Homological Ring & Scheme Invariants

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Abstractions about depth, dimension, derivations, resolutions, matrix factorizations, regular schemes, and homological invariants of rings.

13 abstractions in this family — domain-specific abstractions that sit near one another in structural-signature space (k-means over structural-signature embeddings). Each is shown with its short description.

• Depth (ring theory) — A homological invariant measuring the length of a maximal regular sequence acting on a module, equivalently the first degree of nonvanishing Ext under standard local Noetherian hypotheses.
• Deviation of a local ring — A sequence of nonnegative homological invariants counting generators in an acyclic closure and measuring successive departures of a local ring from regularity and complete-intersection structure.
• Gelfand–Kirillov dimension — An invariant measuring the polynomial growth rate of an algebra or module generated by finite-dimensional subspaces.
• Grade (ring theory) — The least degree in which a module or ideal has nonzero Ext into the base ring, equivalently under suitable hypotheses the maximum length of a regular sequence in its annihilator or ideal.
• Hasse–Schmidt derivation — A sequence of additive maps encoding a formal higher-order derivation through a multiplicative generating-series identity.
• Koszul algebra — A graded algebra whose ground field admits a minimal graded free resolution that is linear in every homological degree.
• Koszul–Tate resolution — A differential graded commutative algebra resolution of a quotient ring that generalizes the Koszul complex by adding generators to kill successive homology.
• Matrix factorization (algebra) — A pair of finite free-module maps whose two composites equal multiplication by a fixed potential, yielding a two-periodic resolution over the hypersurface quotient.
• Matrix factorization of a polynomial — A pair of square matrices over a polynomial ring whose two products both equal multiplication by a fixed polynomial times the identity.
• Morava K-theory — A prime- and height-indexed family of periodic generalized homology theories that isolates chromatic layers of stable homotopy theory.
• Regular scheme — A locally Noetherian scheme whose every local ring is regular, so local dimension equals the minimal number of generators of its maximal ideal.
• Stanley–Reisner ring — The quotient of a polynomial ring by the squarefree monomial ideal generated by nonfaces of a simplicial complex.
• Weak dimension — The least upper bound of the flat dimensions of all modules over a ring, measuring how far the ring is from making every module flat.