The derived category of a locally complete intersection ring

In this paper, we answer a question of Dwyer, Greenlees, and Iyengar by proving a local ring $R$ is a complete intersection if and only if every complex of $R$-modules with finitely generated homology is proxy small. Moreover, we establish that a commutative noetherian ring $R$ is locally a complete intersection if and only if every complex of $R$-modules with finitely generated homology is virtually small.Comment: 14 page

A partial converse ghost lemma for the derived category of a commutative noetherian ring

In this article a condition is given to detect the containment among thick subcategories of the bounded derived category of a commutative noetherian ring. More precisely, for a commutative noetherian ring $R$ and complexes of $R$-modules with finitely generated homology $M$ and $N$, we show $N$ is in the thick subcategory generated by $M$ if and only if the ghost index of $N_\mathfrak{p}$ with respect to $M_\mathfrak{p}$ is finite for each prime $\mathfrak{p}$ of $R$. To do so, we establish a "converse coghost lemma" for the bounded derived category of a non-negatively graded DG algebra with noetherian homology.Comment: 10 pages, comments welcom

Bounds on cohomological support varieties

Over a local ring $R$, the theory of cohomological support varieties attaches to any bounded complex $M$ of finitely generated $R$-modules an algebraic variety $V_R(M)$ that encodes homological properties of $M$. We give lower bounds for the dimension of $V_R(M)$ in terms of classical invariants of $R$. In particular, when $R$ is Cohen-Macaulay and not complete intersection we find that there are always varieties that cannot be realized as the cohomological support of any complex. When $M$ has finite projective dimension, we also give an upper bound for $\dim V_R(M)$ in terms of the dimension of the radical of the homotopy Lie algebra of $R$. This leads to an improvement of a bound due to Avramov, Buchweitz, Iyengar, and Miller on the Loewy lengths of finite free complexes. Finally, we completely classify the varieties that can occur as the cohomological support of a complex over a Golod ring.Comment: 23 pages. Comments welcom

Constructing non-proxy small test modules for the complete intersection property

A local ring $R$ is regular if and only if every finitely generated $R$-module has finite projective dimension. Moreover, the residue field $k$ is a test module: $R$ is regular if and only if $k$ has finite projective dimension. This characterization can be extended to the bounded derived category $\mathsf{D}^f(R)$, which contains only small objects if and only if $R$ is regular. Recent results of Pollitz, completing work initiated by Dwyer-Greenlees-Iyengar, yield an analogous characterization for complete intersections: $R$ is a complete intersection if and only if every object in $\mathsf{D}^f(R)$ is proxy small. In this paper, we study a return to the world of $R$-modules, and search for finitely generated $R$-modules that are not proxy small whenever $R$ is not a complete intersection. We give an algorithm to construct such modules in certain settings, including over equipresented rings and Stanley-Reisner rings.Comment: Comments welcome. Changes in v2: added Example 4.4 and corrected small typo

Exceptional complete intersection maps of local rings

This work concerns surjective maps $\varphi\colon R\to S$ of commutative noetherian local rings with kernel generated by a regular sequence that is part of a minimal generating set for the maximal ideal of $R$. The main result provides criteria for detecting such exceptional complete intersection maps in terms of the lattices of thick subcategories of the derived category of complexes of finite length homology. A key input is a characterization of such maps in terms of the truncated Atiyah class of $\varphi$.Comment: 16 pages; added a missing hypothesis to Lemma 2.8, and minor changes to the proofs of Theorems 3.4 and 5.6. To appear in the Pacific Journal of Mathematic

Locally complete intersection maps and the proxy small property

It is proved that a map $\varphi\colon R\to S$ of commutative noetherian rings that is essentially of finite type and flat is locally complete intersection if and only $S$ is proxy small as a bimodule. This means that the thick subcategory generated by $S$ as a module over the enveloping algebra $S\otimes_RS$ contains a perfect complex supported fully on the diagonal ideal. This is in the spirit of the classical result that $\varphi$ is smooth if and only if $S$ is small as a bimodule, that is to say, it is itself equivalent to a perfect complex. The geometric analogue, dealing with maps between schemes, is also established. Applications include simpler proofs of factorization theorems for locally complete intersection maps.Comment: V2: 19 pages, some substantial simplifications and clarifications, to appear in IMR

The homotopy Lie algebra of a Tor-independent tensor product

In this article we investigate a pair of surjective local ring maps $S_1\leftarrow R\to S_2$ and their relation to the canonical projection $R\to S_1\otimes_R S_2$, where $S_1,S_2$ are Tor-independent over $R$. Our main result asserts a structural connection between the homotopy Lie algebra of $S:=S_1\otimes_R S_2$, denoted $\pi(S)$, in terms of those of $R,S_1$ and $S_2$. Namely, $\pi(S)$ is the pullback of (adjusted) Lie algebras along the maps $\pi(S_i)\to \pi(R)$ in various cases, including when the maps above have residual characteristic zero. Consequences to the main theorem include structural results on Andr\'{e}-Quillen cohomology, stable cohomology, and Tor algebras, as well as an equality relating the Poincar\'{e} series of the common residue field of $R,S_1,S_2$ and $S$.Comment: 20 pages. Corrected a mistake in 1.7; simplified and reorganized Sections 4 and