Abstract Hilbert Schemes

By M. Artin, James J. Zhang. 2001

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Non-commutative Donaldson-Thomas theory and the conifold (2007, background): “A symmetric perfect obstruction theory [4] is a triple (E,ψ, θ) including also a non-degenerate symmetric bilinear form θ : E → E∨[1].”
Deformation theory of abelian categories (2004, unknown): “For M ∈ Mod(R), |M | ≤ sup{|S ⊗R M |, |N|}.”
Projectively simple rings (2004, background): “or each n≥ 0. The algebra Ais called strongly noetherian if for every commutative noetherian k-algebra C, the ring A⊗ k Cis noetherian. If Ais strongly noetherian, then by the work of Artin and Zhang [AZ2] there exists a projective scheme X which parametrizes the point modules for A. More exactly, i”
Noncommutative projective geometry (2003, background): “The best positive result is due to Artin, Small and Zhang [3, 9], for which we need a definition.”
Sheaves of t-structures and valuative criteria for stable complexes (2003, background): “An analogous result can be found in [ AZ ] in their context (it may be that their result implies ours).”
Multigraded Hilbert schemes (2002, background): “ans of a general construction which works in more contexts than just multigraded polynomial rings. It also applies to Quot schemes and to Hilbert schemes arising in noncommutative geometry; see e.g., [1], [4]. Our results resolve several open questions about Hilbert schemes and their equations. Our ”