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For a monomial ideal $I$, we consider the $i$th homological shift ideal of $I$, denoted by $text{HS}_i(I)$, that is, the ideal generated by the $i$th multigraded shifts of $I$. Some algebraic properties of this ideal are studied. It is shown that for any monomial ideal $I$ and any monomial prime ideal $P$, $text{HS}_i(I(P))subseteq text{HS}_i(I)(P)$ for all $i$, where $I(P)$ is the monomial localization of $I$. In particular, we consider the homological shift ideal of some families of monomial ideals with linear quotients. For any $textbf{c}$-bounded principal Borel ideal $I$ and for the edge ideal of complement of any path graph, it is proved that $text{HS}_i(I)$ has linear quotients for all $i$. As an example of $textbf{c}$-bounded principal Borel ideals, Veronese type ideals are considered and it is shown that the homological shift ideal of these ideals are polymatroidal. This implies that for any polymatroidal ideal which satisfies the strong exchange property, $text{HS}_j(I)$ is again a polymatroidal ideal for all $j$. Moreover, for any edge ideal with linear resolution, the ideal $text{HS}_j(I)$ is characterized and it is shown that $text{HS}_1(I)$ has linear quotients.
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We construct a local Cohen-Macaulay ring $R$ with a prime ideal $mathfrak{p}inspec(R)$ such that $R$ satisfies the uniform Auslander condition (UAC), but the localization $R_{mathfrak{p}}$ does not satisfy Auslanders condition (AC). Given any positive integer $n$, we also construct a local Cohen-Macaulay ring $R$ with a prime ideal $mathfrak{p}inspec(R)$ such that $R$ has exactly two non-isomorphic semidualizing modules, but the localization $R_{mathfrak{p}}$ has $2^n$ 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.
Let I be either the ideal of maximal minors or the ideal of 2-minors of a row graded or column graded matrix of linear forms L. In two previous papers we showed that I is a Cartwright-Sturmfels ideal, that is, the multigraded generic initial ideal gin(I) of I is radical (and essentially independent of the term order chosen). In this paper we describe generators and prime decomposition of gin(I) in terms of data related to the linear dependences among the row or columns of the submatrices of L. In the case of 2-minors we also give a closed formula for its multigraded Hilbert series.
It is proved that a module $M$ over a Noetherian local ring $R$ of prime characteristic and positive dimension has finite flat dimension if Tor$_i^R({}^e R, M)=0$ for dim $R$ consecutive positive values of $i$ and infinitely many $e$. Here ${}^e R$ denotes the ring $R$ viewed as an $R$-module via the $e$th iteration of the Frobenius endomorphism. In the case $R$ is Cohen-Macualay, it suffices that the Tor vanishing above holds for a single $egeq log_p e(R)$, where $e(R)$ is the multiplicity of the ring. This improves a result of D. Dailey, S. Iyengar, and the second author, as well as generalizing a theorem due to C. Miller from finitely generated modules to arbitrary modules. We also show that if $R$ is a complete intersection ring then the vanishing of Tor$_i^R({}^e R, M)$ for single positive values of $i$ and $e$ is sufficient to imply $M$ has finite flat dimension. This extends a result of L. Avramov and C. Miller.
Let $G$ be a finite simple connected graph on $[n]$ and $R = K[x_1, ldots, x_n]$ the polynomial ring in $n$ variables over a field $K$. The edge ideal of $G$ is the ideal $I(G)$ of $R$ which is generated by those monomials $x_ix_j$ for which ${i, j}$ is an edge of $G$. In the present paper, the possible tuples $(n, {rm depth} (R/I(G)), {rm reg} (R/I(G)), dim R/I(G), {rm deg} h(R/I(G)))$, where ${rm deg} h(R/I(G))$ is the degree of the $h$-polynomial of $R/I(G)$, arising from Cameron--Walker graphs on $[n]$ will be completely determined.
The commutative and homological algebra of modules over posets is developed, as closely parallel as possible to the algebra of finitely generated modules over noetherian commutative rings, in the direction of finite presentations, primary decompositions, and resolutions. Interpreting this finiteness in the language of derived categories of subanalytically constructible sheaves proves two conjectures due to Kashiwara and Schapira concerning sheaves with microsupport in a given cone. The motivating case is persistent homology of arbitrary filtered topological spaces, especially the case of multiple real parameters. The algebraic theory yields computationally feasible, topologically interpretable data structures, in terms of birth and death of homology classes, for persistent homology indexed by arbitrary posets. The exposition focuses on the nature and ramifications of a suitable finiteness condition to replace the noetherian hypothesis. The tameness condition introduced for this purpose captures finiteness for variation in families of vector spaces indexed by posets in a way that is characterized equivalently by distinct topological, algebraic, combinatorial, and homological manifestations. Tameness serves both the theoretical and computational purposes: it guarantees finite primary decompositions, as well as various finite presentations and resolutions all related by a syzygy theorem, and the data structures thus produced are computable in addition to being interpretable. The tameness condition and its resulting theory are new even in the finitely generated discrete setting, where being tame is materially weaker than being noetherian.
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