The spectrum of a twisted commutative algebra

A counterexample to a Gröbner approach for noetherianity of the twisted commutative algebra ${\rm Sym}({\rm Sym}^2(\mathbf{C}^\infty))$

We resolve an open question posed by the authors of arXiv:1501.06925v2 in 2015 concerning a Gröbner theoretic approach for the noetherianity of the twisted commutative algebra ${\rm Sym}({\rm Sym}^2(\mathbf{C}^\infty))$. We provide a negative answer to their question by producing an explicit antichain. In doing so, we establish a connection to well studied posets of graphs under the subgraph and induced subgraph relation. We then analyze this connection to suggest future paths of investigation, for example a deeper study of initial ideals in twisted commutative algebras.

The twisted inverse image pseudofunctor over commutative DG rings and perfect base change

Let $K$ be a Gorenstein noetherian ring of finite Krull dimension, and consider the category of cohomologically noetherian commutative differential graded rings $A$ over $K$, such that $H^0(A)$ is essentially of finite type over $K$, and $A$ has finite flat dimension over $K$. We extend Grothendieck's twisted inverse image pseudofunctor to this category by generalizing the theory of rigid dualizing complexes to this setup. We prove functoriality results with respect to cohomologically finite and cohomologically essentially smooth maps, and prove a perfect base change result for $f^{!}$ in this…

Noetherianity of some degree two twisted commutative algebras

In recent years, researchers have discovered various large algebraic structures that have surprising finiteness properties, such as FI-modules and Delta-modules. In this paper, we add another example to the growing list: we show that certain degree two twisted commutative algebras are noetherian. This example appears to have some fundamental differences from previous examples, and is therefore especially interesting. Reflective of this, our proof introduces new methods for establishing noetherianity that are likely to be applicable in other situations. The algebras considered in this paper are…

Constructing commutative semifields of square order

The projection construction has been used to construct semifields of odd characteristic using a field and a twisted semifield [Commutative semifields from projection mappings, Designs, Codes and Cryptography, 61 (2011), 187--196]. We generalize this idea to a projection construction using two twisted semifields to construct semifields of odd characteristic. Planar functions and semifields have a strong connection so this also constructs new planar functions.

Stillman问题在扭曲交换代数中的情形

Stillman's question for twisted commutative algebras

本文构造了由二次多项式等变生成的理想族，利用Ananyan-Hochster原理证明其正则性无界，否定了Stillman猜想推广中的一个问题。

有限交换半群的半群代数上的齐次权重

Homogeneous Weights on Semigroup Algebras of Finite Commutative Semigroups

本文将Steinberg行列式理论扩展至有限域扭曲收缩半群代数，建立其弗罗贝尼乌斯判据，构造该类代数的齐次权重，刻画满足平均性质的主理想，拓展了齐次权重的适用范围。

Applications of Differential Graded Algebra Techniques in Commutative Algebra

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.

On monogenic functions defined in different commutative algebras

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.

Isomorphism types of commutative algebras of rank 7 over an arbitrary algebraically closed field of characteristic not 2 or 3

We classify isomorphism types of unital commutative algebras of rank 7 over an algebraically closed field of characteristic not 2 or 3 completely.

Frobenius Splitting in Commutative Algebra

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.

Connection between commutative algebra and topology

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.

交换环上的刚性对偶复形及其函子性质

Rigid Dualizing Complexes over Commutative Rings and their Functorial Properties

本文通过刚性对偶复形处理noetherian环的Grothendieck Duality，证明了正则基环上本质有限型环具有唯一的刚性对偶复形，并利用其函子性质构造了扭曲诱导伪函子，为后续在Deligne-Mumford stacks上建立全局对偶性奠定基础。

Simple graded commutative algebras

We study the notion of $Γ$-graded commutative algebra for an arbitrary abelian group $Γ$. The main examples are the Clifford algebras already treated by Albuquerque and Majid. We prove that the Clifford algebras are the only simple finite-dimensional associative graded commutative algebras over $\mathbb{R}$ or $\mathbb{C}$. Our approach also leads to non-associative graded commutative algebras extending the Clifford algebras.

Homotopical Aspects of Commutative Algebras

This article investigates the homotopy theory of simplicial commutative algebras with a view to homological applications.

Hyperholomorphic functions on commutative algebras

In this paper we study properties of hyperholomorphic functions on commutative finite algebras. It is investigated the Cauchy-Riemann type conditions for hyperholomorphic functions. We prove that a hyperholomorphic function on a commutative finite algebra can be expanded in a Taylor series. We also present a technique for computing zeros of polynomials in some commutative algebras.

Homotopical Aspects of Commutative Algebras I: Freeness Conditions for Crossed Squares

We give an alternative description of the top algebra of the free crossed square of algebras on 2-construction data in terms of tensors and coproducts of crossed modules of commutative algebras.

Homogeneous Buchberger algorithms and Sullivant's computational commutative algebra challenge

We give a variant of the homogeneous Buchberger algorithm for positively graded lattice ideals. Using this algorithm we solve the Sullivant computational commutative algebra challenge.

Universal lifts of chain complexes over non-commutative parameter algebras

We define the notion of universal lift of a projective complex based on non-commutative parameter algebras, and prove its existence and uniqueness. We investigate the properties of parameter algebras for universal lifts.

Length and Multiplicities in Graded Commutative Algebra

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…

关于Siegfried Bosch《代数几何与交换代数》的笔记（献给格罗滕迪克）

Notes on Siegfried Bosch's Algebraic Geometry and Commutative Algebra (dedicated to Grothendieck)

本文系统整理交换代数与代数几何的核心内容，涵盖环模层概形及平展上同调等进阶主题，作为对格罗滕迪克的致敬。