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In this paper, we prove a decomposition result for the Chow groups of projectivizations of coherent sheaves of homological dimension $\le 1$. In this process, we establish the decomposition of Chow groups for the cases of Cayley's trick and standard flips. Moreover, we apply these results to study the Chow groups of symmetric powers of curves, nested Hilbert schemes of surfaces, and the varieties resolving Voisin maps for cubic fourfolds., Comment: v2. Many grammatical problems fixed. References updated

We show a new neutral-fermionic presentation of Ikeda-Naruse's $K$-theoretic $Q$-functions, which represent a Schubert class in the $K$-theory of coherent sheaves on the Lagrangian Grassmannian. Our presentation provides a simple description and yields straightforward proof of two types of Pfaffian formulas for them. We present a dual space of $G\Gamma$, the vector space generated by all $K$-theoretic $Q$-functions, by constructing a non-degenerate bilinear form that is compatible with the neutral fermionic presentation. We give a new family of dual $K$-theoretic $Q$-functions, their neutral-fermionic presentations, and Pfaffian formulas., Comment: 27 pages, 1 figure v4: Major revision. The title has been changed. (The old title was "Pfaffian formulas in K-theory and the Boson-Fermion Correspondence")

We prove that the bounded derived category of coherent sheaves on a quasicompact separated quasiexcellent scheme of finite dimension has a strong generator in the sense of Bondal-Van den Bergh. This extends a recent result of Neeman and is new even in the affine case. The main ingredient includes Gabber's weak local uniformization theorem and the notions of boundedness and descendability of a morphism of schemes., 5 pages

Let Λ be a basic finite dimensional algebra over an algebraically closed field $\Bbbk $ , and let $\widehat {\Lambda }$ be the repetitive algebra of Λ. In this article, we prove that if $\widehat {V}$ is a left $\widehat {\Lambda }$ -module with finite dimension over $\Bbbk $ , then $\widehat {V}$ has a well-defined versal deformation ring $R(\widehat {\Lambda },\widehat {V})$ , which is a local complete Noetherian commutative $\Bbbk $ -algebra whose residue field is also isomorphic to $\Bbbk $ . We also prove that $R(\widehat {\Lambda }, \widehat {V})$ is universal provided that $\underline {\text {End}}_{\widehat {\Lambda }}(\widehat {V})=\Bbbk $ and that in this situation, $R(\widehat {\Lambda }, \widehat {V})$ is stable after taking syzygies. We apply the obtained results to finite dimensional modules over the repetitive algebra of the 2-Kronecker algebra, which provides an alternative approach to the deformation theory of objects in the bounded derived category of coherent sheaves over $\mathbb {P}^{1}_{\Bbbk }$ .

A concept of Cohen–Macaulay in codimension t is defined and characterized for arbitrary finitely generated modules and coherent sheaves by Miller, Novik, and Swartz in 2011. Soon after, Haghighi, Yassemi, and Zaare-Nahandi defined and studied CMt simplicial complexes, which is the pure version of the abovementioned concept and naturally generalizes both Cohen–Macaulay and Buchsbaum properties. The purpose of this paper is to survey briefly recent results of CMt simplicial complexes.

In [Ven16] , Venkatesh conjectures a relation between the action of derived Hecke operators and an action by motivic cohomology groups. In [HV18] , Harris and Venkatesh reformulate this conjecture for the case of cohomology groups of coherent sheaves associated to modular forms of weight one. We refer to the latter as the HV conjecture. Recently, the work of Darmon, Harris, Rotger and Venkatesh in [DHRV] proves that this conjecture holds for dihedral weight one forms. The following article focuses therefore on the case of exotic forms, describes methods to compute explicitly all key ingredients appearing in the HV conjecture and provides further numerical evidence for it.

