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We establish a characterization of dualizing modules among semidualizing modules. Let R be a finite dimensional commutative Noetherian ring with identity and C a semidualizing R-module. We show that C is a dualizing R-module if and only if Tor_i^R(E,E) is C- injective for all C-injective R-modules E and E and all igeq 0.
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We give a version of the usual Jacobian characterization of the defining ideal of the singular locus in the equal characteristic case: the new theorem is valid for essentially affine algebras over a complete local algebra over a mixed characteristic discrete valuation ring. The result makes use of the minors of a matrix that includes a row coming from the values of a $p$-derivation. To study the analogue of modules of differentials associated with the mixed Jacobian matrices that arise in our context, we introduce and investigate the notion of a perivation, which may be thought of, roughly, as a linearization of the notion of $p$-derivation. We also develop a mixed characteristic analogue of the positive characteristic $Gamma$-construction, and apply this to give additional nonsingularity criteria.
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Let $M$ and $N$ be differential graded (DG) modules over a positively graded commutative DG algebra $A$. We show that the Ext-groups $operatorname{Ext}^i_A(M,N)$ defined in terms of semi-projective resolutions are not in general isomorphic to the Yoneda Ext-groups $operatorname{YExt}^i_A(M,N)$ given in terms of equivalence classes of extensions. On the other hand, we show that these groups are isomorphic when the first DG module is semi-projective.
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Let $A$ be a Noetherian flat $K[t]$-algebra, $h$ an integer and let $N$ be a graded $K[t]$-module, we introduce and study $N$-fiber-full up to $h$ $A$-modules. We prove that an $A$-module $M$ is $N$-fiber-full up to $h$ if and only if $mathrm{Ext}^i_A(M, N)$ is flat over $K[t]$ for all $ile h-1$. And we show some applications of this result extending the recent result on squarefree Grobner degenerations by Conca and Varbaro.
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