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Let ( S , 픫 ) {(S,{\mathfrak{n}})} be a commutative noetherian local ring and let ω ∈ 픫 {\omega\in{\mathfrak{n}}} be non-zerodivisor. This paper is concerned with the two categories of monomorphisms between finitely generated (Gorenstein) projective S-modules, such that their cokernels are annihilated by ω. It is shown that these categories, which will be denoted by 햬허헇 ⁢ ( ω , 풫 ) {{\mathsf{Mon}}(\omega,\mathcal{P})} and 햬허헇 ⁢ ( ω , 풢 ) {{\mathsf{Mon}}(\omega,\mathcal{G})} , are both Frobenius categories with the same projective objects. It is also proved that the stable category 햬허헇 ¯ ⁢ ( ω , 풫 ) {\underline{\mathsf{Mon}}(\omega,\mathcal{P})} is triangle equivalent to the category of D-branes of type B, 햣햡 ⁢ ( ω ) {\mathsf{DB}(\omega)} , which has been introduced by Kontsevich and studied by Orlov. Moreover, it will be observed that the stable categories 햬허헇 ¯ ⁢ ( ω , 풫 ) {\underline{\mathsf{Mon}}(\omega,\mathcal{P})} and 햬허헇 ¯ ⁢ ( ω , 풢 ) {\underline{\mathsf{Mon}}(\omega,\mathcal{G})} are closely related to the singularity category of the factor ring R = S / ( ω ) {R=S/({\omega)}} . Precisely, there is a fully faithful triangle functor from the stable category 햬허헇 ¯ ⁢ ( ω , 풢 ) {\underline{\mathsf{Mon}}(\omega,\mathcal{G})} to 햣 헌헀 ⁡ ( R ) {\operatorname{\mathsf{D_{sg}}}(R)} , which is dense if and only if R (and so S) are Gorenstein rings. Particularly, it is proved that the density of the restriction of this functor to 햬허헇 ¯ ⁢ ( ω , 풫 ) {\underline{\mathsf{Mon}}(\omega,\mathcal{P})} , guarantees the regularity of the ring S. [ABSTRACT FROM AUTHOR]

We study the Banach algebras C (X , R) of continuous functions from a compact Hausdorff topological space X to a Banach ring R whose topology is discrete. We prove that the Berkovich spectrum of C (X , R) is homeomorphic to ζ (X) × M (R) , where ζ (X) is the Banaschewski compactification of X and M (R) is the Berkovich spectrum of R. We study how the topology of the spectrum of C (X , R) is related to the notion of homotopy Zariski open embedding used in derived geometry. We find that the topology of ζ (X) can be easily reconstructed from the homotopy Zariski topology associated with C (X , R) . We also prove some results about the existence of Schauder bases on C (X , R) and a generalization of the Stone–Weierstrass Theorem, under suitable hypotheses on X and R. [ABSTRACT FROM AUTHOR]

It is well-known that the generalized Auslander-Reiten condition (GARC) and the symmetric Auslander condition (SAC) are equivalent, and (GARC) implies that the Auslander-Reiten condition (ARC). In this paper we explore (SAC) along with the several canonical change of rings R → S . First, we prove the equivalence of (SAC) for R and R/xR, where x is a non-zerodivisor on R, and the equivalence of (SAC) and (SACC) for rings with positive depth, where (SACC) is the symmetric Auslander condition for modules with constant rank. The latter assertion affirmatively answers a question posed by Celikbas and Takahashi. Secondly, for a ring homomorphism R → S , we prove that if S satisfies (SAC) (resp. (ARC)), then R also satisfies (SAC) (resp. (ARC)) if the flat dimension of S over R is finite. We also prove that (SAC) holds for R implies that (SAC) holds for S when R is Gorenstein and S = R / Q ℓ , where Q is generated by a regular sequence of R and the length of the sequence is at least ℓ . This is a consequence of more general results about Ulrich ideals proved in this paper. Applying these results to determinantal rings and numerical semigroup rings, we provide new classes of rings satisfying (SAC). A relation between (SAC) and an invariant related to the finitistic extension degree is also explored. [ABSTRACT FROM AUTHOR]

