A General Theory of Algebraic Geometry Over Dedekind Domains, I: The Notion of Models

A class of resolutions of objects of an abelian category determines a theory of derived functors if each morphism between objects extends to a morphism, unique to within homotopies, between their resolutions. This paper is primarily concerned with resolutions canonically associated with certain natural classes of extensions (E-functors), and the known examples are constructed by using pairs of adjoint functors. An inclusion between two E-functors on the same category induces natural transformations between functors derived from their associated resolutions, and other relations exist in the form of invariant exact couples. The relations simplify for the special and frequently occurring class of ‘central’ inclusions of E-functors; in particular the operations of forming satellites of a functor on the two resolutions commute. Amongst various applications the general theory provides generalizations of: results on groups of extensions of modules over Dedekind domains; the Hochschild—Serre spectral sequences in the homology theory of groups; the spectral sequences for coherent algebraic sheaves that determine Ext by means of vector bundle resolutions and affine coverings.

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Examples in non-commutative projective geometry

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100072716 ◽

1994 ◽

Vol 116 (3) ◽

pp. 415-433 ◽

Cited By ~ 22

Author(s):

J. T. Stafford ◽

J. J. Zhang

Keyword(s):

Algebraic Geometry ◽

General Theory ◽

Projective Geometry ◽

Projective Variety ◽

Companion Paper ◽

Central Field ◽

Commutative Algebras ◽

Formal Definitions ◽

Noetherian Algebra ◽

Natural Way

Let A = k ⊕ ⊕n ≥ 1An connected graded, Noetherian algebra over a fixed, central field k (formal definitions will be given in Section 1 but, for the most part, are standard). If A were commutative, then the natural way to study A and its representations would be to pass to the associated projective variety and use the power of projective algebraic geometry. It has become clear over the last few years that the same basic idea is powerful for non-commutative algebras; see, for example, [ATV1, 2], [AV], [Sm], [SS] or [TV] for some of the more significant applications. This suggests that it would be profitable to develop a general theory of ‘non-commutative projective geometry’ and the foundations for such a theory have been laid down in the companion paper [AZ]. The results proved there raise a number of questions and the aim of this paper is to provide negative answers to several of these.

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Extreme self-sacrifice beyond fusion: Moral expansiveness and the special case of allyship

Behavioral and Brain Sciences ◽

10.1017/s0140525x18001620 ◽

2018 ◽

Vol 41 ◽

Cited By ~ 2

Author(s):

Daniel Crimston ◽

Matthew J. Hornsey

Keyword(s):

General Theory ◽

Biological Markers ◽

Natural Environment ◽

Relevant Dimension ◽

Special Case

AbstractAs a general theory of extreme self-sacrifice, Whitehouse's article misses one relevant dimension: people's willingness to fight and die in support of entities not bound by biological markers or ancestral kinship (allyship). We discuss research on moral expansiveness, which highlights individuals’ capacity to self-sacrifice for targets that lie outside traditional in-group markers, including racial out-groups, animals, and the natural environment.

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