanishing conjecture for maps of Tor and derived splinters

We say an excellent local domain $(S,n)$ satisfies the vanishing conditions for maps of Tor, if for every $A\to R\to S$ with $A$ regular and $A\to R$ module-finite torsion-free extension, and every $A$-module $M$, the map $Tor^A_i(M, R)\to Tor_i^A(M, S)$ vanishes for every $i\geq 1$. Hochster-Huneke's conjecture (theorem in equal characteristic) thus states that regular rings satisfy such vanishing conditions. The main theorem of this paper shows that, in equal characteristic, rings that satisfy the vanishing conditions for maps of Tor are exactly derived splinters in the sense of Bhatt. In pa…

某些Cohen-Macaulay模的性质测试及局部环的判别准则 via Ext或Tor有限消失

Test properties of some Cohen-Macaulay modules and criteria for local rings via finite vanishing of Ext or Tor

本文通过Ext或Tor的有限消失性质，测试Cohen-Macaulay模的特性，并用于刻画局部环，验证了Auslander-Reiten猜想的扩展结果。

Canonical Filtrations of Gorenstein Injective Modules

The principle "Every result in classical homological algebra should have a counterpart in Gorenstein homological algebra" is given in [3]. There is a remarkable body of evidence supporting this claim (cf. [2] and [3]). Perhaps one of the most glaring exceptions is provided by the fact that tensor products of Gorenstein projective modules need not be Gorenstein projective, even over Gorenstein rings. So perhaps it is surprising that tensor products of Gorenstein injective modules over Gorenstein rings of finite Krull dimension are Gorenstein injective. Our main result is in support of the princ…

交换环上模的广义平方差因子吸收子模

Generalized square-difference factor absorbing submodules of modules over commutative rings

本文研究了交换环上模的广义平方差因子吸收子模的性质与行为，探讨了其在局部化、同态像、直积、理想化和融合等模扩展中的表现，并通过实例区分了该类子模与其他概念。

On the Factorization of Nonlinear Recurrences in Modules

For rings R with identity, we define a class of nonlinear higher order recurrences on unitary left R-modules that include linear recurrences as special cases. We obtain conditions under which a recurrence of order k+1 in this class is equivalent to a pair, known as a semiconjugate factorization, that consists of a recurrence of order k and a recurrence of order 1. We show that such a factorization is possible whenever R contains certain sequences of units. Further, if the coefficients of the original recurrence in R are independent of the index then we show that the semiconjugate factorization…

Cohomological support and the geometric join

Let $M,N$ be finitely generated modules over a local complete intersection $R$. Assume that for each $i>0$, $\mathrm{Tor}^R_i(M,N)=0$. We prove that the cohomological support of $M\otimes_R N$ (in the sense of Avramov-Buchweitz) is equal to the geometric join of the cohomological supports of $M,N$. Such result gives a new connection between two active areas or research, and immediately produces several surprising corollaries. Naturally, it also raises many intriguing new questions about the homological properties of modules over a complete intersection, some of those are investigated in the se…