A cohomological support, $\operatorname{Supp}^*_{\mathcal A}(M)$, is defined for finitely generated modules $M$ over a left noetherian ring $R$, with respect to a ring $\mathcal A$ of central cohomology operations on the derived category of $R$-modules. It is proved that if the $\mathcal A$-module $\operatorname{Ext}^*_R(M,M)$ is noetherian and $\operatorname{Ext}^*_R(M,R)=0$ for $i\gg0$, then every closed subset of $\operatorname{Supp}^*_{\mathcal A}(M)$ is the support of some finitely generated $R$-module. This theorem specializes to known realizability results for varieties of modules over group algebras, over local complete intersections, and over finite dimensional algebras over a field. The theorem is also used to produce large families of finitely generated modules of finite projective dimension over commutative local noetherian rings.

This paper examines the existence of apolar covariants of order greater than or equal to \(n.\) These are shown to exist for all \(n>2.\)

We give a proof of local strong factorization of a birational, monomial extension of regular local rings along a valuation of rank 1 and maximal rational rank. Our proof uses methods from linear algebra, and is in the spirit of Christensen's proof of this result in dimension 3. This has also been proven by Karu using toric geometry.

Let $f: X\to \mathbb P^n$ be a proper map such that dimension of $f(X)\ge 2$. We address the following question: Is $\dim H^o(X,\,f^{\ast}(T_{\mathbb P^n}(-1)) = n + 1$? We provide an affirmative answer under standard mild restrictions on $X$. We also point out that this provides an affirmative answer to a similar question raised via regular alteration of a closed subvariety in a blow-up of a regular local ring at its closed point in the mixed characteristics.

The main result asserts that a local commutative Noetherian ring is Gorenstein, if it possesses a non-zero cyclic module of finite Gorenstein injective dimension. From this follows a classical result by Peskine and Szpiro stating that the ring is Gorenstein, if it admits a non-zero cyclic module of finite (classical) injective dimension. The main result applies to local homomorphisms of local rings and yields the next: if the source is a homomorphic image of a Gorenstein local ring and the target has finite Gorenstein injective dimension over the source, then the source is a Gorenstein ring. This, in turn, applies to the Frobenius endomorphism when the local ring is of prime equicharacteristic and is a homomorphic image of a Gorenstein local ring.

We extend the theory of generalized divisors so as to work on any scheme $X$ satisfying the condition $S_2$ of Serre. We define a generalized notion of Gorenstein biliaison for schemes in projective space. With this we give a new proof in a stronger form of the theorem of Gaeta, that standard determinantal schemes are in the Gorenstein biliaison class of a complete intersection.

We also show, for schemes of codimension three in ${\mathbb P}^n$, that the relation of Gorenstein biliaison is equivalent to the relation of even strict Gorenstein liaison.

A commutative Noetherian local ring $(R,\m)$ is said to be \emph{Dedekind-like} provided $R$ has Krull-dimension one, $R$ has no non-zero nilpotent elements, the integral closure $\overline R$ of $R$ is generated by two elements as an $R$-module, and $\m$ is the Jacobson radical of $\overline R$. A classification theorem due to Klingler and Levy implies that if $M$ is a finitely generated indecomposable module over a Dedekind-like ring, then, for each minimal prime ideal $P$ of $R$, the vector space $M_P$ has dimension $0, 1$ or $2$ over the field $R_P$. The main theorem in the present paper states that if $R$ (commutative, Noetherian and local) has non-zero Krull dimension and is not a homomorphic image of a Dedekind-like ring, then there are indecomposable modules that are free of any prescribed rank at each minimal prime ideal.

We consider injective local maps from a local domain $R$ to a local domain $S$ such that the generic fiber of the inclusion map $R \hookrightarrow S$ is trivial, that is, $P \cap R \ne (0)$ for every nonzero prime ideal $P$ of $S$. We present several examples of injective local maps involving power series that have or fail to have this property. For an extension $R \hookrightarrow S$ having this property, we give some results on the dimension of $S$; in some cases we show $\dim S = 2$ and in some cases $\dim S = 1$.

In this paper we study local cohomology of finitely generated bigraded modules over a standard bigraded ring with respect to the irrelevant bigraded ideals and establish a duality theorem. Several applications are considered.

