arXiv subjects
Publications and source records attributed to Lars Winther Christensen.
Let R be a commutative noetherian local ring and M and N be R-complexes with finitely generated homology. We prove that formulas for the homological dimensions of the derived Hom complex RHom(M,N), in terms of the homological dimensions of M and N, hold without the a priori assumptions on the dimensions of M or N present in classic results.
We discuss an example of a rigid non-principal ideal in a one dimensional (commutative) Gorenstein domain, which contradicts a conjecture of C. Huneke and R. Wiegand. The construction and its analysis were discovered by Codex, when prompted by one of the authors to verify the conjecture or find a counterexample. A proof of the conjecture, also discovered by Codex, is presented when the ring is equicharacteristic and its embedding dimension is at most three.
Peskine and Szpiro's Intersection Theorem for finitely generated modules was generalized by Foxby to all modules and bounded complexes of such. We strengthen Foxby's result and prove a dual result, which for complexes with finitely generated homology recovers Iversen's Amplitude Inequality but also applies to derived complete complexes.
We prove that a commutative noetherian ring $R$ is Gorenstein of dimension at most $d$ if $d+1$ is an upper bound on the G-levels of perfect $R$-complexes. For $R$ local, we prove a formula for levels, with respect to injective or Gorenstein injective $R$-modules, of $R$-complexes with finitely generated homology; it mimics Bass' classic formula for injective dimension of finitely generated $R$-modules.
A 2009 paper by Iacob and Iyengar characterizes noetherian regular rings in terms of properties of complexes of projective modules, flat modules, and injective modules. We show that the relevant properties of such complexes are equivalent without reference to regularity of the ring and that they characterize coherent regular rings and von Neumann regular rings.
Ideals in the ring of power series in three variables can be classified based on algebra structures on their minimal free resolutions. The classification is incomplete in the sense that it remains open which algebra structures actually occur; this realizability question was formally raised by Avramov in 2012. We discuss the outcomes of an experiment performed to shed light on Avramov's question: Using the computer algebra system Macaulay2, we classify a billion randomly generated ideals and build a database with examples of ideals of all classes realized in the experiment. Based on the outcomes, we discuss the status of recent conjectures that relate to the realizability question.
Distinctive characteristics of Iwanaga--Gorenstein rings are typically understood through their intrinsic symmetry. We show that several of those that pertain to the Gorenstein global dimensions carry over to the one-sided situation, even without the noetherian hypothesis. Our results yield new relations among homological invariants related to the Gorenstein property, not only Gorenstein global dimensions but also the suprema of projective/injective dimensions of injective/projective modules and finitistic dimensions.
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.
Artinian quotients R of the local ring Q = k[[x,y,z]] are classified by multiplicative structures on A = Tor_Q^*(R,k); in particular, R is Gorenstein if and only if A is a Poincare duality algebra while R is Golod if and only if all products in A_{>0} are trivial. There is empirical evidence that generic quotient rings with small socle ranks fall on a spectrum between Golod and Gorenstein in a very precise sense: The algebra A breaks up as a direct sum of a Poincare duality algebra P and a graded vector space V, on which P_{>0} acts trivially. That is, A is a trivial extension, A = P \ltimes V, and the extremes A = (k \oplus Σk) \ltimes V and A = P correspond to R being Golod and Gorenstein, respectively. We prove that this observed behavior is, indeed, the generic behavior for graded quotients R of socle rank 2, and we show that the rank of P is controlled by the difference between the order and the degree of the socle polynomial of R.
We prove a generalized version of Evans and Griffith's Improved New Intersection Theorem: Let I be an ideal in a local ring R. If a finite free R-complex, concentrated in nonnegative degrees, has I-torsion homology in positive degrees, and the homology in degree 0 has an I-torsion minimal generator, then the length of the complex is at least dim R - dim R/I. This improves the bound ht I obtained by Avramov, Iyengar, and Neeman in 2018.
We expand on two existing characterizations of rings of Gorenstein (weak) global dimension zero and give two new characterizations of rings of finite Gorenstein (weak) global dimension. We also include the answer to a question of Y.~Xiang on Gorenstein weak global dimension of group rings.
We study singularity categories of exact categories with a focus on those associated to a complete hereditary cotorsion pair. As an application we identify a non-affine analogue of the singularity category of a Gorenstein local ring; with this Buchweitz's classic equivalence of three categories over Gorenstein local rings has been generalized to schemes, a project started by Murfet and Salarian more than ten years ago. As another application we use the framework to characterize rings of finite finitistic dimension.
We are interested in the structure of almost complete intersection ideals of grade 3. We give three constructions of these ideals and their free resolutions: one from the commutative algebra point of view, an equivariant construction giving a nice canonical form, andfinally an interpretation in terms of open sets in certain Schubert varieties.
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.
We introduce a refinement of the Gorenstein flat dimension for complexes over an associative ring--the Gorenstein flat-cotorsion dimension--and prove that it, unlike the Gorenstein flat dimension, behaves as one expects of a homological dimension without extra assumptions on the ring. Crucially, we show that it coincides with the Gorenstein flat dimension for complexes where the latter is finite, and for complexes over right coherent rings--the setting where the Gorenstein flat dimension is known to behave as expected.
Foxby defined the (Krull) dimension of a complex of modules over a commutative Noetherian ring in terms of the dimension of its homology modules. In this note it is proved that the dimension of a bounded complex of free modules of finite rank can be computed directly from the matrices representing the differentials of the complex.
We use Knuth's combinatorial approach to Pfaffians to reprove and clarify a century-old formula, due to Brill. It expresses arbitrary minors of a skew symmetric matrix in terms of Pfaffians.
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.