Let R be a commutative Noetherian ring and I a proper ideal of R. In this paper, we study finitely generated R-modules M with only one non-vanishing local cohomology module H I c (M) where c = c d (I , M) . Let C be a semidualizing R-module. We investigate the conditions under which H I c (M) belongs to either the Auslander class A C (R) or the Bass class B C (R) . [ABSTRACT FROM AUTHOR]

The goal of the article is to develop a theory dual to that of support in the derived category D(R). This is done by introducing 'big' and 'small' cosupport for complexes that are different from the cosupport in D. J. Benson, S. B. Iyengar, H. Krause (2012). We give some properties for cosupport that are similar, or rather dual, to those of support for complexes, study some relations between 'big' and 'small' cosupport and give some comparisons of support and cosupport. Finally, we investigate the dual notion of associated primes. [ABSTRACT FROM AUTHOR]

In 2009, Lipman published his polished book (Lipman in Foundations of Grothendieck duality for diagrams of schemes, Lecture Notes in Mathematics, vol. 1960, Springer, Berlin, pp 1–259, 2009), the product of a decade’s work, giving the definitive, state-of-the-art treatment of Grothendieck duality. In this article, we achieve a sharp improvement: we begin by giving a new proof of the base-change theorem, which can handle unbounded complexes and work in the generality of algebraic stacks (subject to mild technical restrictions). This means that our base-change theorem must be subtle and delicate, the unbounded version is right at the boundary of known counterexamples—counterexamples (in the world of schemes) that had led the experts to believe that major parts of the theory could only be developed in the bounded-below derived category. Having proved our new base-change theorem, we then use it to define the functor f ! on the unbounded derived category and establish its functoriality properties. In Sect. 1, we will use this to clarify the relation among all the various constructions of Grothendieck duality. One illustration of the power of the new methods is that we can improve Lipman (Lecture Notes in Mathematics, vol. 1960, Springer, Berlin, pp 1–259, 2009, Theorem 4.9.4) to handle complexes that are not necessarily bounded. There are also applications to the theory developed by Avramov, Iyengar, Lipman and Nayak on the connection between Grothendieck duality and Hochschild homology and cohomology but, to keep this paper from becoming even longer, these are being relegated to separate articles. See for example (Neeman in K-Theory—Proceedings of the International Colloquium, Mumbai, 2016, Hindustan Book Agency, New Delhi, pp 91–126, 2018). [ABSTRACT FROM AUTHOR]

Let (S , n) be a commutative noetherian local ring and ω ∈ n be non-zerodivisor. This paper deals with the behavior of the category Mon (ω , P) consisting of all monomorphisms between finitely generated projective S-modules with cokernels annihilated by ω . We introduce a homotopy category H Mon (ω , P) , which is shown to be triangulated. It is proved that this homotopy category embeds into the singularity category of the factor ring R = S / (ω) . As an application, not only the existence of almost split sequences ending at indecomposable non-projective objects of Mon (ω , P) is proved, but also the Auslander–Reiten translation, τ Mon (-) , is completely recognized. Particularly, it will be observed that any non-projective object of Mon (ω , P) with local endomorphism ring is invariant under the square of the Auslander–Reiten translation. [ABSTRACT FROM AUTHOR]

In this paper, we prove Faltings’ annihilator theorem for complexes over a CM-excellent ring. As an application, we give a complete classification of the t-structures of the bounded derived category of finitely generated modules over a CM-excellent ring of finite Krull dimension. [ABSTRACT FROM AUTHOR]

