Dualities of resolving subcategories of module categories over rings are introduced and characterized as dualities with respect to Wakamatsu tilting bimodules. By restriction of the dualities to smaller resolving subcategories, sufficient and necessary conditions for these bimodules to be tilting are provided. This leads to the Gorenstein version of both the Miyashita’s duality and Huisgen-Zimmermann’s correspondence. An application of resolving dualities is to show that higher algebraic K-groups and semi-derived Ringel–Hall algebras of finitely generated Gorenstein-projective modules over Artin algebras are preserved under tilting.

For a finitely dominated Poincaré duality space $M$, we show how the first author's total obstruction $\mu _M$ to the existence of a Poincaré embedding of the diagonal map $M \to M \times M$ in [17] relates to the Reidemeister trace of the identity map of $M$. We then apply this relationship to show that $\mu _M$ vanishes when suitable conditions on the fundamental group of $M$ are satisfied.

We define a class of invariants, which we call homological invariants, for persistence modules over a finite poset. Informally, a homological invariant is one that respects some homological data and takes values in the free abelian group generated by a finite set of indecomposable modules. We focus in particular on groups generated by “spread modules,” which are sometimes called “interval modules” in the persistence theory literature. We show that both the dimension vector and rank invariant are equivalent to homological invariants taking values in groups generated by spread modules. We also show that the free abelian group generated by the “single-source” spread modules gives rise to a new invariant which is finer than the rank invariant.

For a not-necessarily commutative ring $R$ we define an abelian group $W(R;M)$ of Witt vectors with coefficients in an $R$-bimodule $M$. These groups generalize the usual big Witt vectors of commutative rings and we prove that they have analogous formal properties and structure. One main result is that $W(R) := W(R;R)$ is Morita invariant in $R$. For an $R$-linear endomorphism $f$ of a finitely generated projective $R$-module we define a characteristic element $\chi _f \in W(R)$. This element is a non-commutative analogue of the classical characteristic polynomial and we show that it has similar properties. The assignment $f \mapsto \chi _f$ induces an isomorphism between a suitable completion of cyclic $K$-theory $K_0^{\mathrm {cyc}}(R)$ and $W(R)$.

Let $k$ be a field, and let ${\mathcal{C}}$ be a $k$-linear, Hom-finite triangulated category with split idempotents. In this paper, we show that under suitable circumstances, the Grothendieck group of ${\mathcal{C}}$, denoted by $K_{0}({\mathcal{C}})$, can be expressed as a quotient of the split Grothendieck group of a higher cluster tilting subcategory of ${\mathcal{C}}$. The results we prove are higher versions of results on Grothendieck groups of triangulated categories by Xiao and Zhu and by Palu. Assume that $n\geqslant 2$ is an integer; ${\mathcal{C}}$ has a Serre functor $\mathbb{S}$ and an $n$-cluster tilting subcategory ${\mathcal{T}}$ such that $\operatorname{Ind}{\mathcal{T}}$ is locally bounded. Then, for every indecomposable $M$ in ${\mathcal{T}}$, there is an Auslander–Reiten $(n+2)$-angle in ${\mathcal{T}}$ of the form $\mathbb{S}\unicode[STIX]{x1D6F4}^{-n}(M)\rightarrow T_{n-1}\rightarrow \cdots \rightarrow T_{0}\rightarrow M$ and

$$\begin{eqnarray}K_{0}({\mathcal{C}})\cong K_{0}^{\text{sp}}({\mathcal{T}})\left/\left\langle -[M]+(-1)^{n}[\mathbb{S}\unicode[STIX]{x1D6F4}^{-n}(M)]+\left.\mathop{\sum }_{i=0}^{n-1}(-1)^{i}[T_{i}]\right|M\in \operatorname{Ind}{\mathcal{T}}\right\rangle .\right.\end{eqnarray}$$

$d$

${\mathcal{C}}$

$d$

${\mathcal{S}}$

$d$

${\mathcal{S}}$

$(d+2)$

$K_{0}({\mathcal{S}})$

$K_{0}^{\text{sp}}({\mathcal{S}})$

$$\begin{eqnarray}K_{0}({\mathcal{C}})\cong K_{0}({\mathcal{S}}).\end{eqnarray}$$

