A bstract We investigate conformal field theories with gauge group U( N ) at arbitrary rank N , focusing on the role of trace relations in determining the structure of the Hilbert space. Working in the free trace algebra without imposing relations, we identify a class of evanescent states that vanish at finite N . Using the Koszul complex of [1], we implement trace relations systematically via ghosts and a fermionic charge Q b . This framework allows us to define and compute transition amplitudes between evanescent and physical states, which we show correspond precisely to ordinary CFT amplitudes analytically continued in N . Our results provide a direct algebraic realization of the proposals which realize trace relations in the bulk as over-maximal giant gravitons [1–3] and establish analytic continuation in N as a powerful tool for understanding finite- N effects.

Abstract In this work, we classify the thick subcategories of the bounded derived category of dg modules over a Koszul complex on any list of elements in a regular ring. This simultaneously recovers a theorem of Stevenson when the list of elements is a regular sequence and the classification of thick subcategories for an exterior algebra over a field (via the BGG correspondence). One of the major ingredients is a classification of thick tensor submodules of perfect curved dg modules over a graded commutative noetherian ring concentrated in even degrees, recovering a theorem of Hopkins and Neeman. We give several consequences of the classification result over a Koszul complex, one being that the lattice of thick subcategories of the bounded derived category is fixed by Grothendieck duality.

In this article we study base change of Poincaré series along a quasi-complete intersection homomorphism φ : Q → R \varphi \colon Q \to R , where Q Q is a local ring with maximal ideal m \mathfrak {m} . In particular, we give a precise relationship between the Poincaré series P M Q ( t ) \mathrm {P}^Q_M(t) of a finitely generated R R -module M M to P M R ( t ) \mathrm {P}^R_M(t) when the kernel of φ \varphi is contained in m a n n Q ( M ) \mathfrak {m}\,\mathrm {ann}_Q(M) . This generalizes a classical result of Shamash for complete intersection homomorphisms. Our proof goes through base change formulas for Poincaré series under the map of dg algebras Q → E Q\to E , with E E the Koszul complex on a minimal set of generators for the kernel of φ \varphi .

On Koszul complex of a supermodule

We study the structure of mod 2 cohomology rings of oriented Grassmannians Gr~k(n)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\widetilde{{\ ext {Gr}}}_k(n)$$\\end{document} of oriented k-planes in Rn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {R}}^n$$\\end{document}. Our main focus is on the structure of the cohomology ring H∗(Gr~k(n);F2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{H}^*(\\widetilde{{\ ext {Gr}}}_k(n);{\\mathbb {F}}_2)$$\\end{document} as a module over the characteristic subring C, which is the subring generated by the Stiefel–Whitney classes w2,…,wk\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$w_2,\\ldots ,w_k$$\\end{document}. We identify this module structure using Koszul complexes, which involves the syzygies between the relations defining C. We give an infinite family of such syzygies, which results in a new upper bound on the characteristic rank of Gr~k(2t)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\widetilde{{\ ext {Gr}}}_k(2^t)$$\\end{document}, k<2t\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$k<2^t$$\\end{document}, and formulate a conjecture on the exact value of the characteristic rank of Gr~k(n)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\widetilde{{\ ext {Gr}}}_k(n)$$\\end{document}. For the case k=3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$k=3$$\\end{document}, we use the Koszul complex to compute a presentation of the cohomology ring H=H∗(Gr~3(n);F2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$H=\ extrm{H}^*(\\widetilde{{\ ext {Gr}}}_3(n);{\\mathbb {F}}_2)$$\\end{document} for 2t-1<n≤2t-4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$2^{t-1}<n\\le 2^t-4$$\\end{document} for t≥4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$t\\ge 4$$\\end{document}, complementing existing descriptions in the cases n=2t-i\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n=2^t-i$$\\end{document}, i=0,1,2,3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$i=0,1,2,3$$\\end{document} for t≥3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$t\\ge 3$$\\end{document}. More precisely, as a C-module, H splits as a direct sum of the characteristic subring C and the anomalous module H/C, and we compute a complete presentation of H/C as a C-module from the Koszul complex. We also discuss various issues that arise for the cases k>3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$k>3$$\\end{document}, supported by computer calculation.

