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- W2016804016 abstract "Recall that, in (II), one always has GK(M) 2 max(GK(N), GK(E’)). If equality holds for all short exact sequences of S-modules, then GK-dimension is said to be exact for S-modules. In general, exactness fails quite drastically, even in situations which are otherwise considered to be well behaved. For example, G. Bergman [3] has constructed an afline PIalgebra S having an ideal I of square 0 such that I is cyclic as right ideal of S, yet GK(S) = 3 > GK(S/1) = 2. Thus quite stringent conditions have to be imposed on the algebra or the modules in question for equality to hold in (II). On the other hand, the integrality question (I) presumably has a positive answer for many classes of algebras and modules that are of interest. Although the GK-dimension of an algebra (module) can be 0, 1, co, or any real number 2 2 (0, co, or any real number 2 1; Warfield [ 17]), it tends to be an integer or co in most cases which arise naturally. Thus no example of an affrne algebra S which is either Noetherian or finitely presented but has finite non-integral GK-dimension seems to be known. However, positive results are rare. Some exceptions, where GK-dimension is known to be an integer or co, are: finitely presented monomial algebras (Govorov [9]), almost commutative algebras (Tauvel [16]), an Noetherian PI-algebras (Lorenz and Small [ 131). GK-dimension, by its definition, measures the rate of growth of the steps in certain canonically defined filtrations on algebras and modules. Often," @default.
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- W2016804016 date "1988-11-01" @default.
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- W2016804016 title "On Gelfand-Kirillov dimension and related topics" @default.
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- W2016804016 doi "https://doi.org/10.1016/0021-8693(88)90031-2" @default.
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