In this paper, we study the structure of the local components of the (shallow, i.e. without $U_{p}$ ) Hecke algebras acting on the space of modular forms modulo $p$ of level $1$ , and relate them to pseudo-deformation rings. In many cases, we prove that those local components are regular complete local algebras of dimension $2$ , generalizing a recent result of Nicolas and Serre for the case $p=2$ .

Let $A$ be an abelian variety over a global field $K$ of characteristic $p\geqslant 0$ . If $A$ has nontrivial (respectively full) $K$ -rational $l$ -torsion for a prime $l\neq p$ , we exploit the fppf cohomological interpretation of the $l$ -Selmer group $\text{Sel}_{l}\,A$ to bound $\#\text{Sel}_{l}\,A$ from below (respectively above) in terms of the cardinality of the $l$ -torsion subgroup of the ideal class group of $K$ . Applied over families of finite extensions of $K$ , the bounds relate the growth of Selmer groups and class groups. For function fields, this technique proves the unboundedness of $l$ -ranks of class groups of quadratic extensions of every $K$ containing a fixed finite field $\mathbb{F}_{p^{n}}$ (depending on $l$ ). For number fields, it suggests a new approach to the Iwasawa ${\it\mu}=0$ conjecture through inequalities, valid when $A(K)[l]\neq 0$ , between Iwasawa invariants governing the growth of Selmer groups and class groups in a $\mathbb{Z}_{l}$ -extension.

Let $A$ be an abelian variety defined over a field $k$ . In this paper we define a descending filtration $\{F^{r}\}_{r\geqslant 0}$ of the group $\mathit{CH}_{0}(A)$ and prove that the successive quotients $F^{r}/F^{r+1}\otimes \mathbb{Z}[1/r!]$ are isomorphic to the group $(K(k;A,\dots ,A)/Sym)\otimes \mathbb{Z}[1/r!]$ , where $K(k;A,\dots ,A)$ is the Somekawa $K$ -group attached to $r$ -copies of the abelian variety  $A$ . In the special case when $k$ is a finite extension of $\mathbb{Q}_{p}$ and $A$ has split multiplicative reduction, we compute the kernel of the map $\mathit{CH}_{0}(A)\otimes \mathbb{Z}[\frac{1}{2}]\rightarrow \text{Hom}(Br(A),\mathbb{Q}/\mathbb{Z})\otimes \mathbb{Z}[\frac{1}{2}]$ , induced by the pairing $\mathit{CH}_{0}(A)\times Br(A)\rightarrow \mathbb{Q}/\mathbb{Z}$ .

A special linear Grassmann variety $\text{SGr}(k,n)$ is the complement to the zero section of the determinant of the tautological vector bundle over $\text{Gr}(k,n)$ . For an $SL$ -oriented representable ring cohomology theory $A^{\ast }(-)$ with invertible stable Hopf map ${\it\eta}$ , including Witt groups and $\text{MSL}_{{\it\eta}}^{\ast ,\ast }$ , we have $A^{\ast }(\text{SGr}(2,2n+1))\cong A^{\ast }(pt)[e]/(e^{2n})$ , and $A^{\ast }(\text{SGr}(k,n))$ is a truncated polynomial algebra over $A^{\ast }(pt)$ whenever $k(n-k)$ is even. A splitting principle for such theories is established. Using the computations for the special linear Grassmann varieties, we obtain a description of $A^{\ast }(\text{BSL}_{n})$ in terms of homogeneous power series in certain characteristic classes of tautological bundles.

We give sufficient conditions for a Frobenius category to be equivalent to the category of Gorenstein projective modules over an Iwanaga–Gorenstein ring. We then apply this result to the Frobenius category of special Cohen–Macaulay modules over a rational surface singularity, where we show that the associated stable category is triangle equivalent to the singularity category of a certain discrepant partial resolution of the given rational singularity. In particular, this produces uncountably many Iwanaga–Gorenstein rings of finite Gorenstein projective type. We also apply our method to representation theory, obtaining Auslander–Solberg and Kong type results.

Let $k$ be an infinite field. Let $R$ be the semi-local ring of a finite family of closed points on a $k$ -smooth affine irreducible variety, let $K$ be the fraction field of $R$ , and let $G$ be a reductive simple simply connected $R$ -group scheme isotropic over $R$ . Our Theorem 1.1 states that for any Noetherian $k$ -algebra $A$ the kernel of the map

$$\begin{eqnarray}H_{\acute{\text{e}}\text{t}}^{1}(R\otimes _{k}A,G)\rightarrow H_{\acute{\text{e}}\text{t}}^{1}(K\otimes _{k}A,G)\end{eqnarray}$$

$R$

$K$

$A=k$

$R$

Let $Q$ be a finite quiver without oriented cycles, and let $k$ be an algebraically closed field. The main result in this paper is that there is a natural bijection between the elements in the associated Weyl group $W_{Q}$ and the cofinite additive quotient closed subcategories of the category of finite dimensional right modules over $kQ$ . We prove this correspondence by linking these subcategories to certain ideals in the preprojective algebra associated to $Q$ , which are also indexed by elements of $W_{Q}$ .