Computation of Finite Free Resolutions and Projective Dimension

2021 ◽  
pp. 87-104
Author(s):  
Huishi Li
Keyword(s):  
Projective Dimension ◽  
Free Resolutions

scholarly journals Adic approximation of complexes, and multiplicities

1974 ◽  
Vol 54 ◽  
pp. 61-67 ◽  
Author(s):  
David Eisenbud
Keyword(s):  
Euler Characteristic ◽  
General Result ◽  
Noetherian Ring ◽  
Approximation Theorem ◽  
Projective Dimension ◽  
Local Rings ◽  
Free Resolutions ◽  
Intersection Multiplicity ◽  
Finite Projective Dimension ◽  
Finitely Generated Modules

In [2, Section 1.6] Peskine and Szpiro prove a theorem on adic approximations of finite free resolutions over local rings which, together with M. Artin’s Approximation Theorem [1], allows them to “descend” modules of finite projective dimension over the completions of certain local rings to modules of finite projective dimension over finite étale extensions of those rings. In this note we will prove a more general result, which deals with the change in homology under an adic approximation of any complex of finitely generated modules over a noetherian ring, and which allows one to descend not only modules of finite projective dimension, but also the Euler characteristic or intersection multiplicity of two such modules.

scholarly journals Minimal free resolutions of 2 × n domino tilings

2019 ◽  
Vol 18 (06) ◽  
pp. 1950118
Author(s):  
Rachelle R. Bouchat ◽  
Tricia Muldoon Brown
Keyword(s):  
Monomial Ideal ◽  
Betti Numbers ◽  
Projective Dimension ◽  
Free Resolution ◽  
Free Resolutions ◽  
Minimal Free Resolution ◽  
Domino Tilings ◽  
Minimal Free Resolutions ◽  
The Ideal

We introduce a squarefree monomial ideal associated to the set of domino tilings of a [Formula: see text] rectangle and proceed to study the associated minimal free resolution. In this paper, we use results of Dalili and Kummini to show that the Betti numbers of the ideal are independent of the underlying characteristic of the field, and apply a natural splitting to explicitly determine the projective dimension and Castelnuovo–Mumford regularity of the ideal.

scholarly journals Free resolutions of some Schubert singularities in the Lagrangian Grassmannian

2015 ◽  
Vol 279 (1-2) ◽  
pp. 329-355
Author(s):  
Venkatramani Lakshmibai ◽  
Reuven Hodges
Keyword(s):  
Free Resolutions ◽  
Lagrangian Grassmannian

Modules of finite projective dimension and cocovers

1996 ◽  
Vol 306 (1) ◽  
pp. 445-457 ◽  
Author(s):  
Dieter Happel ◽  
Luise Unger
Keyword(s):  
Projective Dimension ◽  
Finite Projective Dimension

scholarly journals Projective dimension and Brown-Peterson homology

Topology ◽  
1973 ◽  
Vol 12 (4) ◽  
pp. 327-353 ◽  
Author(s):  
David Copeland Johnson ◽  
W.Stephen Wilson
Keyword(s):  
Projective Dimension

Cellular actions and groups of finite quasi-projective dimension

1980 ◽  
Vol 170 (1) ◽  
pp. 85-90 ◽  
Author(s):  
James Howie ◽  
Hans Rudolf Schneebeli
Keyword(s):  
Projective Dimension

Free resolutions for deformations of maximal cohen-macaulay modules

2000 ◽  
Vol 28 (11) ◽  
pp. 5329-5352 ◽  
Author(s):  
Liam O'Carroll ◽  
Dorin Popescu
Keyword(s):  
Free Resolutions

Homological Dimensions Relative to Special Subcategories

2021 ◽  
Vol 28 (01) ◽  
pp. 131-142
Author(s):  
Weiling Song ◽  
Tiwei Zhao ◽  
Zhaoyong Huang
Keyword(s):  
Abelian Category ◽  
Projective Dimension ◽  
Homological Dimensions ◽  
Category Of Modules ◽  
The Right

Let [Formula: see text] be an abelian category, [Formula: see text] an additive, full and self-orthogonal subcategory of [Formula: see text] closed under direct summands, [Formula: see text] the right Gorenstein subcategory of [Formula: see text] relative to [Formula: see text], and [Formula: see text] the left orthogonal class of [Formula: see text]. For an object [Formula: see text] in [Formula: see text], we prove that if [Formula: see text] is in the right 1-orthogonal class of [Formula: see text], then the [Formula: see text]-projective and [Formula: see text]-projective dimensions of [Formula: see text] are identical; if the [Formula: see text]-projective dimension of [Formula: see text] is finite, then the [Formula: see text]-projective and [Formula: see text]-projective dimensions of [Formula: see text] are identical. We also prove that the supremum of the [Formula: see text]-projective dimensions of objects with finite [Formula: see text]-projective dimension and that of the [Formula: see text]-projective dimensions of objects with finite [Formula: see text]-projective dimension coincide. Then we apply these results to the category of modules.

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