Abstract Under certain conditions, Koszul complexes can be used to calculate relative Betti diagrams of vector space-valued functors indexed by a poset, without the explicit computation of global minimal relative resolutions. In relative homological algebra of such functors, free functors are replaced by an arbitrary family of functors. Relative Betti diagrams encode the multiplicities of these functors in minimal relative resolutions. In this article we provide conditions under which grading the chosen family of functors leads to explicit Koszul complexes whose homology dimensions are the relative Betti diagrams, thus giving a scheme for the computation of these numerical descriptors.

Let [Formula: see text] be a complete intersection local ring, [Formula: see text] be the Koszul complex on a minimal set of generators of [Formula: see text], and [Formula: see text] be its homology algebra. We establish exact sequences involving direct sums of the components of [Formula: see text] and express the images of the maps of these sequences as homologies of iterated mapping cones built on [Formula: see text]. As an application of this iterated mapping cone construction, we recover a minimal free resolution of the residue field [Formula: see text] over [Formula: see text], independent from the well-known resolution constructed by Tate by adjoining variables and killing cycles. Through our construction, the differential maps can be expressed explicitly as blocks of matrices, arranged in some combinatorial patterns.

In this article we study the properties of preprojective algebras of representation finite species. To understand the structure of a preprojective algebra, one often studies its Nakayama automorphism. A complete description of the Nakayama automorphism is given by Brenner, Butler and King when the algebra is given by a path algebra. We generalize this result to the species case.We show that the preprojective algebra of a representation finite species is an almost Koszul algebra. With this we know that almost Koszul complexes exist. It turns out that the almost Koszul complex for a representation finite species is given by the mapping cone of a chain map, which is homogeneous of degree 1 with respect to a certain grading. We also study a higher dimensional analogue of representation finite hereditary algebras called d-representation finite algebras. One source of d-representation finite algebras comes from taking tensor products. By introducing a functor called the Segre product, we manage to give a complete description of the almost Koszul complex of the preprojective algebra of a tensor product of two species with relations with certain properties, in terms of the knowledge of the given species with relations. This allows us to compute the almost Koszul complex explicitly for certain species with relations more easily.

Buchsbaum-Rim multiplicity: A Koszul homology description

Independent sequences and freeness criteria

We extend the results of Deligne and Illusie on liftings modulo $p^2$ and\ndecompositions of the de Rham complex in several ways. We show that for a\nsmooth scheme $X$ over a perfect field $k$ of characteristic $p&gt;0$, the\ntruncations of the de Rham complex in $\\max(p-1, 2)$ consecutive degrees can be\nreconstructed as objects of the derived category in terms of its truncation in\ndegrees at most one (or, equivalently, in terms the obstruction class to\nlifting modulo $p^2$). Consequently, these truncations are decomposable if $X$\nadmits a lifting to $W_2(k)$, in which case the first nonzero differential in\nthe conjugate spectral sequence appears no earlier than on page $\\max(p,3)$\n(these corollaries have been recently strengthened by Drinfeld, Bhatt-Lurie,\nand Li-Mondal). Without assuming the existence of a lifting, we describe the\ngerbes of splittings of two-term truncations and the differentials on the\nsecond page of the conjugate spectral sequence, answering a question of Katz.\n The main technical result used in the case $p&gt;2$ belongs purely to\nhomological algebra. It concerns certain commutative differential graded\nalgebras whose cohomology algebra is the exterior algebra, dubbed by us\n"abstract Koszul complexes", of which the de Rham complex in characteristic $p$\nis an example.\n In the appendix, we use the aforementioned stronger decomposition result to\nprove that Kodaira-Akizuki-Nakano vanishing and Hodge-de Rham degeneration both\nhold for $F$-split $(p+1)$-folds.\n

On the geometry of forms on supermanifolds

In this paper, we define the homological Morse numbers of a filtered cell complex in terms of relative homology of nested filtration pieces, and derive inequalities relating these numbers to the Betti tables of the multi-parameter persistence modules of the considered filtration. Using the Mayer-Vietoris spectral sequence we first obtain strong and weak Morse inequalities involving the above quantities, and then we improve the weak inequalities achieving a sharp lower bound for homological Morse numbers. Furthermore, we prove a sharp upper bound for homological Morse numbers, expressed again in terms of the Betti tables.

