The space of finitely generated rings

The space of marked commutative rings on n given generators is a compact metrizable space. We compute the Cantor-Bendixson rank of any member of this space. For instance, the Cantor-Bendixson rank of the free commutative ring on n generators is omega^n, where omega is the smallest infinite ordinal. More generally, we work in the space of finitely generated modules over a given commutative ring.Comment: 10 pages, no figure. To appear in Internat. J. Algebra Compu

Loops of H-spaces with finitely generated cohomology rings

AbstractLet X be a simply connected ap-space. The mod p cohomology rings of Ω,X are studied. When these rings are finitely generated as algebras, Ω,X hasthe mod p homotopy type of a generalized Eilenberg-MacLane space. If X is just an H-space with H∗(Ω,X;Zp) finitely generated as an algebra, H∗(Ω,X;Zp) is still primitively generated free commutative

Finite generation of Cox rings

In this expository note we discuss a class of graded algebras named Cox rings, which are naturally associated to algebraic varieties generalizing the homogeneous coordinate rings of projective spaces. Whenever the Cox ring is finitely generated, the variety admits a quotient presentation by a quasitorus, which resembles the quotient construction of the projective space. We discuss the problem of the finite generation of Cox rings from a geometric perspective and provide examples of both the finitely and non-finitely generated cases.Comment: 17 pages, 6 figure

Stillman's conjecture via generic initial ideals

Using recent work by Erman-Sam-Snowden, we show that finitely generated ideals in the ring of bounded-degree formal power series in infinitely many variables have finitely generated Gr\"obner bases relative to the graded reverse lexicographic order. We then combine this result with the first author's work on topological Noetherianity of polynomial functors to give an algorithmic proof of the following statement: ideals in polynomial rings generated by a fixed number of homogeneous polynomials of fixed degrees only have a finite number of possible generic initial ideals, independently of the number of variables that they involve and independently of the characteristic of the ground field. Our algorithm outputs not only a finite list of possible generic initial ideals, but also finite descriptions of the corresponding strata in the space of coefficients.Comment: Several minor edit

Polynomial identity rings as rings of functions

We generalize the usual relationship between irreducible Zariski closed subsets of the affine space, their defining ideals, coordinate rings, and function fields, to a non-commutative setting, where "varieties" carry a PGL_n-action, regular and rational "functions" on them are matrix-valued, "coordinate rings" are prime polynomial identity algebras, and "function fields" are central simple algebras of degree n. In particular, a prime polynomial identity algebra of degree n is finitely generated if and only if it arises as the "coordinate ring" of a "variety" in this setting. For n = 1 our definitions and results reduce to those of classical affine algebraic geometry.Comment: 24 pages. This is the final version of the article, to appear in J. Algebra. Several proofs have been streamlined, and a new section on Brauer-Severi varieties has been adde