Good tilting modules and recollements of derived module categories

Let $T$ be an infinitely generated tilting module of projective dimension at most one over an arbitrary associative ring $A$, and let $B$ be the endomorphism ring of $T$. In this paper, we prove that if $T$ is good then there exists a ring $C$, a homological ring epimorphism B\ra C and a recollement among the (unbounded) derived module categories \D{C} of $C$, \D{B} of $B$, and \D{A} of $A$. In particular, the kernel of the total left derived functor $T\otimes_B^{\mathbb L}-$ is triangle equivalent to the derived module category \D{C}. Conversely, if the functor $T\otimes_B^{\mathbb L}-$ admits a fully faithful left adjoint functor, then $T$ is a good tilting module. We apply our result to tilting modules arising from ring epimorphisms, and can then describe the rings $C$ as coproducts of two relevant rings. Further, in case of commutative rings, we can weaken the condition of being tilting modules, strengthen the rings $C$ as tensor products of two commutative rings, and get similar recollements. Consequently, we can produce examples (from commutative algebra and $p$-adic number theory, or Kronecker algebra) to show that two different stratifications of the derived module category of a ring by derived module categories of rings may have completely different derived composition factors (even up to ordering and up to derived equivalence),or different lengths. This shows that the Jordan-H\"older theorem fails even for stratifications by derived module categories, and also answers negatively an open problem by Angeleri-H\"ugel, K\"onig and Liu

Optimal CSMA-based Wireless Communication with Worst-case Delay and Non-uniform Sizes

Carrier Sense Multiple Access (CSMA) protocols have been shown to reach the full capacity region for data communication in wireless networks, with polynomial complexity. However, current literature achieves the throughput optimality with an exponential delay scaling with the network size, even in a simplified scenario for transmission jobs with uniform sizes. Although CSMA protocols with order-optimal average delay have been proposed for specific topologies, no existing work can provide worst-case delay guarantee for each job in general network settings, not to mention the case when the jobs have non-uniform lengths while the throughput optimality is still targeted. In this paper, we tackle on this issue by proposing a two-timescale CSMA-based data communication protocol with dynamic decisions on rate control, link scheduling, job transmission and dropping in polynomial complexity. Through rigorous analysis, we demonstrate that the proposed protocol can achieve a throughput utility arbitrarily close to its offline optima for jobs with non-uniform sizes and worst-case delay guarantees, with a tradeoff of longer maximum allowable delay