$n=2d$

${\mathcal{T}}\subseteq {\mathcal{S}}$

$K_{0}({\mathcal{S}})$

$K_{0}^{\text{sp}}({\mathcal{T}})$

We show that a directed graph $E$ is a finite graph with no sinks if and only if, for each commutative unital ring $R$, the Leavitt path algebra $L_{R}(E)$ is isomorphic to an algebraic Cuntz–Krieger algebra if and only if the $C^{\ast }$-algebra $C^{\ast }(E)$ is unital and $\text{rank}(K_{0}(C^{\ast }(E)))=\text{rank}(K_{1}(C^{\ast }(E)))$. Let $k$ be a field and $k^{\times }$ be the group of units of $k$. When $\text{rank}(k^{\times })<\infty$, we show that the Leavitt path algebra $L_{k}(E)$ is isomorphic to an algebraic Cuntz–Krieger algebra if and only if $L_{k}(E)$ is unital and $\text{rank}(K_{1}(L_{k}(E)))=(\text{rank}(k^{\times })+1)\text{rank}(K_{0}(L_{k}(E)))$. We also show that any unital $k$-algebra which is Morita equivalent or stably isomorphic to an algebraic Cuntz–Krieger algebra, is isomorphic to an algebraic Cuntz–Krieger algebra. As a consequence, corners of algebraic Cuntz–Krieger algebras are algebraic Cuntz–Krieger algebras.

A quantum Frobenius map a la Lusztig for $\mathfrak{s}\mathfrak{l}_{2}$ is categorified at a prime root of unity.

In this paper we demonstrate that non-commutative localizations of arbitrary additive categories (generalizing those defined by Cohn in the setting of rings) are closely (and naturally) related to weight structures. Localizing an arbitrary triangulated category $\text{}\underline{C}$ by a set $S$ of morphisms in the heart $\text{}\underline{Hw}$ of a weight structure $w$ on it one obtains a triangulated category endowed with a weight structure $w^{\prime }$. The heart of $w^{\prime }$ is a certain version of the Karoubi envelope of the non-commutative localization $\text{}\underline{Hw}[S^{-1}]_{\mathit{add}}$ (of $\text{}\underline{Hw}$ by $S$). The functor $\text{}\underline{Hw}\rightarrow \text{}\underline{Hw}[S^{-1}]_{\mathit{add}}$ is the natural categorical version of Cohn’s localization of a ring, i.e., it is universal among additive functors that make all elements of $S$ invertible. For any additive category $\text{}\underline{A}$, taking $\text{}\underline{C}=K^{b}(\text{}\underline{A})$ we obtain a very efficient tool for computing $\text{}\underline{A}[S^{-1}]_{\mathit{add}}$; using it, we generalize the calculations of Gerasimov and Malcolmson (made for rings only). We also prove that $\text{}\underline{A}[S^{-1}]_{\mathit{add}}$ coincides with the ‘abstract’ localization $\text{}\underline{A}[S^{-1}]$ (as constructed by Gabriel and Zisman) if $S$ contains all identity morphisms of $\text{}\underline{A}$ and is closed with respect to direct sums. We apply our results to certain categories of birational motives $DM_{gm}^{o}(U)$ (generalizing those defined by Kahn and Sujatha). We define $DM_{gm}^{o}(U)$ for an arbitrary $U$ as a certain localization of $K^{b}(Cor(U))$ and obtain a weight structure for it. When $U$ is the spectrum of a perfect field, the weight structure obtained this way is compatible with the corresponding Chow and Gersten weight structures defined by the first author in previous papers. For a general $U$ the result is completely new. The existence of the corresponding adjacent$t$-structure is also a new result over a general base scheme; its heart is a certain category of birational sheaves with transfers over $U$.

We study the K′-theory of a CM Henselian local ring R of finite Cohen-Macaulay type. We first describe a long exact sequence involving the groups K′i(R) and the K-groups of certain other rings, including the Auslander algebra. By examining the terms and maps in the sequence, we obtain information about K′(R).