We prove non-degeneracy of the cohomological bulk and boundary traces for general open-closed Landau-Ginzburg models associated to a pair $(X,W)$, where $X$ is a non-compact complex manifold with trivial canonical line bundle and $W$ is a complex-valued holomorphic function defined on $X$, assuming only that the critical locus of $W$ is compact (but may not consist of isolated points). These results can be viewed as certain "deformed" versions of Serre duality. The first amounts to a duality property for the hypercohomology of the sheaf Koszul complex of $W$, while the second is equivalent with the statement that a certain power of the shift functor is a Serre functor on the even subcategory of the $\mathbb{Z}_2$-graded category of topological D-branes of such models.

Vanishing of Avramov obstructions for products of sequentially transverse ideals

For a complex algebraic variety X $X$ , we introduce higher p $p$ -Du Bois singularity by imposing canonical isomorphisms between the sheaves of Kähler differential forms Ω X q $\Omega _X^q$ and the shifted graded pieces of the Du Bois complex Ω ̲ X q $\underline{\Omega }_X^q$ for q ⩽ p $q\leqslant p$ . If X $X$ is a reduced hypersurface, we show that higher p $p$ -Du Bois singularity coincides with higher p $p$ -log canonical singularity, generalizing a well-known theorem for p = 0 $p=0$ . The assertion that p $p$ -log canonicity implies p $p$ -Du Bois has been proved by Mustata, Olano, Popa, and Witaszek quite recently as a corollary of two theorems asserting that the sheaves of reflexive differential forms Ω X [ q ] $\Omega _X^{[q]}$ ( q ⩽ p $q\leqslant p$ ) coincide with Ω X q $\Omega _X^q$ and Ω ̲ X q $\underline{\Omega }_X^q$ , respectively, and these are shown by calculating the depth of the latter two sheaves. We construct explicit isomorphisms between Ω X q $\Omega _X^q$ and Ω ̲ X q $\underline{\Omega }_X^q$ applying the acyclicity of a Koszul complex in a certain range. We also improve some non-vanishing assertion shown by them using mixed Hodge modules and the Tjurina subspectrum in the isolated singularity case. This is useful for instance to estimate the lower bound of the maximal root of the reduced Bernstein–Sato polynomial in the case where a quotient singularity is a hypersurface and its singular locus has codimension at most 4.

This works concerns cohomological support varieties of modules over commutative local rings. The main result is that the support of a derived tensor product of a pair of differential graded modules over a Koszul complex is the join of the supports of the modules. This generalizes, and gives another proof of, a result of Dao and the third author dealing with Tor-independent modules over complete intersection rings. The result for Koszul complexes has a broader applicability, including to exterior algebras over local rings.

Computation of Koszul homology and application to involutivity of partial differential systems

For a two-periodic complex of vector bundles, Polishchuk and Vaintrob have constructed its localized Chern character. We explore some basic properties of this localized Chern character. In particular, we show that the cosection localization defined by Kiem and Li is equivalent to a localized Chern character operation for the associated two-periodic Koszul complex, strengthening a work of Chang, Li, and Li. We apply this equivalence to the comparison of virtual classes of moduli of \({\varepsilon }\)-stable quasimaps and moduli of the corresponding LG \({\varepsilon }\)-stable quasimaps, in full generality.

We introduce a derived representation scheme associated with a quiver, which may be thought of as a derived version of a Nakajima variety. We exhibit an explicit model for the derived representation scheme as a Koszul complex and by doing so we show that it has vanishing higher homology if and only if the moment map defining the corresponding Nakajima variety is flat. In this case we prove a comparison theorem relating isotypical components of the representation scheme to equivariant K-theoretic classes of tautological bundles on the Nakajima variety. As a corollary of this result we obtain some integral formulas present in the mathematical and physical literature since a few years, such as the formula for Nekrasov partition function for the moduli space of framed instantons on S^4. On the technical side we extend the theory of relative derived representation schemes by introducing derived partial character schemes associated with reductive subgroups of the general linear group and constructing an equivariant version of the derived representation functor for algebras with a rational action of an algebraic torus.