Global Dimensions of Some Artinian Algebras
نویسنده
چکیده
The structure of arbitrary associative commutative unital artinian algebras is well-known: they are finite products of associative commutative unital local algebras [6, pg.351, Cor. 23.12]. In the semi-simple case, we have the Artin-Wedderburn Theorem which states that any semi-simple artinian algebra (which is assumed to be associative and unital but not necessarily commutative) is a direct product of matrix algebras over division rings [6, pg.35, Par. 3.5]. Along these lines, we observe a simple classification of artinian algebras and their representations in Proposition 1.3.2 (hereby referred as the Classification Lemma) in terms of a category in which each object has a local artinian endomorphism algebra. This category is constructed using a fixed set of primitive (not necessarily central) idempotents in the underlying algebra. The Classification Lemma is a version of Freyd’s Representation Theorem [4, Sect. 5.3]: from an artinian algebra A we create a category CA on finitely many objects, and then the category of A-modules can be realized as a category of functors which admit CA as their domain. This construction can also be thought as a higher dimensional analogue of the semi-trivial extensions of [10] for artinian algebras.
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