Two distinguished types of semidualizing complexes are the shifts of the underlying rings and dualizing complexes. Let C be a semidualizing complex for an analytically irreducible local ring R and set n : = sup C and d : = dim R C. We show that C is quasi-isomorphic to a shift of R if and only if the nth Betti number of C is one. Also, we show that C is a dualizing complex for R if and only if the dth Bass number of C is one. [ABSTRACT FROM AUTHOR]

A celebrated result by Orlov states that any fully faithful exact functor between the bounded derived categories of coherent sheaves on smooth projective varieties is of geometric origin, i.e. it is a Fourier–Mukai functor. In this paper we prove that any smooth projective variety of dimension $\ge 3$ equipped with a tilting bundle can serve as the source variety of a non-Fourier–Mukai functor between smooth projective schemes. [ABSTRACT FROM AUTHOR]

Let A be an abelian category with enough projective objects, and let X be a quasi-resolving subcategory of A . In this paper, we investigate the affinity of the Spanier–Whitehead category S W (X) of the stable category of X with the singularity category D s g (A) of A . We construct a fully faithful triangle functor from S W (X) to D s g (A) , and we prove that it is dense if and only if the bounded derived category D b (A) of A is generated by X . Applying this result to commutative rings, we obtain characterizations of the isolated singularities, the Gorenstein rings and the Cohen–Macaulay rings. Moreover, we classify the Spanier–Whitehead categories over complete intersections. Finally, we explore a method to compute the (Rouquier) dimension of the triangulated category S W (X) in terms of generation in X . [ABSTRACT FROM AUTHOR]

Let (R , m) be a local ring such that the going-up theorem holds for the extension R ⊆ R ̂ , I an ideal of R and X ∈ D □ f (R) , X ≃ 0 such that H I c (X) is Artinian I-cofinite, where cd (I , X) = c. Then Att R H I c (X) = { p ∈ Supp R X | dim R / p − inf X p = c and I + p = m }. We also provide a generalization of Lichtenbaum-Hartshorne Vanishing Theorem on derived category. [ABSTRACT FROM AUTHOR]

Let R be a commutative noetherian ring, L be a Serre subcategory of the category of finitely generated R-modules and P be the category of finitely generated projective R-modules. We define invariants based on the categories of chain complexes from P homologically supported in L with support between 0 and n ( { C h L [ 0 , n ] (P) } n ∈ N ) and the K-group with support K 0 (R on L) similar to the stable range in classical K-theory. We introduce a notion called a reducer which we use to express the class of a complex in terms of classes of complexes of smaller amplitude and use these to study and bound the above defined invariants by standard invariants like arithmetic rank, grade and projective dimension. We also give conditions for K 0 (R on L) to be isomorphic to K 0 (modules in L with finite projective dimension). [ABSTRACT FROM AUTHOR]

We study smooth maps that arise in derived algebraic geometry. Given a map A → B between non-positive commutative noetherian DG-rings which is of flat dimension 0, we show that it is smooth in the sense of Toën–Vezzosi if and only if it is homologically smooth in the sense of Kontsevich. We then show that B, being a perfect DG-module over B ⊗ A L B has, locally, an explicit semi-free resolution as a Koszul complex. As an application we show that a strong form of Van den Bergh duality between (derived) Hochschild homology and cohomology holds in this setting. [ABSTRACT FROM AUTHOR]

We continue our work on adic semidualizing complexes over a commutative noetherian ring R by investigating the associated Auslander and Bass classes (collectively known as Foxby classes), following Foxby and Christensen. Fundamental properties of these classes include Foxby Equivalence, which provides an equivalence between the Auslander and Bass classes associated to a given adic semidualizing complex. We prove a variety of stability results for these classes, for instance, with respect to F ⊗ R L − where F is an R-complex finite flat dimension, including special converses of these results. We also investigate change of rings and local-global properties of these classes. [ABSTRACT FROM AUTHOR]

