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We introduce the notion of residual intersections of modules and prove their existence. We show that projective dimension one modules have Cohen-Macaulay residual intersections, namely they satisfy the relevant Artin-Nagata property. We then establish a formula for the core of orientable modules satisfying certain homological conditions, extending previous results of Corso, Polini, and Ulrich on the core of projective one modules. Finally, we provide examples of classes of modules that satisfy our assumptions.
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The primary goals of this study are to determine if the datasets of positive COVID-19 test cases and CO2 emissions from Connecticut over the span of March 24th, 2020-October 31, 2021 are in any ways correlated. With climate change a prominent issue facing the entire world today, it is important to explore methods of providing records of past patterns of greenhouse gas emissions in order to inform decision making that could reduce future ones. Autoregressive integrated moving average (ARIMA) modeling is also implemented in this paper to provide forecasting based on CO2 emissions in CT starting from 2019. The most significant results from this paper are as follows: the CO2 emission data of transportation sectors including ground transportation, domestics aviation, and international aviation and weekly COVID-19 positive test cases data has a strong relationship during the first 28 weeks of the pandemic with a correlation of -86.34%. The CO2 emissions experienced on average a -22.96% change of pre-pandemic vs during initial quarantine conditions and at most a - 44.48% change when comparing the pre-pandemic mean to the during initial quarantine minimum value. Lastly, the ARIMA model found to have the lowest Akaike information criterion (AIC) was ARIMA (4,0,4). In conclusion, in the event of a collective global pandemic and lockdown conditions, less traveling resulting in a correlated decrease of CO2 emissions. This means that perhaps concentrated efforts on reducing unnecessary travel could help mitigate the levels of carbon dioxide emissions as a more long-term solution to climate change opposed to the pandemic’s short-term example.
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Given commutative, unital rings $$\mathcal {A}$$and $$\mathcal {B}$$with a ring homomorphism $$\mathcal {A}\rightarrow \mathcal {B}$$making $$\mathcal {B}$$free of finite rank as an $$\mathcal {A}$$-module, we can ask for a “trace” or “norm” homomorphism taking algebraic data over $$\mathcal {B}$$to algebraic data over $$\mathcal {A}$$. In this paper we we construct a norm functor for the data of a quadratic algebra: given a locally-free rank-2 $$\mathcal {B}$$-algebra $$\mathcal {D}$$, we produce a locally-free rank-2 $$\mathcal {A}$$-algebra $$\textrm{Nm}_{\mathcal {B}/\mathcal {A}}(\mathcal {D})$$in a way that is compatible with other norm functors and which extends a known construction for étale quadratic algebras. We also conjecture a relationship between discriminant algebras and this new norm functor.
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We discuss invariants of Cohen-Macaulay local rings that admit a canonical module $$\omega$$. Attached to each such ring R, when $$\omega$$is an ideal, there are integers–the type of R, the reduction number of $$\omega$$–that provide valuable metrics to express the deviation of R from being a Gorenstein ring. In (Ghezzi et al. in JMS 589:506–528, 2017) and (Ghezzi et al. in JMS 571:55–74, 2021) we enlarged this list with the canonical degree and the bi-canonical degree. In this work we extend the bi-canonical degree to rings where $$\omega$$is not necessarily an ideal. We also discuss generalizations to rings without canonical modules but admitting modules sharing some of their properties.
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