We discuss a number of the local homological conjectures, many of which are now theorems in equal characteristic and conjectures in mixed characteristic. One focus is the syzygy theorem of Evans and Griffith, and its connection with the direct summand conjecture, the existence of big Cohen-Macaulay modules and algebras, and tight closure theory.

A fundamental property connecting the symbolic powers and the usual powers of ideals in regular rings was discovered by Ein, Lazarsfeld, and Smith in 2001, and later extended by Hochster and Huneke in 2002. In this paper we give further generalizations which give better results in case the quotient of the regular ring by the ideal is F-pure or F-pure type. Our methods also give insight into a conjecture of Eisenbud and Mazur concerning the existence of evolutions. The methods used come from tight closure and reduction to positive characteristic.

Let $k$ be a field of positive characteristic $p$, $R$ be a Gorenstein graded $k$-algebra, and $S=R/J$ be an artinian quotient of $R$ by a homogeneous ideal. We ask how the socle degrees of $S$ are related to the socle degrees of $F_R^e(S)=R/J^{[q]}$. If $S$ has finite projective dimension as an $R$-module, then the socles of $S$ and $F_R^e(S)$ have the same dimension and the socle degrees are related by the formula $D_i=qd_i-(q-1)a(R)$, where $d_1\le \dots\le d_{\ell}$ and $D_1\le \dots \le D_{\ell}$ are the socle degrees of $S$ and $F_R^e(S)$, respectively, and $a(R)$ is the $a$-invariant of the graded ring $R$, as introduced by Goto and Watanabe. We prove the converse when $R$ is a complete intersection.

For a map $f\colon X\to Y$ of quasi-compact quasi-separated schemes, we discuss quasi-perfection, i.e., the right adjoint $f^\times$ of $\mathbf Rf_*$ respects small direct sums. This is equivalent to the existence of a functorial isomorphism $f^\times\mathcal O_{Y}\otimes^{\mathbf L} \mathbf Lf^*(\<-\<)\! {\longrightarrow{}^\sim} f^\times (-)$; to quasi-properness (preservation by $\Rf$ of pseudo-coherence, or just properness in the noetherian case) plus boundedness of $\mathbf Lf^*\<$ (finite tor-dimensionality), or of the functor $f^\times\<$; and to some other conditions. We use a globalization, previously known only for divisorial schemes, of the local definition of pseudo-coherence of complexes, as well as a refinement of the known fact that the derived category of complexes with quasi-coherent homology is generated by a single perfect complex.

This paper discusses the problem of whether it is possible to annihilate elements of local cohomology modules by elements of arbitrarily small order under a fixed valuation. We first discuss the general problem and its relationship to the Direct Summand Conjecture, and next present two concrete examples where annihilators with small order are shown to exist. We then prove a more general theorem, where the existence of such annihilators is established in some cases using results on abelian varieties and the Albanese map.

Among the finitely generated modules over a Noetherian ring $R$, the semidualizing modules have been singled out due to their particularly nice duality properties. When $R$ is a normal domain, we exhibit a natural inclusion of the set of isomorphism classes of semidualizing $R$-modules into the divisor class group of $R$. After a description of the basic properties of this inclusion, it is employed to investigate the structure of the set of isomorphism classes of semidualizing $R$-modules. In particular, this set is described completely for determinantal rings over normal domains.

We study a question raised by Eisenbud, Mustaţa, and Stillman regarding the injectivity of natural maps from $\Ext$ modules to local cohomology modules. We obtain some positive answers to this question which extend earlier results of Lyubeznik. In the process, we also prove a vanishing theorem for local cohomology modules which connects theorems previously known in the case of positive characteristic and in the case of monomial ideals.

Let $R$ be a graded subalgebra of a polynomial ring $S$ over a field so that $S$ is algebraic over $R$. The goal of this paper is to relate the generator degrees of $R$ to the degree $[S:R]$ of the underlying quotient field extension, and to provide a numerical criterion for $S$ to be integral over $R$ that is based on this relationship. As an application we obtain a condition guaranteeing that a ring of invariants of a finite group is a polynomial ring.

If $A$ is a local, normal approximation domain with finite divisor class group, then $Cl(A) \cong Cl(\widehat{A})$.