Gorenstein weak global dimension is symmetric Christensen, Lars Winther; Estrada, Sergio; Thompson, Peder
Mathematische Nachrichten,
November 2021, Letnik:
294, Številka:
11
Journal Article
Recenzirano
Odprti dostop
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.
Rigidity of Ext and Tor via flat–cotorsion theory Christensen, Lars Winther; Ferraro, Luigi; Thompson, Peder
Proceedings of the Edinburgh Mathematical Society,
11/2023, Letnik:
66, Številka:
4
Journal Article
Recenzirano
Odprti dostop
Let $\mathfrak{p}$ be a prime ideal in a commutative noetherian ring R and denote by $k(\mathfrak{p})$ the residue field of the local ring $R_\mathfrak{p}$. We prove that if an R-module M satisfies ...$\operatorname{Ext}_R^{n}(k(\mathfrak{p}),M)=0$ for some $n\geqslant\dim R$, then $\operatorname{Ext}_R^i(k(\mathfrak{p}),M)=0$ holds for all $i \geqslant n$. 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.
For a semiseparated 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 acy- clic complexes of flat quasi-coherent sheaves as the natural non-affine analogue of the homotopy category of totally acyclic complexes of projective modules.
Let
R
→
S
be a local ring homomorphism and
N
a finitely generated
S
-module. We prove that if the Gorenstein injective dimension of
N
over
R
is finite, then it equals the depth of
R
.
Let R be an R is an injective R \operatorname {Hom}_R(S,N)-module. The converse is not true, not even if R is its completion, but it is close: It is a special case of our main theorem that, in this ...setting, an R with \operatorname {Ext}^{>0}_R(S,N) =0 \operatorname {Hom}_R(S,N)-module.
Tate (co)homology via pinched complexes CHRISTENSEN, LARS WINTHER; JORGENSEN, DAVID A.
Transactions of the American Mathematical Society,
02/2014, Letnik:
366, Številka:
2
Journal Article
Recenzirano
Odprti dostop
For complexes of modules we study two new constructions which we call the pinched tensor product and the pinched Hom. They provide new methods for computing Tate homology \operatorname {\widehat ...{Tor}} \operatorname {\widehat {Ext}} Another application we consider is in local algebra. Under conditions of the vanishing of Tate (co)homology, the pinched tensor product of two minimal complete resolutions yields a minimal complete resolution.
We identify minimal cases in which a power
m
i
≠
0
of the maximal ideal of a local ring
R
is not Golod, i.e. the quotient ring
R
/
m
i
is not Golod. Complementary to a 2014 result by Rossi and Şega, ...we prove that for a generic artinian Gorenstein local ring with
m
4
=
0
≠
m
3
, the quotient
R
/
m
3
is not Golod. This is provided that
m
is minimally generated by at least 3 elements. Indeed, we show that if
m
is 2-generated, then every power
m
i
≠
0
is Golod.
Stable homology over associative rings CELIKBAS, OLGUR; CHRISTENSEN, LARS WINTHER; LIANG, LI ...
Transactions of the American Mathematical Society,
11/2017, Letnik:
369, Številka:
11
Journal Article
Recenzirano
Odprti dostop
stable homology over associative rings and obtain results over Artin algebras and commutative noetherian rings. Our study develops similarly for these classes; for simplicity we only discuss the ...latter here.
Let be a prime ideal in a commutative noetherian ring R. It is proved that if an R-module M satisfies ${\rm Tor}_n^R $(k ( ), M) = 0 for some n ⩾ R , where k( ) is the residue field at , then ${\rm ...Tor}_i^R $(k ( ), M) = 0 holds for all i ⩾ n. Similar rigidity results concerning ${\rm Tor}_R^{\ast} $(k ( ), M) are proved, and applications to the theory of homological dimensions are explored.
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