We introduce and study a class of objects that encompasses Christensen and Foxby’s semidualizing modules and complexes and Kubik’s quasi-dualizing modules: the class of [Formula: see text]-adic semidualizing modules and complexes. We give examples and equivalent characterizations of these objects, including a characterization in terms of the more familiar semidualizing property. As an application, we give a proof of the existence of dualizing complexes over complete local rings that does not use the Cohen Structure Theorem.

Gerko proves that if an artinian local ring $(R,\mathfrak{m}_R)$ possesses a sequence of strongly Tor-independent modules of length $n$, then $\mathfrak{m}_R^n\neq 0$. This generalizes readily to Cohen-Macaulay rings. We present a version of this result for non-Cohen-Macaulay rings.

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.

We explicitly describe the divisor class groups and semidualizing modules for ladder determinantal rings with coefficients in an arbitrary normal domain for arbitrary ladders, not necessarily connected, and all sizes of minors.

We continue our work on adic semidualizing complexes over a commutative noetherian ring $R$ by investigating the associated Auslander and Bass classes (collectively known as Foxby classes), following Foxby and Christensen. Fundamental properties of these classes include Foxby Equivalence, which provides an equivalence between the Auslander and Bass classes associated to a given adic semidualizing complex. We prove a variety of stability results for these classes, for instance, with respect to $F\otimes^{\mathbf{L}}_R-$ where $F$ is an $R$-complex finite flat dimension, including special converses of these results. We also investigate change of rings and local-global properties of these classes.

We introduce and investigate a new injective version of the complete intersection dimension of Avramov, Gasharov, and Peeva. It is like the complete intersection injective dimension of Sahandi, Sharif, and Yassemi in that it is built using quasi-deformations. Ours is different, however, in that we use a Hom functor in place of a tensor product. We show that (a) this invariant characterizes the complete intersection property for local rings, (b) it fits between the classical injective dimension and the G-injective dimension of Enochs and Jenda, (c) it provides modules with Bass numbers that are bounded by polynomials, and (d) it improves a theorem of Peskine, Szpiro, and Roberts (Bass' conjecture).

We introduce and investigate the category of factorization of a multiplicative, commutative, cancellative, pre-ordered monoid A, which we denote $$\mathcal {F}(A)$$. The objects of $$\mathcal {F}(A)$$ are factorizations of elements of A, and the morphisms in $$\mathcal {F}(A)$$ encode combinatorial similarities and differences between the factorizations. We pay particular attention to the divisibility pre-order and to the monoid $$A=D{\setminus }\{0\}$$ where D is an integral domain. Among other results, we show that $$\mathcal {F}(A)$$ is a symmetric and strict monoidal category with weak equivalences and compute the associated category of fractions obtained by inverting the weak equivalences. Also, we use this construction to characterize various factorization properties of integral domains: atomicity, unique factorization, and so on.

We investigate some general machinery for describing semidualizing modules over generic constructions like ladder determinantal rings with coefficients in a normal domain. We also pose and investigate natural localization questions that arise in the process.

Conditions on the Koszul complex of a noetherian local ring $R$ guarantee that $\mathrm{Tor}^{R}_{i}(M,N)$ is non-zero for infinitely many $i$, when $M$ and $N$ are finitely generated $R$-modules of infinite projective dimension. These conditions are obtained from results concerning Tor of differential graded modules over certain trivial extensions of commutative differential graded algebras.

We construct a local Cohen–Macaulay ring R with a prime ideal p∈Spec(R) such that R satisfies the uniform Auslander condition (UAC), but the localization Rp does not satisfy Auslander's condition (AC). Given any positive integer n, we also construct a local Cohen–Macaulay ring R with a prime ideal p∈Spec(R) such that R has exactly two non-isomorphic semidualizing modules, but the localization Rp has 2n non-isomorphic semidualizing modules. Each of these examples is constructed as a fiber product of two local rings over their common residue field. Additionally, we characterize the non-trivial Cohen–Macaulay fiber products of finite Cohen–Macaulay type.

Over a noetherian ring, it is a classic result of Matlis that injective modules admit direct sum decompositions into injective hulls of quotients by prime ideals. We show that over a Cohen-Macaulay ring admitting a dualizing module, Gorenstein injective modules admit similar filtrations. We also investigate Tor-modules of Gorenstein injective modules over such rings. This extends work of Enochs and Huang over Gorenstein rings. Furthermore, we give examples showing the following: (1) the class of Gorenstein injective R-modules need not be closed under tensor products, even when R is local and artinian; (2) the class of Gorenstein injective R-modules need not be closed under torsion products, even when R is a local, complete hypersurface; and (3) the filtrations given in our main theorem do not yield direct sum decompositions, even when R is a local, complete hypersurface.

We continue our study of ladder determinantal rings over a field k from the perspective of semidualizing modules. In particular, given a ladder of variables Y, we show that the associated ladder determinantal ring k[Y]/I2(Y) admits exactly 2n non-isomorphic semidualizing modules where n is determined from the combinatorics of the ladder Y: the number n is essentially the number of non-Gorenstein factors in a certain decomposition of Y. From this, for each n, we show explicitly how to find ladders Y such that k[Y]/I2(Y) admits exactly 2n non-isomorphic semidualizing modules. This is in contrast to our previous work, which demonstrates that large classes of ladders have exactly 2 non-isomorphic semidualizing modules.

We introduce and study a new class of integral domains which we call irreducible divisor pair domains (IDPDs). In particular, we show how IDPDs fit in with other classes of integral domains defined in terms of factorization conditions. For instance, every UFD is an IDPD, and every IDPD is an HFD, but the converses fail in general. We also show that many familiar examples of HFDs are also IDPDs.

We identify all semidualizing modules over certain classes of ladder determinantal rings over a field 𝗄. Specifically, given a ladder of variables Y, we show that the ring 𝗄[Y]/I_t(Y) has only trivial semidualizing modules up to isomorphism in the following cases: (1) Y is a one-sided ladder, and (2) Y is a two-sided ladder with t=2 and no coincidental inside corners.

This textbook in combinatorial commutative algebra concentrates on properties of monomial ideals in polynomial rings and their connections with other areas of mathematics such as combinatorics, electrical engineering, topology, geometry, and homological algebra.

This chapter deals with another case of monomial ideals with a reasonable algorithm for computing m-irreducible decompositions. These are the monomial ideals I with monomial radical $${\text {m-rad}}\left( I\right) $$ equal to the ideal $$\mathfrak {X}$$ generated by all the variables in R. See Section for properties of the monomial radical; the exercises of that section are particularly relevant.

Motivated by work of Hochster and Huneke, we investigate several constructions related to the $S_2$-ification $T$ of a complete equidimensional local ring $R$: the canonical module, the top local cohomology module, topological spaces of the form $\operatorname{Spec}(R)-V(J)$, and the (finite simple) graph $\Gamma_R$ with vertex set $\operatorname{Min}(R)$ defined by Hochster and Huneke. We generalize one of their results by showing, e.g., that the number of maximal ideals of $T$ is equal to the number of connected components of $\Gamma_R$. We further investigate this graph by exhibiting a technique for showing that a given graph $G$ can be realized as one of the form $\Gamma_R$.

The computer algebra system Macaulay2Macaulay2 is a powerful tool for understanding ringsring andideal ideals, among other things. This appendix introduces the basic syntax for this system, roughly following the topics of the previous appendix.