In this article we consider the exterior power and the symmetric tensors of the polynomial ring in one variable. The structure of an associative semigraded algebra of this polynomial ring induces on the symmetric tensors the structure of an associative semigraded algebra, and on the exterior power induces structure of a semigraded module over semigraded algebra of symmetric tensors. The algebra of symmetric polynomials is isomorphic to the algebra of the symmetric tensors of polynomial ring in one variables. We obtained the explicit expression for symmetric polynomials via elementary symmetric polynomials and the explicit expression for elements of the exterior power via elementary symmetric polynomials and elements of the exterior power of the lower polynomial degree.
设$S$是一个具有剩余域$k$的正则局部环，$M$是一个正则生成的$S$ -模.设$f_1,\dots ,f_c\in S$是使$M$零化的正则序列，$E$是$k$上由$c$元生成的外代数. $M$自由分解上的$f_{i}$的同伦导出了${\rm Tor}^{S}(M,k)$上分次$E$ -模的自然结构.在$M$在整个交点R上是高合点的情况下：=S/（f_{1}，\dots，f_{c}）$ we describe this $ E $-module structure in detail, including its minimal free resolution over $ E $. Turning to $ { \rm Ext}_{R}（M，I，k）$ we show that, when $ M $ is a high syzygy over $ R $, the minimal free resolution of $ { \rm Ext}_{R}（M，I，k）$ as a module over the ring of CI operators is the Bernstein-Gel'fand-Gel'fand dual of the $ E $-module $ { \rm Tor}^{S}（M，I，k）$. For the proof we introduce \…
