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Abstract
The Hochschild cohomological dimension of any commutative k-algebra is lower-bounded by the least-upper bound of the flat-dimension difference and its global dimension. Our result is used to show that for a smooth affine scheme X satisfying Pointcaré duality, there must exist a vector bundle with section M and suitable n which the module of algebraic differential n-forms Ωn(X,M). Further restricting the notion of smoothness, we use our result to show that most k-algebras fail to be smooth in the quasi-free sense. This consequence, extends the currently known results, which are restricted to the case where k=C.
Highlights
Non-commutative geometry is a rapidly developing area of contemporary mathematical research that studies non-commutative algebras using formal geometric tools
The Hochschild cohomological dimension of any commutative k-algebra is lower-bounded by the least-upper bound of the flat-dimension difference and its global dimension
The trend of understanding geometric properties via algebraic dual theories is echoed throughout mathematics; with notable examples coming from the duality between finitely generated algebras and affine schemes, the description of any smooth manifold M through its commutative algebra C∞(M), and culminating with the work of [3,4] describing the duality relationship between algebra and geometry in full generality
Summary
Non-commutative geometry is a rapidly developing area of contemporary mathematical research that studies non-commutative algebras using formal geometric tools. It has become a key tool and object of study in non-commutative geometry since the results of [10] (and more recently generalized in [11] to characteristic p fields); which identifies the Hochschild homology of commutative k-algebras over a characteristic 0 field k, to the module of Khäler differentials over their associated affine scheme. The result identifies Hochschild’s cohomology theory with the modules of derivations and, with the tangential structure over the commutative algebra’s associated affine scheme. Likewise, in these cases, Pointcaré dualitylike results can be entirely formulated between these structures and the Hochschild (co)homology theories as shown in [12]. The paper’s proofs and any auxiliary technical lemma is relegated to Appendices B–D
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