We give three proofs that valuation rings are derived splinters: a geometric proof using absolute integral closure, a homological proof which reduces the problem to checking that valuation rings are splinters (which is done in the second author's PhD thesis and which we reprise here), and a proof by approximation which reduces the problem to Bhatt's proof of the derived direct summand conjecture. The approximation property also shows that smooth algebras over valuation rings are splinters. [ABSTRACT FROM AUTHOR]

We introduce and investigate a new injective version of the complete intersection dimension of Avramov, Gasharov, and Peeva. It is like the complete intersection injective dimension of Sahandi, Sharif, and Yassemi in that it is built using quasi-deformations. Ours is different, however, in that we use a Hom functor in place of a tensor product. We show that (a) this invariant characterizes the complete intersection property for local rings, (b) it fits between the classical injective dimension and the G-injective dimension of Enochs and Jenda, (c) it provides modules with Bass numbers that are bounded by polynomials, and (d) it improves a theorem of Peskine, Szpiro, and Roberts (Bass' conjecture). [ABSTRACT FROM AUTHOR]

We define a category of perverse coherent sheaves as the abelian category corresponding to the category of modules under Bondal–Rickard equivalence which arises from a tilting bundle for a projective morphism. The purpose of this paper is to determine versal non-commutative deformations of simple collections in the categories of perverse coherent sheaves in some cases. In general we prove that the non-commutative structure algebra is recovered as the parameter algebra of the versal non-commutative deformation of the simple collection consisting of all simple objects over a closed point of the base space. In the case where the fiber dimensions are at most 1 and the structure sheaf is relatively acyclic, we determine the versal deformations of some partial simple collections consisting of vanishing simple objects. In particular it is proved that the parameter algebra of the versal non-commutative deformation is isomorphic to its opposite algebra in this case. [ABSTRACT FROM AUTHOR]

We classify all compactly generated t-structures in the unbounded derived category of an arbitrary commutative ring, generalizing the result of Alonso Tarrío et al. (J Algebra 324(3):313–346, 2010) for noetherian rings. More specifically, we establish a bijective correspondence between the compactly generated t-structures and infinite filtrations of the Zariski spectrum by Thomason subsets. Moreover, we show that in the case of a commutative noetherian ring, any bounded below homotopically smashing t-structure is compactly generated. As a consequence, all cosilting complexes are classified up to equivalence. [ABSTRACT FROM AUTHOR]

We compute the Ext group of the (filtered) Ogus category over a number field K. In particular we prove that the filtered Ogus realisation of mixed motives is not fully faithful. [ABSTRACT FROM AUTHOR]

The Orlov spectrum and Rouquier dimension are invariants of a triangulated category to measure how big the category is, and they have been studied actively. In this paper, we investigate the singularity category D sg (R) of a hypersurface R of countable representation type. For a thick subcategory T of D sg (R) and a full subcategory X of T , we calculate the Rouquier dimension of T with respect to X . Furthermore, we prove that the level in D sg (R) of the residue field of R with respect to each nonzero object is at most one. [ABSTRACT FROM AUTHOR]

Orlov's famous representability theorem asserts that any fully faithful exact functor between the bounded derived categories of coherent sheaves on smooth projective varieties is a Fourier–Mukai functor. In this paper we show that this result is false without the fully faithfulness hypothesis. We also show that our functor does not lift to the homotopy category of spectral categories if the ground field is Q . [ABSTRACT FROM AUTHOR]

By virtue of Balmer's celebrated theorem, the classification of thick tensor ideals of a tensor triangulated category T is equivalent to the topological structure of its Balmer spectrum Spc T . Motivated by this theorem, we discuss connectedness, irreducibility, and Noetherianity of the Balmer spectrum of a right bounded derived category of finitely generated modules over a commutative ring. [ABSTRACT FROM AUTHOR]

Let R be a commutative Noetherian ring. We introduce the notion of colocalization functors γ W with supports in arbitrary subsets W of Spec R . If W is a specialization-closed subset, then γ W coincides with the right derived functor R Γ W of the section functor Γ W with support in W . We prove that the local duality theorem and the vanishing theorem of Grothendieck type hold for γ W with W being an arbitrary subset. [ABSTRACT FROM AUTHOR]

