Related papers: Tensor product of dualizing complexes over a field

Let $\fa$ be an ideal of a Noetherian local ring $R$ and let $C$ be a semidualizing $R$-module. For an $R$-module $X$, we denote any of the quantities $\fd_R X$, $\Gfd_R X$ and $\GCfd_RX$ by $\T(X)$. Let $M$ be an $R$-module such that…

In this paper, we define and prove basic properties of complement polyhedral product spaces, dual complexes and polyhedral product complexes. Then we compute the universal algebra of polyhedral product complexes under certain split…

We give a response to a question posed by Groethendieck on the transfert of the properties: reduced, normal, domain, regular, complete intersection, Gorenstein, Cohen-Macaulay, to the completed tensor product of two Noetherian algebras on a…

In this paper, we study two-term tilting complexes for preprojective algebras of non-Dynkin type. We show that there exist two families of two-term tilting complexes, which are respectively parameterized by the elements of the corresponding…

Suppose C is a bounded chain complex of finitely generated free modules over the Laurent polynomial ring L = R[x,1/x]. Then C is R-finitely dominated, ie, homotopy equivalent over R to a bounded chain complex of finitely generated…

Let $R$ be a two-sided noetherian ring and $M$ be a nilpotent $R$-bimodule, which is finitely generated on both sides. We study Gorenstein homological properties of the tensor ring $T_R(M)$. Under certain conditions, the ring $R$ is…

I review the proposal of Berenstein-Douglas for a completely general definition of Seiberg duality. To give evidence for their conjecture I present the first example of a physical dual pair and explicitly check that it satisfies the…

For a commutative noetherian ring $R$, we classify all the hereditary cotorsion pairs cogenerated by pure-injective modules of finite injective dimension. The classification is done in terms of integer-valued functions on the spectrum of…

We consider a non-standard totalisation functor to produce a cochain complex from a given double complex: instead of sums or products, totalisation is defined via truncated products of modules. We give an elementary proof of the fact that a…

Let $R$ be a valuation ring with fraction field $K$ and $2\in R^\times$. We give an elementary proof of the following known result: Two unimodular quadratic forms over $R$ are isometric over $K$ if and only if they are isometric over $R$.…

Let $T$ be a complete local ring. We present necessary and sufficient conditions for $T$ to be the completion of a local (Noetherian) unique factorization domain $A$ such that there exist height one prime ideals $\{J_k\}_{k = 1}^{\infty}$…

In the derived category of modules over a commutative noetherian ring a complex $G$ is said to generate a complex $X$ if the latter can be obtained from the former by taking summands and finitely many cones. The number of cones required in…

We investigate the set S(R) of shift-isomorphism classes of semidualizing R-complexes, ordered via the reflexivity relation, where R is a commutative noetherian local ring. Specifically, we study the question of whether S(R$ has cardinality…

If $k$ is a field, $A$ and $B$ $k$-algebras, $M$ a faithful left $A$-module, and $N$ a faithful left $B$-module, we recall the proof that the left $A\otimes_k B$-module $M\otimes_k N$ is again faithful. If $k$ is a general commutative ring,…

In 1980, Faltings proved, by deep local algebra methods, a local result regarding formal functions which has the following global geometric fact as a consequence. Theorem: Let k be an algebraically closed field (of any characteristic). Let…

We give a constructive elementary proof for the fact that any K-automorphism of the full nxn matrix algebra over a field K is conjugation by some invertible nxn matrix A over K.

We show that the Rouquier dimension of the category of perfect complexes over a regular ring is precisely the Krull dimension of the ring. Previously, it was known that the Krull dimension is an upper bound, the lower bound however was not…

We show that every exact fully faithful functor from the category of perfect complexes on the spectrum of dual numbers to the bounded derived category of a noetherian separated scheme is of Fourier-Mukai type. The kernel turns out to be an…

Let $k$ be an uncountable algebraically closed field and let $A$ be a countably generated left Noetherian $k$-algebra. Then we show that $A \otimes_k K$ is left Noetherian for any field extension $K$ of $k$. We conclude that all subfields…

Let $A$ be a right noetherian algebra over a field $k$. If the base field extension $A \otimes_k K$ remains right noetherian for all extension fields $K$ of $k$, then $A$ is called stably right noetherian over $k$. We develop an inductive…