由线性最优化问题产生的环的对偶、a-不变量和典范模

Generalized local duality, canonical modules, and prescribed bound on projective dimension

We present various approaches to J. Herzog's theory of generalized local cohomology and explore its main aspects, e.g., (non-)vanishing results as well as a general local duality theorem which extends, to a much broader class of rings, previous results by Herzog-Zamani and Suzuki. As an application, we establish a prescribed upper bound for the projective dimension of a module satisfying suitable cohomological conditions, and we derive some freeness criteria and questions of Auslander-Reiten type. Along the way, we prove a new characterization of Cohen-Macaulay modules which truly relies on ge…

Cohen-Macaulay局部环的对偶性

Duality for a Cohen-Macaulay local ring

设$(R,\fm)$是Cohen-Macaulay局部环.如果$R$有一个典范模，那么在这种情况下有一些关于对偶的有趣结果。在本文中，我们证明了在$R$没有正则模的情况下确实可以得到类似的结果。同时，我们给出了有限内射维数的完备大Cohen-Macaulay模的一些刻画，并利用它得到了$R$的$\fm$ -adic完备上Gorenstein模的一些刻画.

Some duality and equivalence results

Let $(R,\fm)$ be a relative Cohen-Macaulay local ring with respect to an ideal $\fa$ of $R$ and set $c:=\h\fa$. In this paper, we investigate some properties of the Matlis dual $\H_{\fa}^c(R)^{\vee}$ of the $R$-module $\H_{\fa}^c(R)$ and we show that such modules treat like canonical modules over Cohen-Macaulay local rings. Also, we provide some duality and equivalence results with respect to the module $\H_{\fa}^c(R)^{\vee}$ and so these results lead to achieve generalizations of some known results, such as the Local Duality Theorem, which have been provided over a Cohen-Macaulay local ring w…

Matrix Factorisations Arising From Well-Generated Complex Reflection Groups

We discuss an interesting duality known to occur for certain complex reflection groups, namely the duality groups. Our main construction yields a concrete, representation theoretic realisation of this duality. This allows us to naturally identify invariant vector fields with vector fields on the orbit space, for the action of a duality group. As another application, we construct matrix factorisations of the highest degree basic invariant which give free resolutions of the module of Kähler differentials of the coinvariant algebra $A$ associated to such a reflection group. From this one can expl…

An explicit self-duality

We provide an exposition of the canonical self-duality associated to a presentation of a finite, flat, complete intersection over a Noetherian ring, following work of Scheja and Storch.

On the Triality Theory in Global Optimization

Triality theory is proved for a general unconstrained global optimization problem. The method adopted is simple but mathematically rigorous. Results show that if the primal problem and its canonical dual have the same dimension, the triality theory holds strongly in the tri-duality form as it was originally proposed. Otherwise, both the canonical min-max duality and the double-max duality still hold strongly, but the double-min duality holds weakly in a super-symmetrical form as it was expected. Additionally, a complementary weak saddle min-max duality theorem is discovered. Therefore, an open…