We revisit the work of Lehn–Lehn–Sorger–van Straten on twisted cubic curves in a cubic fourfold not containing a plane in terms of moduli spaces. We show that the blow-up Z ′ along the cubic of the irreducible holomorphic symplectic eightfold Z , described by the four authors, is isomorphic to an irreducible component of a moduli space of Gieseker stable torsion sheaves or rank three torsion free sheaves. For a very general such cubic fourfold, we show that Z is isomorphic to a connected component of a moduli space of tilt-stable objects in the derived category and to a moduli space of Bridgeland stable objects in the Kuznetsov component. Moreover, the contraction between Z ′ and Z is realized as a wall-crossing in tilt-stability. Finally, Z is birational to an irreducible component of a moduli space of Gieseker stable aCM bundles of rank six. [ABSTRACT FROM AUTHOR]

We investigate torsion exceptional sheaves on a weak del Pezzo surface of degree greater than two whose anticanonical model has at most A n -singularities. We show that every torsion exceptional sheaf can be obtained from a line bundle on a ( − 1 ) -curve by spherical twists. [ABSTRACT FROM AUTHOR]

Given a Fourier–Mukai functor Φ in the general setting of singular schemes, under various hypotheses we provide both left and a right adjoints to Φ, and also give explicit formulas for them. These formulas are simple and natural, and recover the usual formulas when the Fourier–Mukai kernel is a perfect complex. This extends previous work of [1,12,13] and has applications to the twist autoequivalences of [9] . [ABSTRACT FROM AUTHOR]

A theorem by Orlov states that any equivalence $$F:D^{b}_{\mathrm {Coh}}(X) \rightarrow D^{b}_{\mathrm {Coh}}(Y)$$ between the bounded derived categories of coherent sheaves of two smooth projective varieties X and Y is isomorphic to a Fourier-Mukai transform $$\Phi _{E}(-)=R\pi _{2*}(E\mathop {\otimes }\limits ^{L} L\pi _1^{*}(-))$$ , where the kernel E is in $$D^{b}_{\mathrm {Coh}}(X\times Y)$$ . In the case of an exact functor which is not necessarily fully faithful, we compute some sheaves that play the role of the cohomology sheaves of the kernel, and that are isomorphic to the latter whenever an isomorphism $$F\cong \Phi _{E}$$ exists. We then exhibit a class of functors that are not full or faithful and still satisfy the above result. [ABSTRACT FROM AUTHOR]

LetMandNbe differential graded (DG) modules over a positively graded commutative DG algebraA. We show that the Ext-groupsdefined in terms of semi-projective resolutions are not in general isomorphic to the Yoneda Ext-groupsgiven 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. [ABSTRACT FROM AUTHOR]

We prove the versions of amplitude inequalities of Iversen, Foxby and Iyengar, and Frankild and Sather-Wagstaff that replace finite generation conditions with adic finiteness conditions. As an application, we prove that a local ringRof prime characteristic is regular if and only if for some proper ideal𝔟the derived local cohomology complexRΓ𝔟(R) has finite flat dimension when viewed through some positive power of the Frobenius endomorphism. [ABSTRACT FROM AUTHOR]

We apply geometric techniques from representation theory to the study of homologically finite differential graded (DG) modules [ABSTRACT FROM AUTHOR]

We survey some parts of the vast literature on holomorphic vector bundles on compact complex manifolds, focusing on the rank-two case vector bundles on non-Kähler elliptic fibrations. It is by no means intended to be a complete overview of this wide topic, but we rather focus on results obtained by the author and his collaborators. [ABSTRACT FROM AUTHOR]