Let B = A[[t;σ,δ]] be a skew power series ring such that σ is given by an inner automorphism of B. We show that a certain Waldhausen localisation sequence involving the K-theory of B splits into short split exact sequences. In the case that A is noetherian we show that this sequence is given by the localisation sequence for a left denominator set S in B. If B = ℤp[[G]] happens to be the Iwasawa algebra of a p-adic Lie group G ≅ H ⋊ ℤp, this set S is Venjakob's canonical Ore set. In particular, our result implies that

is split exact for each n ≥ 0. We also prove the corresponding result for the localisation of ℤp[[G]][] with respect to the Ore set S*. Both sequences play a major role in non-commutative Iwasawa theory.

We compute the Grothendieck and Picard groups of a smooth toric DM stack by using a suitable category of graded modules over a polynomial ring. The polynomial ring with a suitable grading and suitable irrelevant ideal functions is a homogeneous coordinate ring for the stack.

In [BMR] we observed that, on the level of derived categories, representations of the Lie algebra of a semisimple algebraic group over a field of finite characteristic with a given (generalized) regular central character can be identified with coherent sheaves on the formal neighborhood of the corresponding (generalized) Springer fiber. In the present paper we treat singular central characters.

The basic step is the Beilinson-Bernstein localization of modules with a fixed (generalized) central character λ as sheaves on the partial flag variety corresponding to the singularity of λ. These sheaves are modules over a sheaf of algebras which is a version of twisted crystalline differential operators. We discuss translation functors and intertwining functors. The latter generate an action of the affine braid group on the derived category of modules with a regular (generalized) central character, which intertwines different localization functors. We also describe the standard duality on Lie algebra modules in terms of D-modules and coherent sheaves.

We investigate categorical and amalgamation properties of the functor $\operatorname{Id_c}$ assigning to every partially ordered abelian group $G$ its $\langle \vee, 0 \rangle$-semilattice of compact ideals $\operatorname{Id_c} G$. Our main result is the following.

Theorem 1. Every diagram of finite Boolean semilattices indexed by a finite dismantlable partially ordered set can be lifted, with respect to the$\operatorname{Id_c}$functor, by a diagram of pseudo-simplicial vector spaces.

Pseudo-simplicial vector spaces are a special kind of finite-dimensional partially ordered vector spaces with interpolation over the field of rational numbers. The methods introduced also make it possible to prove the following ring-theoretical result.

Theorem 2. For any countable distributive$\langle \vee, 0 \rangle$-semilattices$S$and$T$and any field$K$, any$\langle \vee, 0 \rangle$-homomorphism$f \colon S \to T$can be lifted, with respect to the$\operatorname{Id_c}$functor on rings, by a homomorphism $f \colon A \to B$of$K$-algebras, for countably dimensional locally matricial algebras$A$and$B$over$K$.

We also state a lattice-theoretical analogue of Theorem 2 (with respect to the $\operatorname{Con_c}$ functor), and we provide counterexamples to various related statements. In particular, we prove that the result of Theorem 1 cannot be achieved with simplicial vector spaces alone.

This article studies algebras $R$ over a simple artinian ring $A$, presented by a quiver and relations and graded by a semigroup $\Sigma $. Suitable semigroups often arise from a presentation of $R$. Throughout, the algebras need not be finite dimensional. The graded ${{K}_{0}}$, along with the $\Sigma $-graded Cartan endomorphisms and Cartan matrices, is examined. It is used to study homological properties.

A test is found for finiteness of the global dimension of a monomial algebra in terms of the invertibility of the Hilbert $\Sigma $-series in the associated path incidence ring.

The rationality of the $\Sigma $-Euler characteristic, the Hilbert $\Sigma $-series and the Poincaré-Betti $\Sigma $-series is studied when $\Sigma $ is torsion-free commutative and $A$ is a division ring. These results are then applied to the classical series. Finally, we find new finite dimensional algebras for which the strong no loops conjecture holds.