The aim of this article is to describe several applications of the theory of cycles to problems in Commutative Algebra. The main topic is the use of the theory of local Chern characters defined in the Chow group of a ring to answer some questions on modules of finite homological dimension and to clarify others. In the first section, we describe the origins of these problems in Intersection Theo...

1 A compilation of two sets of notes at the University of Kansas; one in the Spring of 2002 by ?? and the other in the Spring of 2007 by Branden Stone. These notes have been typed

In this paper we show that a strongly homotopy commutative (or C∞-) algebra with an invariant inner product on its cohomology can be uniquely extended to a symplectic C∞-algebra (an ∞-generalisation of a commutative Frobenius algebra introduced by Kontsevich). This result relies on the algebraic Hodge decomposition of the cyclic Hochschild cohomology of a C∞-algebra and does not generalize to a...

We prove that a Jordan homomorphism from a Banach algebra into a semisimple commutative Banach algebra is a ring homomorphism. Using a signum effectively, we can give a simple proof of the Hyers-Ulam-Rassias stability of a Jordan homomorphism between Banach algebras. As a direct corollary, we show that to each approximate Jordan homomorphism f from a Banach algebra into a semisimple commutative...

for making many comments and corrections concerning these notes. All rings are commutative and contain multiplicative identity, moreover we will always insist that ring homomorphisms respect the multiplicative identity element. Local rings are assumed to be Noetherian. Additionally, all modules are unitary modules. We have made an attempt to be consistent with our notation: (1) Rings are often ...

These notes closely follow Matsumura’s book [Mat80] on commutative algebra. Proofs are the ones given there, sometimes with slightly more detail. Our focus is on the results needed in algebraic geometry, so some topics in the book do not occur here or are not treated in their full depth. In particular material the reader can find in the more elementary [AM69] is often omitted. References on dim...

We compactify the spaces K(m,n) introduced by Maxim Kontsevich. The initial idea was to construct an L∞ algebra governing the deformations of a (co)associative bialgebra. However, this compactification leads not to a resolution of the PROP of (co)associative bialgebras, but to a new algebraic structure we call here a CROC. It turns out that these constructions are related to the non-commutative...

The Hopf algebra of renormalization in quantum field theory is described at a general level. The products of fields at a point are assumed to form a bialgebra B and renormalization endows T (T (B)+), the double tensor algebra of B, with the structure of a noncommutative bialgebra. When the bialgebra B is commutative, renormalization turns S(S(B)+), the double symmetric algebra of B, into a comm...

In this paper we show that a strongly homotopy commutative (or C∞-) algebra with an invariant inner product on its cohomology can be uniquely extended to a symplectic C∞-algebra (an ∞-generalisation of a commutative Frobenius algebra introduced by Kontsevich). This result relies on the algebraic Hodge decomposition of the cyclic Hochschild cohomology of a C∞-algebra and does not generalize to a...