We describe several applications of the theory of cycles to questions in Commutative Algebra. The main topic is the use of the theory of local Chern characters to answer some questions on modules of finite homological dimension. This paper is based on talks given at the conference on cycles in Morelia, Mexico in June 2003.
This is an expository survey on the theory of Bernstein-Sato polynomials with special emphasis in its recent developments and its importance in commutative algebra.
Combining recent results on noetherianity of twisted commutative algebras by Draisma and the resolution of Stillman's conjecture by Ananyan-Hochster, we prove a broad generalization of Stillman's conjecture. Our theorem yields an array of boundedness results in commutative algebra that only depend on the degrees of the generators of an ideal, and not the number of variables in the ambient polynomial ring.
Differential graded (DG) algebras are powerful tools from rational homotopy theory. We survey some recent applications of these in the realm of homological commutative algebra.
A correspondence between a monogenic function in an arbitrary finite-dimensional commutative associative algebra and a finite set of monogenic functions in a special commutative associative algebra is established.
We introduce a category of dual pairs of finite locally free algebras over a ring. This gives an efficient way to represent finite locally free commutative group schemes. We give a number of algorithms to compute with dual pairs of algebras, and we apply our results to Galois representations on finite Abelian groups.
This is a survey of Frobenius splitting techniques in commutative algebra, based on the first author's lectures at the introductory workshop for the special year in commutative algebra at MSRI in fall 2012.
The main aim of this paper to show how commutative algebra is connected to topology. We give underlying topological idea of some results on completable unimodular rows.
In this paper we state and prove ad hoc "Separation Theorems" of the so-called Smooth Commutative Algebra, the Commutative Algebra of \(\mathcal{C}^{\infty}-\)rings. These results are formally similar to the ones we find in (ordinary) Commutative Algebra. However, their proof is not so straightforward, since it depends on the introduction of the concept of "smooth saturation". As an application of these theorems we present an interesting result that sheds light on the connections between the smooth Zariski spectrum and the real smooth spectrum of a \(\mathcal{C}^{\infty}-\)ring.
This paper is a review of concepts from graded commutative algebra with specific attention given to length and multiplicity. The author's motivation for this paper comes from the study of equivariant cohomology in algebraic topology where the graded commutative algebra of the module is intimately connected to topological properties of the space (as shown by Quillen). Results (and their proofs) which are often left as exercises, or considered 'folklore' in the commutative algebra community are included in this paper, as are references to relevant applications in topology. As such, this paper is…
In this paper, we begin a systematic study of modified Rota-Baxter algebras, as an associative analogue of the modified classical Yang-Baxter equation. We construct free commutative modified Rota-Baxter algebras by a variation of the shuffle product and describe the structure both recursively and explicitly. We then provide these algebras with a Hopf algebra structure by applying a Hochschild cocycle.
The Rota-Baxter algebra and the shuffle product are both algebraic structures arising from integral operators and integral equations. Free commutative Rota-Baxter algebras provide an algebraic framework for integral equations with the simple Riemann integral operator. The Zinbiel algebras form a category in which the shuffle product algebra is the free object. Motivated by algebraic structures underlying integral equations involving multiple integral operators and kernels, we study commutative matching Rota-Baxter algebras and construct the free objects making use of the shuffle product with m…
A twisted commutative algebra is (for us) a commutative $\mathbf{Q}$-algebra equipped with an action of the infinite general linear group. In such algebras the "$\mathbf{GL}$-prime" ideals assume the duties fulfilled by prime ideals in ordinary commutative algebra, and so it is crucial to understand them. Unfortunately, distinct $\mathbf{GL}$-primes can have the same radical, which obstructs one from studying them geometrically. We show that this problem can be eliminated by working with super vector spaces: doing so provides enough geometry to distinguish $\mathbf{GL}$-primes. This yields an …
These notes illustrates the power of formulating ideas of commutative algebra in a homotopy invariant form. They can then be applied to derived categories of rings or ring spectra. These ideas are powerful in classical algebra, in representation theory of groups, in classical algebraic topology and elsewhere. The notes grew out of a series of lectures given during the `Interactions between Representation Theory, Algebraic Topology and Commutative Algebra' (IRTATCA) at the CRM (Barcelona) in Spring 2015.
Motivated by the Hopf algebra structures established on free commutative Rota-Baxter algebras, we explore Hopf algebra related structures on free commutative Nijenhuis algebras. Applying a cocycle condition, we first prove that a free commutative Nijenhuis algebra on a left counital bialgebra (in the sense that the right-sided counicity needs not hold) can be enriched to a left counital bialgebra. We then establish a general result that a connected graded left counital bialgebra is a left counital right antipode Hopf algebra in the sense that the antipode is also only right-sided. We finally a…
For a given graph $G$, we construct an associated commutative algebra, whose dimension is equal to the number of $t$-labeled forests of $G$. We show that the dimension of the $k$-th graded component of this algebra also has a combinatorial meaning and that its Hilbert polynomial can be expressed through the Tutte polynomial of $G$.
