The classical Pieri formula is an explicit rule for determining the coefficients in the expansion s1m · sλ = ∑ c 1,λ sμ , where sν is the Schur polynomial indexed by the partition ν. Since the Schur polynomials represent Schubert classes in the cohomology of the complex Grassmannian, this gives a partial description of the cup product in this cohomology. Pieri’s formula was generalized to the c...

Abstract We prove that the rational cohomology group $H^{11}(\overline {\mathcal {M}}_{g,n})$ vanishes unless $g = 1$ and $n \geq 11$ . show furthermore $H^k(\overline is pure Hodge–Tate for all even $k \leq 12$ deduce $\# \overline {M}}_{g,n}(\mathbb {F}_q)$ surprisingly well approximated by a polynomial in q In addition, we use {M}}_{1,11})$ its image under Gysin push-forward tautological map...

Cofiniteness of the generalized local cohomology modules $H^{i}_{mathfrak{a}}(M,N)$ of two $R$-modules $M$ and $N$ with respect to an ideal $mathfrak{a}$ is studied for some $i^{,}s$ witha specified property. Furthermore, Artinianness of $H^{j}_{mathfrak{b}_{0}}(H_{mathfrak{a}}^{i}(M,N))$ is investigated by using the above result, in certain graded situations, where $mathfrak{b}_{0}$ is an idea...