In this article, we prove some results on Witt, Grothendieck–Witt ( GW ) and K -theory of noetherian quasi-projective schemes X , over affine schemes Spec ( A ) . For integers k ≥ 0 , let C M k ( X ) denote the category of coherent O X -modules F , with locally free dimension dim V ( X ) ⁡ ( F ) = k = grade ( F ) . We prove that there is an equivalence D b ( C M k ( X ) ) → D k ( V ( X ) ) of the derived categories. It follows that there is a sequence of zig-zag maps K ( C M k + 1 ( X ) ) ⟶ K ( C M k ( X ) ) ⟶ ∐ x ∈ X ( k ) K ( C M k ( X x ) ) of the K -theory spectra that is a homotopy fibration. In fact, this is analogous to the homotopy fiber sequence of the G -theory spaces of Quillen (see proof of [16, Theorem 5.4] ). We also establish similar homotopy fibrations of GW -spectra and G W -bispectra, by application of the same equivalence theorem. [ABSTRACT FROM AUTHOR]

Let $$\mathcal {X}$$ be a resolving subcategory of an abelian category. In this paper we investigate the singularity category $$\mathsf {D_{sg}}(\underline{\mathcal {X}})=\mathsf {D^b}({\mathsf {mod}}\,\underline{\mathcal {X}})/\mathsf {K^b}({\mathsf {proj}}({\mathsf {mod}}\,\underline{\mathcal {X}}))$$ of the stable category $$\underline{\mathcal {X}}$$ of $$\mathcal {X}$$ . We consider when the singularity category is triangle equivalent to the stable category of Gorenstein projective objects, and when the stable categories of two resolving subcategories have triangle equivalent singularity categories. Applying this to the module category of a Gorenstein ring, we prove that the complete intersections over which the stable categories of resolving subcategories have trivial singularity categories are the simple hypersurface singularities of type $$(\mathsf {A}_1)$$ . We also generalize several results of Yoshino on totally reflexive modules. [ABSTRACT FROM AUTHOR]

In this paper we study the derived categories of coherent sheaves on Grassmannians Gr ( k , n ) , defined over the ring of integers. We prove that the category D b ( Gr ( k , n ) ) has a semi-orthogonal decomposition, with components being full subcategories of the derived category of representations of G L k . This in particular implies existence of a full exceptional collection, which is a refinement of Kapranov's collection [13] , which was constructed over a field of characteristic zero. We also describe the right dual semi-orthogonal decomposition which has a similar form, and its components are full subcategories of the derived category of representations of G L n − k . The resulting equivalences between the components of the two decompositions are given by a version of Koszul duality for strict polynomial functors. We also construct a tilting vector bundle on Gr ( k , n ) . We show that its endomorphism algebra has two natural structures of a split quasi-hereditary algebra over Z , and we identify the objects of D b ( Gr ( k , n ) ) , which correspond to the standard and costandard modules in both structures. All the results automatically extend to the case of arbitrary commutative base ring and the category of perfect complexes on the Grassmannian, by extension of scalars (base change). Similar results over fields of arbitrary characteristic were obtained independently in [7] , by different methods. [ABSTRACT FROM AUTHOR]

We present an in-depth exploration of the module structures of local (co)homology modules (moreover, for complexes) over the completion $\widehat {R}^{\mathcal {a}}$ of a commutative noetherian ring R with respect to a proper ideal $\mathcal {a}$ . In particular, we extend Greenlees-May Duality and MGM Equivalence to track behavior over $\widehat {R}^{\mathcal {a}}$ , not just over R. We apply this to the study of two recent versions of homological finiteness for complexes, and to certain isomorphisms, with a view toward further applications. We also discuss subtleties and simplifications in the computations of these functors. [ABSTRACT FROM AUTHOR]

We answer a question of Celikbas, Dao, and Takahashi by establishing the following characterization of Gorenstein rings: a commutative noetherian local ring $$(R,\mathfrak m)$$ is Gorenstein if and only if it admits an integrally closed $$\mathfrak m$$ -primary ideal of finite Gorenstein dimension. This is accomplished through a detailed study of certain test complexes. Along the way we construct such a test complex that detect finiteness of Gorenstein dimension, but not that of projective dimension. [ABSTRACT FROM AUTHOR]

