arXiv subjects

Giovanna Le Gros

Publications and source records attributed to Giovanna Le Gros.

9 recordsLinked to original sources

arXiv2026

Cotorsion pairs and Tor-pairs over commutative noetherian rings

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 the ring. Each such function gives rise to a system of local depth conditions which describes the left-hand class in the corresponding cotorsion pair. Furthermore, we show that these cotorsion pairs correspond by explicit duality to hereditary Tor-pairs generated by modules of finite flat dimension.

math.AC↗

arXiv2026

Proxy smallness meets $t$-structures

We introduce a notion of proxy smallness for $t$-structures on triangulated categories associated to a Noetherian scheme. Specifically, the theory is developed in the presence of tensor actions. Consequently, our results yield a new characterization of schemes that are locally complete intersections in terms of $t$-structures, as well as a topological classification of preaisles on the bounded derived category of coherent sheaves.

math.AG↗

arXiv2026

Semi-Bousfield classes and nonmonotone perversities

In the generality of a rigidly-compactly generated tensor triangulated category, we introduce semi-Bousfield classes in terms of the vanishing of the tensor product in positive degrees with respect to a fixed reasonable $t$-structure. We show that semi-Bousfield classes provide a common generalisation of Bousfield classes and compactly generated tensor-compatible $t$-structures. Then we specialise to the setting of the unbounded derived category $\mathcal{D}_{\mathrm{qc}}(X)$ of a Noetherian scheme $X$ and show that the stratification bijection naturally extends to an assignment which takes a (not necessarily monotone) perversity on $X$ to a semi-Bousfield class in $\mathcal{D}_{\mathrm{qc}}(X)$. If $X$ is regular, this assignment constitutes a stratification of the whole semi-Bousfield lattice, while in the singular case, its image consists precisely of those semi-Bousfield classes arising from objects of finite Tor-dimension. Restricting this bijection to monotone perversities recovers the recent classification of compactly generated tensor-compatible $t$-structures of Dubey and Sahoo, (arXiv:2204.05015).

math.CT↗

arXiv2025

Govorov--Lazard and finite deconstructibility for Gorenstein and restricted homological dimensions

Over Cohen--Macaulay rings admitting a pointwise dualizing module, we show that the class of modules of restricted projective dimension bounded by any integer is finitely deconstructible and that the class of modules of restricted flat dimension bounded by any integer satisfies the Govorov-Lazard property. Along the way, we prove the corresponding result for Gorenstein projective and flat dimensions over (locally) Gorenstein rings. Outside of Cohen--Macaulay rings, we consider analogous properties for restricted projective dimension zero and restricted flat dimension zero and establish them for commutative noetherian rings of finite Krull dimension. This has consequences for the corresponding classes of finitely generated modules being preenveloping in certain cases and provides generalizations of Holm's results on structure of balanced big Cohen--Macaulay modules in various directions.

math.AC↗

arXiv2023

The finite type of modules of bounded projective dimension and Serre's conditions

Let $R$ be a commutative noetherian ring. We prove that the class of modules of projective dimension bounded by $k$ is of finite type if and only if $R$ satisfies Serre's condition $(S_k)$. In particular, this answers positively a question of Bazzoni and Herbera in the specific setting of a Gorenstein ring. Applying similar techniques, we also show that the $k$-dimensional version of the Govorov-Lazard Theorem holds if and only if $R$ satisfies the "almost" Serre condition $(C_{k+1})$.

math.AC↗

arXiv2023

Restricted injective dimensions over Cohen-Macaulay rings

We show that the small and large restricted injective dimensions coincide for Cohen-Macaulay rings of finite Krull dimension. Based on this, and inspired by the recent work of Sather-Wagstaff and Totushek, we suggest a new definition of Cohen-Macaulay Hom injective dimension. We show that the class of Cohen-Macaulay Hom injective modules is the right constituent of a perfect cotorsion pair. Our approach relies on tilting theory, and in particular, on the explicit construction of the tilting module inducing the minimal tilting class recently obtained in arXiv:2207.01309.

math.AC↗

arXiv2020

Covering classes and $1$-tilting cotorsion pairs over commutative rings

We are interested in characterising the commutative rings for which a $1$-tilting cotorsion pair $(\mathcal{A}, \mathcal{T})$ provides for covers, that is when the class $\mathcal{A}$ is a covering class. We use Hrbek's bijective correspondence between the $1$-tilting cotorsion pairs over a commutative ring $R$ and the faithful finitely generated Gabriel topologies on $R$. Moreover, we use results of Bazzoni-Positselski, in particular a generalisation of Matlis equivalence and their characterisation of covering classes for $1$-tilting cotorsion pairs arising from flat injective ring epimorphisms. Explicitly, if $\mathcal{G}$ is the Gabriel topology associated to the $1$-tilting cotorsion pair $(\mathcal{A}, \mathcal{T})$, and $R_\mathcal{G}$ is the ring of quotients with respect to $\mathcal{G}$, we show that if $\mathcal{A}$ is covering then $\mathcal{G}$ is a perfect localisation (in Stenström's sense) and the localisation $R_\mathcal{G}$ has projective dimension at most one. Moreover, we show that $\mathcal{A}$ is covering if and only if both the localisation $R_\mathcal{G}$ and the quotient rings $R/J$ are perfect rings for every $J \in \mathcal{G}$. Rings satisfying the latter two conditions are called $\mathcal{G}$-almost perfect.

math.AC↗

arXiv2020

Enveloping Classes over Commutative Rings

Given a $1$-tilting cotorsion pair over a commutative ring, we characterise the rings over which the $1$-tilting class is an enveloping class. To do so, we consider the faithful finitely generated Gabriel topology $\mathcal{G}$ associated to the $1$-tilting class $\mathcal{T}$ over a commutative ring as illustrated by Hrbek. We prove that a $1$-tilting class $\mathcal{T}$ is enveloping if and only if $ \mathcal{G}$ is a perfect Gabriel topology (that is, it arises from a perfect localisation) and $R/J$ is a perfect ring for each $J \in \mathcal{G}$, or equivalently $\mathcal{G}$ is a perfect Gabriel topology and the discrete quotient rings of the topological ring $\mathfrak R=$End$(R_ \mathcal{G}/R)$ are perfect rings where $R_\mathcal{G}$ denotes the ring of quotients with respect to $\mathcal{G}$. Moreover, if the above equivalent conditions hold it follows that pdim$R_\mathcal{G} \leq 1$ and $\mathcal{T}$ arises from a flat ring epimorphism.

math.AC↗

arXiv2020

$\mathcal{P}_1$-covers over commutative rings

In this paper we consider the class $\mathcal{P}_1(R)$ of modules of projective dimension at most one over a commutative ring $R$ and we investigate when $\mathcal{P}_1(R)$ is a covering class. More precisely, we investigate Enochs' Conjecture for this class, that is the question of whether $\mathcal{P}_1(R)$ is covering necessarily implies that $\mathcal{P}_1(R)$ is closed under direct limits. We answer the question affirmatively in the case of a commutative semihereditary ring $R$. This gives an example of a cotorsion pair $(\mathcal{P}_1(R), \mathcal{P}_1(R)^\perp)$ which is not necessarily of finite type such that $\mathcal{P}_1(R)$ satisfies Enochs' Conjecture. Moreover, we describe the class $\varinjlim \mathcal{P}_1(R)$ over (not-necessarily commutative) rings which admit a classical ring of quotients.

math.AC↗