Srikanth B. Iyengar
Homological invariants of modules over contracting endomorphisms.
Mon, 10/18/2010 - 15:01 — SomNambulAuthors: Srikanth B. Iyengar, Luchezar L. Avramov, Yongwei Yao, Melvin Hochster
Subjects: Commutative Algebra
AbstractIt is proved that when R is a local ring of positive characteristic, $\phi$
is its Frobenius endomorphism, and some non-zero finite R-module has finite
flat dimension or finite injective dimension for the R-module structure induced
through $\phi$, then R is regular.Detecting flatness over smooth bases.
Mon, 02/22/2010 - 16:01 — SomNambulAuthors: Srikanth B. Iyengar, Luchezar L. Avramov
Subjects: Algebraic Geometry
AbstractIt is proved that when R is an (essentially) smooth algebra over a field and
M is an R-module that is finitely generated over some R-algebra A (essentially)
of finite type, if the d-fold tensor product of M over R is torsion-free, as an
R-module, for some integer d greater than or equal to dim R, then M is flat.
This is an analog--but not a consequence--of a classical criterion of Auslander
and Lichtenbaum for the freeness of finitely generated modules over regular
local rings.Reflexivity and rigidity for complexes, II: Schemes.
Thu, 01/21/2010 - 16:01 — SomNambulAuthors: Srikanth B. Iyengar, Luchezar L. Avramov, Joseph Lipman
Subjects: Algebraic Geometry
AbstractWe prove basic facts about reflexivity in derived categories over noetherian
schemes; and about related notions such as semidualizing complexes, invertible
complexes, and Gorenstein-perfect maps. Also, we study a notion of rigidity
with respect to semidualizing complexes, in particular, relative dualizing
complexes for Gorenstein-perfect maps. Our results include theorems of
Yekutieli and Zhang concerning rigid dualizing complexes on schemes. This work
is a continuation of part I, which dealt with commutative rings.Reflexivity and rigidity for complexes. I. Commutative rings.
Wed, 09/16/2009 - 13:15 — SomNambulAuthors: Srikanth B. Iyengar, Luchezar L. Avramov, Joseph Lipman
Subjects: Commutative Algebra
AbstractA notion of rigidity with respect to an arbitrary semidualizing complex C
over a commutative noetherian ring R is introduced and studied. One of the main
result characterizes C-rigid complexes. Specialized to the case when C is the
relative dualizing complex of a homomorphism of rings of finite Gorenstein
dimension, it leads to broad generalizations of theorems of Yekutieli and Zhang
concerning rigid dualizing complexes, in the sense of Van den Bergh. Along the
way, a number of new results concerning derived reflexivity with respect to C
are established.Reflexivity and rigidity for complexes. I. Commutative rings.
Wed, 09/16/2009 - 13:15 — SomNambulAuthors: Srikanth B. Iyengar, Luchezar L. Avramov, Joseph Lipman
Subjects: Commutative Algebra
AbstractA notion of rigidity with respect to an arbitrary semidualizing complex C
over a commutative noetherian ring R is introduced and studied. One of the main
result characterizes C-rigid complexes. Specialized to the case when C is the
relative dualizing complex of a homomorphism of rings of finite Gorenstein
dimension, it leads to broad generalizations of theorems of Yekutieli and Zhang
concerning rigid dualizing complexes, in the sense of Van den Bergh. Along the
way, a number of new results concerning derived reflexivity with respect to C
are established.Homological dimensions and regular rings.
Wed, 08/19/2009 - 19:30 — SomNambulAuthors: Alina Iacob, Srikanth B. Iyengar
Subjects: Commutative Algebra
AbstractA question of Avramov and Foxby concerning injective dimension of complexes
is settled in the affirmative for the class of noetherian rings. A key step in
the proof is to recast the problem on hand into one about the homotopy category
of complexes of injective modules. Analogous results for flat dimension and
projective dimension are also established.
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