Let $(\mathcal C,\otimes,1)$ be an abelian symmetric monoidal category satisfying certain conditions and let $X$ be a scheme over $(\mathcal C,\otimes,1)$ in the sense of Toen and Vaquie. In this paper we show that when $X$ is quasi-compact and semi-separated, any quasi-coherent sheaf on $X$ may be expressed as a directed colimit of its finitely generated quasi-coherent submodules. Thereafter, we introduce a notion of "field objects" in $(\mathcal C,\otimes,1)$ that satisfy several properties similar to those of fields in usual commutative algebra. Finally we show that the points of a Noetherian, quasi-compact and semi-separated scheme $X$ over such a field object $K$ in $(\mathcal C,\otimes,1)$ can be recovered from certain kinds of functors between categories of quasi-coherent sheaves. The latter is a partial generalization of some recent results of Brandenburg and Chirvasitu.

We develop a version of Hodge theory for a large class of smooth formally proper quotient stacks $X/G$ analogous to Hodge theory for smooth projective schemes. We show that the noncommutative Hodge-de Rham sequence for the category of equivariant coherent sheaves degenerates. This spectral sequence converges to the periodic cyclic homology, which we canonically identify with the topological equivariant K-theory of $X$ with respect to a maximal compact subgroup of $G$, equipping the latter with a canonical pure Hodge structure. We also establish Hodge-de Rham degeneration for categories of matrix factorizations for a large class of equivariant Landau-Ginzburg models., Comment: 47 pages, updated to match the published version, to avoid confusion. Following referee's suggestion, we reorganized the paper so that all matrix factorization material appears in its own separate section

This paper aims at studying the homotopy category of cotorsion flat left modules $${{\mathbb {K}}({\mathrm{CotF}}\text {-}R)}$$ over a ring R. We prove that if R is right coherent, then the homotopy category $${\mathbb {K}}(\mathrm{dg}\text {-}\mathrm{CotF}\text {-}R)$$ of dg-cotorsion complexes of flat R-modules is compactly generated. This uses firstly the existence of cotorsion flat preenvelopes over such rings and, secondly, the existence of a complete cotorsion pair $$({{\mathbb {K}}_{\mathrm{p}}({\mathrm{Flat}}\text {-}R)}, {\mathbb {K}}(\mathrm{dg}\text {-}\mathrm{CotF}\text {-}R))$$ in the homotopy category $${{\mathbb {K}}({\mathrm{Flat}}\text {-}R)}$$ of complexes of flat R-modules, for arbitrary R. In the setting of quasi coherent sheaves over a Noetherian scheme, this cotorsion pair was discovered in the literature. However, we use a more elementary argument that gives this cotorsion pair for arbitrary R. Next we deal with cotorsion flat resolutions of complexes and define and study the notion of cotorsion flat dimension for complexes of flat R-modules. We also obtain an equivalence $${\mathbb {K}}(\mathrm{dg}\text {-}\mathrm{CotF}\text {-}R)\approx {{\mathbb {K}}({\mathrm{Proj}}\text {-}R)}$$ of triangulated categories where $${{\mathbb {K}}({\mathrm{Proj}}\text {-}R)}$$ is the homotopy category of projective R-modules. Combined with the aforementioned result, this recovers a result from Neeman, asserting the compact generation of $${{\mathbb {K}}({\mathrm{Proj}}\text {-}R)}$$ over right coherent R. Also we get the unbounded derived category $${\mathbb {D}} (R)$$ of R as a Verdier quotient of $${\mathbb {K}}(\mathrm{dg}\text {-}\mathrm{CotF}\text {-}R)$$ .

We construct a minimal Lefschetz decomposition of the bounded derived category of coherent sheaves on the isotropic Grassmannian IGr(3, 7). Moreover, we show that IGr(3, 7) admits a full exceptional collection consisting of equivariant vector bundles.

Let $f$ and $g$ be two cuspidal modular forms and let $\mathcal{F}$ be a Coleman family passing through $f$, defined over an open affinoid subdomain $V$ of weight space $\mathcal{W}$. Using ideas of Pottharst, under certain hypotheses on $f$ and $g$ we construct a coherent sheaf over $V \times \mathcal{W}$ which interpolates the Bloch-Kato Selmer group of the Rankin-Selberg convolution of two modular forms in the critical range (i.e. the range where the $p$-adic $L$-function $L_p$ interpolates critical values of the global $L$-function). We show that the support of this sheaf is contained in the vanishing locus of $L_p$., Comment: Final version. To appear in Canadian Jour. Math