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1. Locally Complete Intersection Maps and the Proxy Small Property

   https://doi.org/10.1093/imrn/rnab041  

   Briggs, Benjamin; Iyengar, Srikanth B; Letz, Janina C; Pollitz, Josh (April 2021, International Mathematics Research Notices)

   null (Ed.)

   Abstract 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 if $$S$$ 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. 

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2. Morphisms of Character Varieties

   https://doi.org/10.1093/imrn/rnae124  

   Cotner, Sean (June 2024, International Mathematics Research Notices)

   Abstract Let $$k$$ be a field, let $$H \subset G$$ be (possibly disconnected) reductive groups over $$k$$, and let $$\Gamma $$ be a finitely generated group. Vinberg and Martin have shown that the induced morphism $$\underline{\operatorname{Hom}}_{k\textrm{-gp}}(\Gamma , H)//H \to \underline{\operatorname{Hom}}_{k\textrm{-gp}}(\Gamma , G)//G$$ is finite. In this note, we generalize this result (with a significantly different proof) by replacing $$k$$ with an arbitrary locally Noetherian scheme, answering a question of Dat. Along the way, we use Bruhat–Tits theory to establish a few apparently new results about integral models of reductive groups over discrete valuation rings. 

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3. Rigidity properties of the cotangent complex

   https://doi.org/10.1090/jams/1000  

   Briggs, Benjamin; Iyengar, Srikanth (January 2023, Journal of the American Mathematical Society)

   This work concerns a map φ : R → S \varphi \colon R\to S of commutative noetherian rings, locally of finite flat dimension. It is proved that the André-Quillen homology functors are rigid, namely, if D n ( S / R ; − ) = 0 \mathrm {D}_n(S/R;-)=0 for some n ≥ 1 n\ge 1 , then D i ( S / R ; − ) = 0 \mathrm {D}_i(S/R;-)=0 for all i ≥ 2 i\ge 2 and φ {\varphi } is locally complete intersection. This extends Avramov’s theorem that draws the same conclusion assuming D n ( S / R ; − ) \mathrm {D}_n(S/R;-) vanishes for all n ≫ 0 n\gg 0 , confirming a conjecture of Quillen. The rigidity of André-Quillen functors is deduced from a more general result about the higher cotangent modules which answers a question raised by Avramov and Herzog, and subsumes a conjecture of Vasconcelos that was proved recently by the first author. The new insight leading to these results concerns the equivariance of a map from André-Quillen cohomology to Hochschild cohomology defined using the universal Atiyah class of φ \varphi . 

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4. A uniform treatment of Grothendieck's localization problem

   https://doi.org/10.1112/S0010437X21007715  

   Murayama, Takumi (January 2022, Compositio Mathematica)

   Let $$f\colon Y \to X$$ be a proper flat morphism of locally noetherian schemes. Then the locus in $$X$$ over which $$f$$ is smooth is stable under generization. We prove that, under suitable assumptions on the formal fibers of $$X$$ , the same property holds for other local properties of morphisms, even if $$f$$ is only closed and flat. Our proof of this statement reduces to a purely local question known as Grothendieck's localization problem. To answer Grothendieck's problem, we provide a general framework that gives a uniform treatment of previously known cases of this problem, and also solves this problem in new cases, namely for weak normality, seminormality, $$F$$ -rationality, and the ‘Cohen–Macaulay and $$F$$ -injective’ property. For the weak normality statement, we prove that weak normality always lifts from Cartier divisors. We also solve Grothendieck's localization problem for terminal, canonical, and rational singularities in equal characteristic zero. 

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5. Rees algebras and the reduced fiber cone of divisorial filtrations on two dimensional normal local rings

   https://doi.org/10.1080/00927872.2025.2459254  

   Cutkosky, Steven Dale (August 2025, Communications in Algebra)

   Unknown (Ed.)

   Let I = {In} be a Q-divisorial filtration on a two dimensional normal excellent local ring (R, mR). Let R[I] = ⊕n≥0In be the Rees algebra of I and τ : ProjR[I]) → Spec(R) be the natural morphism. The reduced fiber cone of I is the R- algebra R[I]/ p mRR[I], and the reduced exceptional fiber of τ is Proj(R[I]/ p mRR[I]). In [7], we showed that in spite of the fact that R[I] is often not Noetherian, mRR[I] always has only finitely many minimal primes, so τ −1 (mR) has only finitely many ir- reducible components. In Theorem 1.2, we give an explicit description of the scheme structure of Proj(R[I]). As a corollary, we obtain in Theorem 1.3 a new proof of a theorem of F. Russo, showing that Proj(R[I]) is always Noetherian and that R[I] is Noetherian if and only if Proj(R[I]) is a proper R-scheme. In Corollary 1.4 to Theorem 1.2, we give an explicit description of the scheme structure of the reduced exceptional fiber Proj(R[I]/ p mRR[I]) of τ , in terms of the possible values 0, 1 or 2 of the analytic spread l(I) = dim R[I]/mRR[I]. In the case that l(I) = 0, τ −1 (mR) is the emptyset; this case can only occur if R[I] is not Noetherian. At the end of the introduction, we give a simple example of a graded filtration J of a two dimensional regular local ring R such that Proj(R[J ]) is not Noetherian. This filtration is necessarily not divisorial 

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