The stable category of Gorenstein flat sheaves on a noetherian scheme

For a semi-separated noetherian scheme, we show that the category of cotorsion Gorenstein flat quasi-coherent sheaves is Frobenius and a natural non-affine analogue of the category of Gorenstein projective modules over a noetherian ring. We show that this coheres perfectly with the work of Murfet and Salarian that identifies the pure derived category of F-totally acyclic complexes of flat quasi-coherent sheaves as the natural non-affine analogue of the homotopy category of totally acyclic complexes of projective modules.Comment: Final version, to appear in Proc. Amer. Math. Soc.; 14 p

RIO Country Report 2017: Denmark

The R&I Observatory country report 2017 provides a brief analysis of the R&I system covering the economic context, main actors, funding trends & human resources, policies to address R&I challenges, and R&I in national and regional smart specialisation strategies. Data is from Eurostat, unless otherwise referenced and is correct as at January 2018. Data used from other international sources is also correct to that date. The report provides a state-of-play and analysis of the national level R&I system and its challenges, to support the European Semester.JRC.B.7-Knowledge for Finance, Innovation and Growt

Pure-minimal chain complexes

We introduce a notion of pure-minimality for chain complexes of modules and show that it coincides with (homotopic) minimality in standard settings, while being a more useful notion for complexes of flat modules. As applications, we characterize von Neumann regular rings and left perfect rings.Comment: Old Section 6 removed and minor edits. Final version, to appear in Rend. Semin. Mat. Univ. Padova; 18 p

RIO Country Report 2016: Denmark

The 2016 series of the RIO Country Report analyses and assesses the development and performance of the national research and innovation system of the EU-28 Member States and related policies with the aim of monitoring and evaluating the EU policy implementation as well as facilitating policy learning in the Member States.JRC.B.7-Knowledge for Finance, Innovation and Growt

Rigidity of Ext and Tor via flat-cotorsion theory

Let p be a prime ideal in a commutative noetherian ring R and denote by k(p) the residue field of the local ring R_p. We prove that if an R-module M satisfies Ext_R^n(k(p),M) = 0 for some n >= dim R, then Ext_R^i(k(p),M) = 0 holds for all i >= n. This improves a result of Christensen, Iyengar, and Marley by lowering the bound on n. We also improve existing results on Tor-rigidity. This progress is driven by the existence of minimal semi-flat-cotorsion replacements in the derived category as recently proved by Nakamura and Thompson.Comment: 10 p

Homotopy categories of totally acyclic complexes with applications to the flat-cotorsion theory

We introduce a notion of total acyclicity associated to a subcategory of an abelian category and consider the Gorenstein objects they define. These Gorenstein objects form a Frobenius category, whose induced stable category is equivalent to the homotopy category of totally acyclic complexes. Applied to the flat-cotorsion theory over a coherent ring, this provides a new description of the category of cotorsion Gorenstein flat modules; one that puts it on equal footing with the category of Gorenstein projective modules.Comment: Added Proposition 4.2, updated after review. Final version, to appear in Contemp. Math.; 20 p

Gorenstein weak global dimension is symmetric

We study the Gorenstein weak global dimension of associative rings and its relation to the Gorenstein global dimension. In particular, we prove the conjecture that the Gorenstein weak global dimension is a left-right symmetric invariant -- just like the (absolute) weak global dimension.Comment: Minor revisions. Final version, to appear in Math. Nachr.; 9 p