Ziegler spectrum
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In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the (right) Ziegler spectrum of a
ring Ring may refer to: * Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry * To make a sound with a bell, and the sound made by a bell :(hence) to initiate a telephone connection Arts, entertainment and media Film and ...
''R'' is a
topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called points ...
whose points are (isomorphism classes of) indecomposable pure-injective right ''R''-modules. Its closed subsets correspond to theories of modules closed under arbitrary products and direct summands. Ziegler spectra are named after Martin Ziegler, who first defined and studied them in 1984.


Definition

Let ''R'' be a ring (associative, with 1, not necessarily commutative). A (right) pp-''n''-formula is a formula in the language of (right) ''R''-modules of the form : \exists \overline \ (\overline,\overline) A=0 where \ell,n,m are natural numbers, A is an (\ell+n)\times m matrix with entries from ''R'', and \overline is an \ell-tuple of variables and \overline is an n-tuple of variables. The (right) Ziegler spectrum, \operatorname_R, of ''R'' is the topological space whose points are isomorphism classes of indecomposable pure-injective right modules, denoted by \operatorname_R, and the topology has the sets : (\varphi/\psi) = \ as
subbasis In topology, a subbase (or subbasis, prebase, prebasis) for a topological space X with topology T is a subcollection B of T that generates T, in the sense that T is the smallest topology containing B. A slightly different definition is used by so ...
of open sets, where \varphi,\psi range over (right) pp-1-formulae and \varphi(N) denotes the subgroup of N consisting of all elements that satisfy the one-variable formula \varphi. One can show that these sets form a basis.


Properties

Ziegler spectra are rarely Hausdorff and often fail to have the T_0-property. However they are always
compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact * Blood compact, an ancient ritual of the Philippines * Compact government, a type of colonial rule utilized in British ...
and have a basis of compact open sets given by the sets (\varphi/\psi) where \varphi,\psi are pp-1-formulae. When the ring ''R'' is countable \operatorname_R is
sober In cryptography, SOBER is a family of stream ciphers initially designed by Greg Rose of QUALCOMM Australia starting in 1997. The name is a contrived acronym for ''S''eventeen ''O''ctet ''B''yte ''E''nabled ''R''egister. Initially the cipher wa ...
. It is not currently known if all Ziegler spectra are sober.


Generalization

Ivo Herzog showed in 1997 how to define the Ziegler spectrum of a locally coherent
Grothendieck category In mathematics, a Grothendieck category is a certain kind of abelian category, introduced in Alexander Grothendieck's TĂ´hoku paper of 1957English translation in order to develop the machinery of homological algebra for modules and for sheaves in ...
, which generalizes the construction above.


References

{{reflist Model theory