In this note we relate the notions of Lefschetz type, decomposability, and isomorphism for Chow motives with the notions of trivial type, decomposability, and isomorphism for noncommutative motives. Some examples, counter-examples, and applications are also described. [ABSTRACT FROM AUTHOR]

We investigate how one can detect the dualizing property for a chain complex over a commutative local Noetherian ring R. Our focus is on homological properties of contracting endomorphisms of R, e.g., the Frobenius endomorphism when R contains a field of positive characteristic. For instance, in this case, when R is F-finite and C is a semidualizing R-complex, we prove that the following conditions are equivalent: (i) C is a dualizing R-complex; (ii) C ∼ RHom( R, C) for some n > 0; (iii) G-dim R < ∞ and C is derived RHom( R, C)-reflexive for some n > 0; and (iv) G-dim R < ∞ for infinitely many n > 0. [ABSTRACT FROM AUTHOR]

Orlov's famous representability theorem asserts that any fully faithful exact functor between the bounded derived categories of coherent sheaves on smooth projective varieties is a Fourier–Mukai functor. This result has been extended by Lunts and Orlov to include functors from perfect complexes to quasi-coherent complexes. In this paper we show that the latter extension is false without the full faithfulness hypothesis. Our results are based on the properties of scalar extensions of derived categories, whose investigation was started by Pawel Sosna and the first author. [ABSTRACT FROM AUTHOR]

Simple-minded systems of objects in a stable module category are defined by common properties with the set of simple modules, whose images under stable equivalences do form simple-minded systems. Over a representation-finite self-injective algebra, it is shown that all simple-minded systems are images of simple modules under stable equivalences of Morita type, and that all simple-minded systems can be lifted to Nakayama-stable simple-minded collections in the derived category. In particular, all simple-minded systems can be obtained algorithmically using mutations. [ABSTRACT FROM AUTHOR]

A result of Foxby states that if there exists a complex with finite depth, finite flat dimension, and finite injective dimension over a local ring R, then R is Gorenstein. In this paper we investigate some homological dimensions involving a semidualizing complex and improve on Foxby's result by answering a question of Takahashi and White. In particular, we prove for a semidualizing complex C, if there exists a complex with finite depth, finite $${\mathcal{F}_{C}}$$ -projective dimension, and finite $${\mathcal{I}_{C}}$$ -injective dimension over a local ring R, then R is Gorenstein. [ABSTRACT FROM AUTHOR]

Let R be a standard graded algebra over an F -finite field of characteristic p > 0 . Let ϕ : R → R be the Frobenius endomorphism. For each finitely generated graded R -module M , let M ϕ be the abelian group M with the R -module structure induced by the Frobenius endomorphism. The R -module M ϕ has a natural grading given by deg ⁡ x = j if x ∈ M j p + i for some 0 ≤ i ≤ p − 1 . In this paper, we prove that R is Koszul if and only if there exists a non-zero finitely generated graded R -module M such that reg R M ϕ < ∞ . This result supplies another instance for the ability of the Frobenius in detecting homological properties, as exemplified by Kunz's famous regularity criterion. The main technical tool is the notion of Castelnuovo–Mumford regularity over certain homomorphisms between N -graded algebras. The latter notion is a common generalization of the relative and absolute Castelnuovo–Mumford regularity of modules. [ABSTRACT FROM AUTHOR]

LetAbe a differential graded (DG) algebra with a trivial differential over a commutative unital ring. This paper investigates the image of the totaling functor, defined from the category of complexes of gradedA-modules to the category of DGA-modules. Specifically, we exhibit a special class of semifree DGA-modules which can always be expressed as the totaling of some complex of graded freeA-modules. As a corollary, we also provide results concerning the image of the totaling functor whenAis a polynomial ring over a field. [ABSTRACT FROM AUTHOR]